Chapter 10

Dependent samples:

occur when each observation in one sample is connected to an observation in the other sampleDependent samples are also called paired samples or paired data

In a sample of 35 randomly selected vehicles in City A, 7 use alternative energy sources. In a sample of 45 randomly selected vehicles in City B, 17 use alternative energy sources. Test the alternative hypothesis that the population proportion for City A is different from the population proportion for City B. The test statistic is z=−1.72. What is the corresponding p-value? Compute your answer using a value from the table below.

0.086 Using a normal distribution table, the area to the left of z=−1.72 is approximately 0.043. Twice this area is 2(0.043)=0.086. Therefore, the p-value is approximately 0.086.

A researcher wants to show that the proportions of mosquitoes that carry a certain disease in Region 1 and Region 2 are different. In a sample of 1,392 mosquitoes trapped in Region 1, 1,173 test positive for the disease. In a sample of 1,457 mosquitoes trapped in Region 2, 1,196 test positive for the disease. Test the alternative hypothesis that the population proportion for Region 1 is different from the population proportion for Region 2. The test statistic is z=1.55. What is the corresponding p-value? Compute your answer using a value from the table below.

0.122

Leah Peschel is the bottling department manager for a bottling company that produces various soft drinks and juices. The company uses two different machines from different manufacturers to fill the bottles of its popular cola. Leah periodically verifies that the population mean amount of cola in the bottles filled by Machine 1 is the same as the population mean amount in the bottles filled by Machine 2. The manufacturers calibrated the machines at the time of installation and provided that information to the bottling company. Based on this information, Leah assumes that the population standard deviation for Machine 1 is 0.021 ounce and the population standard deviation for Machine 2 is 0.019 ounce. Leah randomly selects samples of bottles filled by Machine 1 and Machine 2. The amount of cola in each bottle is recorded for both samples, and the results are shown in the table. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, find the null and alternative hypotheses, where μ1 is the population mean of Machine 1 and μ2 is the population mean of Machine 2. Machine 1 x1=12.524 n1=244 Machine2 x2=12.518 n2=251

All of the conditions to conduct the hypothesis test have been met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2≠0. All of the conditions that are required for the hypothesis test have been met. The population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large. Since the company is testing whether the amount of cola in the bottles filled by Machine 1 is different than the amount of cola in the bottles filled by Machine 2, the alternative hypothesis is that the difference of means is not equal to 0. Therefore, the hypotheses for this hypothesis test are H0:μ1−μ2=0 Ha:μ1−μ2≠0.

Lydia Hayward is an external research consultant hired by FirstCard, a major credit card company, to find evidence that the population mean total credit card balance for adults ages 25 to 34 with a college degree in the region is greater than for adults ages 25 to 34 without a college degree. Due to research already conducted by FirstCard, Lydia assumes that the population standard deviation is $2,047.24 for adults ages 25 to 34 with a college degree and $1,749.65 for those without a college degree. She conducts a survey at a local shopping center and selects every tenth adult ages 25 to 34 with a college degree and every tenth adult ages 25 to 34 without a college degree to ask for the total credit card balance. The results of the samples are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test. Let μ1 be the population mean total credit card balance for adults ages 25 to 34 with a college degree and μ2 be the population mean total credit card balance for adults ages 25 to 34 without a college degree. With College Degree x=$6,312.91 n1=271 Without College Degree x2=$5,483.15 n2=315

Although the population standard deviations are known and the sample sizes are sufficiently large, the samples are not randomly selected and are not independent of each other. Lydia did not select a random sample for both groups because she selected her samples from patrons at a local shopping center. Since not all adults ages 25 to 34 in the region have an equal opportunity to be selected, the sample does not satisfy the randomization condition.

Kayla Greene is a team lead for an environmental group for a certain region. She is investigating whether the population mean monthly number of kilowatt hours (kWh) used per residential customer in the region has changed from 2006 to 2017. She is concerned that changes such as more efficient lighting and the increased use of electronics and air conditioners are affecting how much electricity residential customers now consume. Kayla assumes the population standard deviation for 2006 and 2017 using the data that were provided to her by local utility companies. Using data that were collected by her organization, Kayla selects a random sample of residential customers who were active for all of 2006 and a separate sample of residential customers who were active for all of 2017. The results from the samples and the population standard deviations that are assumed for this hypothesis test are provided in the accompanying table. Let μ1 be the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and μ2 be the population mean monthly number of kilowatt hours consumed per residential customer in 2017. The p-value rounded to three decimal places is 0.250, the significance level is α=0.01, the null hypothesis is H0:μ1−μ2=0, and the alternative hypothesis is Ha:μ1−μ2≠0. 2006 x1=894.7kWh n1=361 σ1=193.1kWH 2017 x2=910.2 kWh n2=424 σ2=182.9kWH Which of the following statements are accurate for this hypothesis test to evaluate the claim that the true difference between the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and the population mean monthly number of kilowatt hours consumed per residential customer in 2017 is not equal to zero?

Fail to reject the null hypothesis that the true difference between the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and the population mean monthly number of kilowatt hours consumed per residential customer in 2017 is equal to zero. Based on the results of the hypothesis test, there is not enough evidence at the α=0.01 level of significance to suggest that the true difference between the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and the population mean monthly number of kilowatt hours consumed per residential customer in 2017 is not equal to zero. Compare the p-value, 0.250, to α=0.01. Since the p-value is greater than α, fail to reject H0. Therefore, there is not enough evidence to suggest that the true difference between the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and the population mean monthly number of kilowatt hours consumed per residential customer in 2017 is not equal to zero.

To determine if computer 1 starts up faster than computer 2, the mean startup time of the two competing computers is compared (the shorter the startup time, the faster the computer). Twenty-five startup times are randomly sampled from each computer. Both populations have normal distributions with known standard deviations.

H0:μ1≥μ2; Ha:μ1<μ2 In order for computer 1 to have a faster startup speed than computer 2, the mean startup time of computer 1 must be less than the mean startup time of computer 2: μ1<μ2. The null hypothesis must have some form of equality, so Ha is μ1<μ2 and H0 is μ1≥μ2.

Suppose you were asked to determine if the population proportions of women who do yoga is different in two different countries. State the null and alternative hypotheses, where p1 is the proportion of women in the population of the first country and p2 is the proportion of women in the population of the second country.

Null Hypothesis: H0: p1−p2=0 Alternative Hypothesis: Ha: p1−p2≠0

A coin produces 3 heads in 20 flips. Another coin produces 5 heads in 15 flips. Are all the conditions for the hypothesis test for the difference between the proportions met? (Let the first coin correspond to sample 1, the second coin correspond to sample 2, and a success be a head.)

No. The value n1pˆ1 is less than 5.

Researchers in a country randomly select 100 males and 100 females. The researchers determine that 2% of the males and 3% of the females suffer from some type of eating disorder. Check that the conditions are met for a hypothesis test to compare the population proportion of males that suffer from some type of eating disorder to the proportion of females.

Not all the values n1p⎯, n1q⎯ n2p⎯⎯ and n2q⎯⎯are at least 5.

Derby Leicester is a city planner preparing for a meeting with the mayor. He would like to show that the population mean age of the houses on Lincoln Street is less than the population mean age of the houses on Maple Street so that more resources are allotted to repair Maple Street. Derby uses information from a previous study and assumes that the population standard deviation for the ages of the houses on Lincoln Street is 7.72 years and 8.39 years for the houses on Maple Street. Due to limited time, Derby randomly selects the houses on Lincoln Street and the houses on Maple Street from the city's property records, and then records the age of each house in years. The results of the samples are shown in the table below. Let μ1 be the population mean age, in years, of the houses on Lincoln Street and μ2 be the population mean age, in years, of the houses on Maple Street. The p-value is less than 0.001, the significance level is α=0.05, the null hypothesis is H0:μ1−μ2=0, and the alternative hypothesis is Ha:μ1−μ2<0. Lincoln Street x1= 50.91 years n1=41 Maple Street x2= 59.27 years n2= 37 Which of the following statements are accurate for this hypothesis test to evaluate the claim that the difference between the population mean age of the houses on Lincoln Street and the population mean age of the houses on Maple Street is less than zero?

Reject the null hypothesis that the true difference between the population mean age of the houses on Lincoln Street and the population mean age of the houses on Maple Street is equal to zero. Based on the results of the hypothesis test, there is enough evidence at the α=0.05 level of significance to support the claim that the true difference between the population mean age of the houses on Lincoln Street and the population mean age of the houses on Maple Street is less than zero. Compare the p-value that is less than 0.001 to α=0.05. Since the p-value is less than α, reject H0. Therefore, there is enough evidence at the α=0.05 level of significance to support the claim that the true difference between the population mean age of the houses on Lincoln Street and the population mean age of the houses on Maple Street is less than zero.

For a sample of 120 randomly selected adult couples, 62% of the men and 79% of the women prefer travel by train over travel by bus. Check that the conditions are met for a hypothesis test to compare the population proportion of men that prefer travel by train over travel by bus to the proportion of women.

The samples are not independent.

Find the p-value for a left-tailed hypothesis test when the test statistic is z=−1.5. Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=−1.5 is approximately 0.067. Therefore, the p-value is approximately 0.067.

In a survey of 100 randomly selected taxi drivers in a city, 72 support increased government spending on roads and bridges. In a survey of 100 randomly selected bus drivers in the same city, 89 support such spending. Test the alternative hypothesis that the population proportion of taxi drivers in the city that support such spending is different from the population proportion of bus drivers in the city. The test statistic is z=−3.03. What is the corresponding p-value? Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=−3.03 is approximately 0.001. Twice this area is 2(0.001)=0.002. Therefore, the p-value is approximately 0.002.

Andy Thompson is a dean at a local college. He is trying to determine whether the population mean GPA of undergraduate students at his college who have at least one job during the semester is different than the population mean GPA of undergraduate students at his college who do not have a job during the semester. Andy conducts research on the topic and assumes that the population standard deviation is 0.59 for undergraduates who have at least one job during the semester and 0.68 for undergraduates who don't have a job during the semester. The results of the survey are displayed in the table below. Let μ1 be the population mean GPA of students who have at least one job during the semester and μ2 be the population mean GPA of students who do not have a job during the semester. If the test statistic is z=0.71, what is the p-value for this hypothesis test? Have at Least One Job x1=2.79 n1=56 Do Not Have a Job x2=2.71 n2=73

p-value=0.478

Arianna Estefan manages Salmon Falls Park, which has two flagship roller coasters, the Flyer and the Destroyer. When Arianna walks around the park during hours of operation, she notices that there are fewer people waiting in line for the Flyer than for the Destroyer. Since both roller coasters have similar capacities and similar run times, both roller coasters should have about the same number of riders. Based on the anecdotal evidence, Arianna claims that the population mean number of riders per hour on the Flyer is less than the population mean number of riders per hour on the Destroyer. Arianna reviews data collected in the past and assumes that the population standard deviation is 48.29 for the Flyer and 51.86 for the Destroyer. Arianna randomly selects hours during which the attendant of each ride counts the number of riders. The sampling occurs over the course of several weeks. The results are provided in the table shown below. Assuming the conditions needed for the hypothesis test have been met, what is the z test statistic for this hypothesis test, rounded to two decimal places? Flyer x1=109.62 n1=54 Destroyer x2=124.71 n2=51

z=-1.54

Casey Fitch is writing a report for a statistics class. He is investigating whether the mean price for a gallon of regular gasoline at stations of Brand A are currently higher than the mean price for a gallon of regular gasoline at stations of Brand B in the region. Casey randomly selects 37 service stations of Brand A and 35 service stations of Brand B and records the price for a gallon of regular gasoline. The sample mean price for Brand A is $2.836 with a sample standard deviation of $0.083, while Brand B has a sample mean price of $2.798 with a standard deviation of $0.049. Let μ1 be the population mean price for a gallon of regular gasoline at Brand A and μ2 be the population mean price for a gallon of regular gasoline at Brand B. Casey uses the alternative hypothesis Ha:μ1−μ2>0. He assumes that the population standard deviations of the two groups are not equal, so uses 34 degrees of freedom. If the t-test statistic is t≈2.38, what is the p-value for this hypothesis test?

0.01< p-value <0.05

Jordan Scarff works for a state's department of transportation. Recently, an inquiry has been made in Jordan's department about the number of vehicles that cross a certain bridge within the state. The engineers of the bridge decided to install tolls on the southbound lanes only. It appears that there are fewer vehicles traveling on the southbound lanes per day than on the northbound lanes. Jordan looks into the inquiry and tests the claim that the mean number of vehicles per day traveling in the southbound lanes is less than the mean number of vehicles per day traveling in the northbound lanes. Due to limited time, Jordan selects a random sample of days for the northbound lanes and another random sample of days for the southbound lanes, where the number of vehicles traveled each day was recorded. The sample statistics are shown in the table below. Let μ1 be the population mean number of vehicles traveling in the southbound lanes per day and μ2 be the population mean number of vehicles traveling in the northbound lanes per day. Jordan uses the alternative hypothesis Ha:μ1−μ2<0, assuming that the population standard deviations of the two groups are not equal and using 60 degrees of freedom. If the t-test statistic is t≈−2.15, what is the p-value for this hypothesis test?

0.01< p-value <0.05

The quality manager at a breakfast cereal producer claimed that one of the two production lines in the factory, Production Line 1, was over-filling the boxes compared to Production Line 2. A random sample of 40 boxes produced by Production Line 1 had a mean weight of 20.062 ounces with a standard deviation of 0.035 ounce. A sample of 40 boxes produced by Production Line 2 had a mean weight of 20.047 ounces with a standard deviation of 0.033 ounce. Let μ1 be the population mean weight of boxes filled by Production Line 1, and let μ2 be the population mean weight of boxes filled by Production Line 2. The manager used the alternative hypothesis Ha:μ1−μ2>0. If the test statistic is t≈1.97 with 78 degrees of freedom, what is the p-value for this hypothesis test?

0.025< p-value <0.05

A pet food manufacturer has a facility with two lines that blend, can, and cook cat food. Nolan Munroe is the production manager of the facility and curious whether there is a difference in the mean number of cans produced each day by the two lines, A and B. Nolan randomly selects 300 daily production reports, 141 of which were for production line A and 159 of which were for production line B. The sample statistics are shown in the table below. Let μ1 be the population mean number of cans produced per day on production line A, and let μ2 be the population mean number of cans produced per day on production line B. Nolan assumes the population standard deviations are not equal and tests the alternative hypothesis Ha:μ1−μ2≠0. If the test statistic is t≈−1.93 and the number of degrees of freedom is 140, what is the p-value for this hypothesis test?

0.05< p-value <0.10

Dylan Rieder is a statistics student investigating whether athletes have better balance than non-athletes for a thesis project. Dylan randomly selected 32 student athletes and 45 students who did not play any sports to walk along a board that was 16 feet long and 2 inches wide and raised 6 inches off the ground. Dylan recorded the number of times each participant touched the ground. The athletes had a mean of 3.7 touches with a standard deviation of 1.1. The non-athletes had a mean of 4.1 touches with a standard deviation of 1.3. Let μ1 be the population mean number of touches for student athletes, and let μ2 be the population mean number of students who did not play any sports. Dylan assumes the population standard deviations are equal and is testing the alternative hypothesis Ha:μ1−μ2<0. If the test statistic is t≈−1.42 and the number of degrees of freedom is 75, what is the p-value for this hypothesis test?

0.05< p-value <0.10

A survey asked randomly selected people whether they thought it was rude for people to interact with their smartphones while in a restaurant. Among the 92 females surveyed, 57 responded "Yes." Of the 108 males surveyed, 62 responded "Yes." Test the alternative hypothesis that the population proportion for females is greater than the population proportion for males. The test statistic is z=0.65. What is the corresponding p-value? Compute your answer using a value from the table below.

0.258 Using a normal distribution table, the area to the left of z=0.65 is approximately 0.742. This means the area to the right of z=0.65 is approximately 1−0.742=0.258. Therefore, the p-value is approximately 0.258.

Independent samples

: occur when the results for one sample do not affect the results of the other sample

For samples of 60 randomly selected men and 60 randomly selected women, 65% of the men and 75% of the women prefer travel by train over travel by bus. Check that the conditions are met for a hypothesis test to compare the population proportion of men that prefer travel by train over travel by bus to the proportion of women.

All of the conditions are met.

In a survey of 100 randomly selected taxi drivers in City A, 76 support increased government spending on roads and bridges. In a survey of 100 randomly selected bus drivers in City B, 82 support such spending. Check that the conditions are met for a hypothesis test to compare the population proportion of taxi drivers in City A that support such spending to the proportion of bus drivers in City B.

All of the conditions are met.

In a survey of 150 randomly selected coal miners in State A, 7 support government subsidies to encourage producing solar energy. In a survey of 150 randomly selected farmers in State B, 132 support such subsidies. Check that the conditions are met for a hypothesis test to compare the population proportion of coal miners in State A that support such subsidies to the proportion of farmers in State B.

All of the conditions are met.

Suppose 90% of a sample of 100 randomly selected homes in Town A and 80% of a sample of 50 randomly selected homes in Town B rely on public water supply for drinking water. Check that the conditions are met for a hypothesis test to compare the population proportion of homes that rely on public water supply for drinking water in Town A to the proportion in Town B.

All of the conditions are met.

To test a drug intended to increase memory, 80 subjects were randomly given a pill that either contained the drug or was a placebo. After ten minutes, the subjects were asked to look at a poster with pictures of 20 common objects for five minutes. After waiting ten minutes, the subjects were asked to list as many of the objects as they could recall. For the subjects that took the drug, 75% were able to list at least half of the objects. For the subjects that took the placebo, 70% were able to list at least half of the objects. Check that the conditions are met for a hypothesis test to compare the population proportion of subjects that took the drug that could list at least half of the objects to the proportion that took the placebo that could.

All of the conditions are met.

An automobile manufacturer claims that the population mean braking distance on its premiere vehicle, the Hawk, is less than the population mean braking distance on its main competitor, the Wildcat. Ryan Pottier is a writer for a national automotive magazine and would like to verify the claim made by the manufacturer for an article. He contacts the manufacturer of each vehicle and assumes that the population standard deviation for the stopping distance is 4.59ft for the Hawk and 4.38ft for the Wildcat. Ryan randomly selects new vehicles of each model and conducts a brake test on each car, where each vehicle is stopped from 60 miles per hour in a controlled environment. The results of the test are provided in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean stopping distance, in feet, of the Hawk and μ2 is the population mean stopping distance, in feet, of the Wildcat.

All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2<0. All of the conditions that are required for the hypothesis test have been met. The population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large. Since Ryan is testing whether the mean stopping distance of the Hawk is less than the mean stopping distance of the Wildcat, μ1 is less than μ2 and the difference, μ1−μ2, would be less than 0. The null and alternative hypotheses are shown below. H0:μ1−μ2=0 Ha:μ1−μ2<0.

Derby Leicester is a city planner preparing for a meeting with the mayor. He would like to show that the population mean age of the houses on Lincoln Street is less than the population mean age of the houses on Maple Street so that more resources are allotted to repair Maple Street. Derby uses a previous study to assume that the population standard deviation for the ages of the houses is 7.72 years for Lincoln Street and 8.39 years for Maple Street. Due to limited time, Derby randomly selects a sample of houses on Lincoln Street and the houses on Maple Street from the city's property records and then records the age of each house in years. The results of the samples are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean age of the homes on Lincoln Street and μ2 is the population mean age of the homes on Maple Street. Lincoln Street x1=50.91 years n1=41 MapleStreet x2=59.27 years n2=37

All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2<0. All of the conditions that are required for the hypothesis test have been met. The population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large. Since Dalton would like to show that the mean age of the houses on Lincoln Street is less than the mean age of the houses on Maple Street, μ1 is less than μ2 and the difference, μ1−μ2, would be less than 0. The null and alternative hypotheses are shown below.

Caleb Satterfield is a student in an automotive technology class. He is writing a report about trends in passenger vehicles, including the age. Caleb would like to claim in the report that the population mean age, in years, of a passenger vehicle owned by a city resident is greater in 2017 than it was in 2012. Caleb goes to the town's vehicle registration office to gather the information. Due to restrictions on the information that can be released, the city clerk provided Caleb with the population standard deviations and a random sample of the ages of passenger vehicles owned by the city residents, where the same resident was not selected in both samples. The results of the samples and population standard deviations that Caleb assumes for this hypothesis test are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean age of passenger vehicles owned by city residents in 2017 and μ2 is the population mean age of passenger vehicles owned by city residents in 2012. 2017 x1=11.57 years n1=349 σ1=4.06 years 2012 x2=11.21 years n2=328 σ2=3.84 years

All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2>0 All of the conditions that are required for the hypothesis test have been met. The population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large. Since Caleb is claiming that the mean age of passenger vehicles owned by city residents is greater in 2017 than in 2012, μ1 is greater than μ2 and the difference, μ1−μ2, would be greater than 0. The null and alternative hypotheses are shown below.

Simone Lahey is a district vice president of a bank. She is receiving complaints from bank customers about the amount of time they have to wait in the Mineola branch. She is investigating whether the population mean wait time of the Mineola branch is greater than the population mean wait time of the Westbury branch. Simone carefully reviews studies of the mean wait times conducted in the past and assumes that the population standard deviation is 1.09 minutes in the Mineola branch and 0.96 minute in the Westbury branch. Simone conducts a survey at each branch over a period of time by randomly selecting customers who wait in line and then recording each customer's wait time in minutes. The results of the survey are displayed in the table shown below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test. Let μ1 be the population mean wait time for customers in the Mineola branch and μ2 be the population mean wait time for customers in the Westbury branch. Mineola x1=3.06 minutes n1=194 Westbury x2=2.87 minutes n2=205

All of the conditions to conduct the hypothesis test are met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2>0. Since the population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large, all of the conditions that are required for the hypothesis test have been met. Since Simone is investigating whether the mean wait time of the Mineola branch is greater than the mean wait time of the Westbury branch, μ1 is greater than μ2 and the difference, μ1−μ2, is positive. The null and alternative hypotheses are shown below.

Kayla Greene is a team lead for an environmental group for a certain region. She is investigating whether the population mean monthly number of kilowatt hours (kWh) used per residential customer in the region has changed from 2006 to 2017. She is concerned that changes such as more efficient lighting and the increased use of electronics and air conditioners are affecting the mean monthly number of kilowatt hours consumed per residential customer. Kayla investigates the data and assumes the population standard deviation for 2006 and 2017 using the data that were provided to her by local utility companies. Using data that was collected by her company, Kayla selects a random sample of residential customers who were active for all of 2006 and a separate sample of residential customers who were active for all of 2017. The population standard deviations and the results from the samples are provided in the accompanying table. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, determine the null and alternative hypotheses for this hypothesis test. Let μ1 be the population mean monthly number of kilowatt hours consumed per residential customer in 2006 and μ2 be the population mean monthly number of kilowatt hours consumed per residential customer in 2017. 2006 x1=894.7kWh n1=361 σ1=193.1kWh 2017 x2=910.2kWh n2=424 σ2=182.9kWh

All of the conditions to conduct the hypothesis test have been met. The null and alternative hypotheses are H0:μ1−μ2=0 Ha:μ1−μ2≠0. All of the conditions that are required for the hypothesis test have been met. The population standard deviations are known, the samples are randomly selected and are independent, and the sample sizes are sufficiently large. Since Kayla wants to find out whether the mean monthly number of kilowatt hours (kWh) used per residential customer has changed from 2006 to 2017, μ1 is not equal to μ2 and the difference, μ1−μ2, would be not equal to 0. The null and alternative hypotheses are shown below.

Dalton Castene is a public school teacher in Burlington. After a conversation with a colleague in Sudbury, Dalton realizes that his salary is much lower than his colleague's, even though they have similar backgrounds and work experiences. Dalton theorizes that the population mean annual salary for a public school teacher in Burlington is less than the population mean annual salary for a public school teacher in Sudbury. He conducts research on the inquiry and assumes that the population standard deviation is $2,409 for Burlington and $3,192 for Sudbury. Dalton randomly selects the salaries of public school teachers in Burlington and in Sudbury. The results of Dalton's findings are shown in the table below. Explain whether a hypothesis test for the difference between two means of independent samples is appropriate, and if so, find the null and alternative hypotheses, where μ1 is the population mean annual salary for a public school teacher in Burlington and μ2 is the population mean annual salary for a public school teacher in Sudbury. Burlington x1=$54,906 n1=26 Sudbury x2=$56,561 n2=22

Although the population standard deviations are known and the samples are randomly selected and independent, the sample sizes are not sufficiently large. Although the population standard deviations are known and the samples are randomly selected and are independent, the sample sizes for Burlington and Sudbury are 26 and 22, respectively. Since both sample sizes are less than 30, the sample sizes are not sufficiently large. Therefore, not all of the conditions to conduct the hypothesis test have been met and a hypothesis test for the difference between two means of independent samples is not appropriate.

Brandon Blake is a real estate agent working in several towns in a local area, mainly Littleton and Whitefield. Brandon has worked in the real estate business for several years and believes that the population mean selling price of a single-family home in Littleton during 2017 was greater than the population mean selling price of a single-family home in Whitefield during 2017. Brandon randomly selects a sample of selling prices for single-family homes for each of the two towns during 2017. The results of the samples are shown in the table below. What are the null and alternative hypotheses for this hypothesis test, where μ1 is the population mean selling price of a single-family home in Littleton during 2017 and μ2 is the population mean selling price of a single-family home in Whitefield during 2017? Littleton: x1=$214,601 n1=84 Whitefield x2=$211,862 n2=91

Although the samples are randomly selected, independent, and sufficiently large, the population standard deviation is not known for both towns. Although the samples are randomly selected, independent, and sufficiently large, the population standard deviation of the selling price is not known for both towns. Therefore, not all of the conditions to conduct the hypothesis test have been met and a hypothesis test for the difference between two means of independent samples is not appropriate.

A cell phone manufacturer claims that the population mean battery life of its flagship smartphone model, the Black Bear, is greater than the population mean battery of the largest competitor, the Grizzly. A consumer advocacy publication tests this claim by purchasing a random sample of Black Bear smartphones and a random sample of Grizzly smartphones. Members of the publication charged each smartphone to full capacity and then had the smartphones play the same videos until the batteries were completely depleted. The sample statistics are in the table below. Based on data from the manufacturers, the publication assumes that the population standard deviation for the Black Bear is 0.71 hour and the population standard deviation for the Grizzly is 0.63 hour. Let μ1 be the population mean battery life for the Black Bear and μ2 be the population mean battery life for the Grizzly. The p-value rounded to three decimal places is 0.084, the significance level is α=0.01, the null hypothesis is H0:μ1−μ2=0, and the alternative hypothesis is Ha:μ1−μ2>0. Black Bear x1=11.24 hours n1=54 Grizzly x2=11.06 hours n2=51

Fail to reject the null hypothesis that the true difference between the population mean battery life of the Black Bear and the population mean battery life of the Grizzly is equal to zero. Based on the results of the hypothesis test, there is not enough evidence at the α=0.01 level of significance to suggest that the true difference between the population mean battery life of the Black Bear and the population mean battery life of the Grizzly is greater than zero. Compare the p-value, 0.084, to α=0.01. Since the p-value is greater than α, fail to reject the null hypothesis H0. Therefore, there is not enough evidence at the α=0.01 level of significance to suggest that the true difference between the population mean battery life of the Black Bear and the population mean battery life of the Grizzly is greater than zero.

A researcher wants to test to see if husbands are significantly older than their wives. To do this, he collects the ages of husbands and pairs them with the ages of their respective wives for a random set of married couples. Suppose that data were collected for a random sample of 12 couples, where each difference is calculated by subtracting the age of the wife from the age of the husband. Assume that the ages are normally distributed. Using a test statistic of t≈1.434, the significance level α=0.05, and the corresponding p-value between 0.05 and 0.10, draw a conclusion for the appropriate hypothesis test, where the null hypothesis is H0:μd=0 and the alternative hypothesis is Ha:μd>0.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that husbands are significantly older than their wives. Compare the p-value to α=0.05. Since the p-value is between 0.05 and 0.10, it is greater than α, so we fail to reject H0. Therefore, there is insufficient evidence at the α=0.05 level of significance to suggest that husbands are significantly older than their wives.

A researcher wants to compare the heights of males between generations to see if they differ. To do this, he samples random pairs of males who are at least 18 years old and their fathers. He then splits them into a sample of fathers and a sample of sons. Suppose that data were collected for a random sample of 11 pairs, where each difference is calculated by subtracting the height of the son from the height of the father. Assume that the heights are normally distributed. Using a test statistic of t≈1.971, the significance level α=0.05, and the corresponding p-value between 0.05 and 0.10, draw a conclusion for the appropriate hypothesis test, where the null hypothesis is H0:μd=0 and the alternative hypothesis is Ha:μd≠0.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the heights of males between generations are different. Compare the p-value to α=0.05. Since the p-value is between 0.05 and 0.10, it is greater than α, so we fail to reject H0. Therefore, there is insufficient evidence at the α=0.05 level of significance to suggest that the heights of males between generations are different.

A farmer wants to test if a new fertilizer will produce more massive crops. Suppose that data were collected for a randomly selected set of 6 crops, where each difference is calculated by subtracting the mass of the crop the previous year with the old fertilizer from the mass of the crop this year with the new fertilizer. Assume that the masses are normally distributed. Using a test statistic of t≈1.266, the significance level α=0.10, and the corresponding p-value greater than 0.10, draw a conclusion for the appropriate hypothesis test, where the null hypothesis is H0:μd=0 and the alternative hypothesis is Ha:μd>0.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the new fertilizer will produce more massive crops. Compare the p-value to α=0.10. Since the p-value is greater than α, fail to reject H0. Therefore, there is insufficient evidence at the α=0.10 level of significance to suggest that the fertilizer produces more massive crops.

Quinn Napoli is looking to purchase a vehicle and is deciding between two models, the Glacier and the Ravine. One of Quinn's friends suggests that perhaps the mean annual maintenance costs are different for the two models, so Quinn obtains random samples of the annual maintenance costs from 50 owners of each model of car. The sample statistics are in the table shown below. Let μ1 be the population mean annual maintenance cost for the Glacier, and let μ2 be the population mean annual maintenance cost for the Ravine. Quinn tests the alternative hypothesis Ha:μ1−μ2≠0 and assumes that the population standard deviations of the two models are not equal. If the p-value of the hypothesis test is greater than 0.10 and the significance level is α=0.10, what conclusion could be made about the population mean annual maintenance costs of the two models?

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the population mean annual maintenance costs of the Glacier is different than the population mean annual maintenance costs of the Ravine. Compare the p-value that is greater than 0.10 to α=0.10. Since the p-value is greater than α, fail to reject H0. Therefore, there is insufficient evidence at the α=0.10 level of significance to suggest that the population mean annual maintenance costs of the Glacier is different than the population mean annual maintenance costs of the Ravine.

A pet food manufacturer has a facility with two lines that cook, blend, and can cat food. Nolan Munroe is the production manager of the facility and curious whether there is a difference in the mean number of cans produced per day by the two lines, A and B. Nolan randomly selects 300 daily production reports, 141 of which were for production line A and 159 of which were for production line B. The sample statistics are shown in the table below. Let μ1 be the population mean number of cans produced per day on production line A, and let μ2 be the population mean number of cans produced per day on production line B. Nolan tests the alternative hypothesis Ha:μ1−μ2≠0 and assumes that the population standard deviations of the two production lines are not equal. If the p-value of the hypothesis test is greater than 0.05 and less than 0.10 and the significance level is α=0.05, what conclusion could be made about the population mean number of cans produced per day by the two production lines?

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the population mean number of cans produced by production line A is different than the population mean number of cans produced by production line B. Compare the p-value that is greater than 0.05 and less than 0.10 to α=0.05. Since the p-value is greater than α, fail to reject H0. Therefore, there is insufficient evidence at the α=0.05 level of significance to suggest that the population mean number of cans produced by production line A is different than the population mean number of cans produced by production line B.

A farm has a large cornfield with two artificial ponds along the east and west borders. Brennan Castillo is an environmental science student who is investigating whether the mean concentrations of a particular pesticide are the same in the two ponds. Brennan collects water samples from random locations and depths in each pond. The sample mean concentration in the 35 samples Brennan collected from the pond to the east of the field is 8.52 parts per billion (ppb) with a standard deviation of 2.71 ppb. The 35 samples Brennan collected from the pond to the west of the field have a sample mean concentration of 9.84 ppb and a standard deviation of 2.93 ppb. Let μ1 be the population mean concentration of the pesticide in the pond to the east of the field, and let μ2 be the population mean concentration of the pesticide in the pond to the west of the field. Brennan tests the alternative hypothesis Ha:μ1−μ2≠0 and assumes that the population standard deviations of the two ponds are equal. If the p-value of the hypothesis test is greater than 0.05 and less than 0.10 and the significance level is α=0.01, what conclusion could be made about the population mean pesticide concentration of the two artificial ponds? Identify all of the appropriate conclusions to the hypothesis test below.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the population mean pesticide concentration of the pond to the east of the field is different than the population mean pesticide concentration of the pond to the west of the field. Compare the p-value that is greater than 0.05 and less than 0.10 to α=0.01. Since the p-value is greater than α, fail to reject H0. Therefore, there is insufficient evidence at the α=0.01 level of significance to suggest that the population mean pesticide concentration of the pond to the east of the field is different than the population mean pesticide concentration of the pond to the west of the field.

Dylan Rieder is a statistics student investigating whether athletes have better balance than non-athletes for a thesis project. Dylan randomly selects 32 student athletes and 45 students who do not play any sports to walk along a board that was 16 feet long and 2 inches wide and raised 6 inches off the ground. Dylan records the number of times each participant touched the ground. The sample of athletes had a mean of 3.7 touches with a standard deviation of 1.1. The sample of non-athletes had a mean of 4.1 touches with a standard deviation of 1.3. Let μ1 be the population mean number of touches for student athletes, and let μ2 be the population mean number of students who do not play any sports. Dylan is testing the alternative hypothesis Ha:μ1−μ2<0 and assumes that the population standard deviation of the two groups of students are equal. If the p-value is greater than 0.05 and less than 0.10 and the significance level is α=0.01, what conclusion could be made about the balance of student athletes and the balance of students who do not play sports? Identify all of the appropriate conclusions to the hypothesis test below.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the student athletes have a different population mean number of touches than students who do not play sports. Compare the p-value that is greater than 0.05 and less than 0.10 to α=0.01. Since the p-value is greater than α, fail to reject H0. Therefore, there is insufficient evidence at the α=0.01 level of significance to suggest that the student athletes have a different population mean number of touches than students who do not play sports.

A survey asked randomly selected drivers if they would be willing to pay higher taxes for gasoline as long as all of the additional revenue went to improving roadways. Among the 48 males surveyed, 14 responded "Yes." Of the 52 females surveyed, 21 responded "Yes." Test the alternative hypothesis that the population proportion for males is different from the population proportion for females. Use the level of significance α=0.02. The test statistic is z≈−1.18, and the p-value is approximately 0.238. Identify all of the appropriate conclusions to the hypothesis test below.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion for males is different from the population proportion for females. The p-value is greater than the level of significance, so fail to reject the null hypothesis. Conclude that there is not enough evidence to support the claim that the population proportion for males is different from the population proportion for females.

In a survey of 100 randomly selected people in City A, 73 support increased government spending on education. In a survey of 100 randomly selected people in City B, 81 support such spending. Test the alternative hypothesis that the population proportion of people in City A that support such spending is different from the population proportion of people in City B. Use the level of significance α=0.05. The test statistic is z≈−1.34, and the p-value is approximately 0.180. Identify all of the appropriate conclusions to the hypothesis test below.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of people in City A that support such spending is different from the population proportion of people in City B. The p-value is greater than the level of significance, so fail to reject the null hypothesis. Conclude that there is not enough evidence to support the claim that the population proportion of people in City A that support such spending is different from the population proportion of people in City B.

In a survey of 100 randomly selected taxi drivers in a city, 76 support increased government spending on roads and bridges. In a survey of 100 randomly selected bus drivers in the same city, 82 support such spending. Test the alternative hypothesis that the population proportion of taxi drivers in the city that support such spending is less than the population proportion of bus drivers in the city that do. Use the level of significance α=0.05. The test statistic is z≈−1.04 and the p-value is approximately 0.149. Identify all of the appropriate conclusions to the hypothesis test below.

Fail to reject the null hypothesis. The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of taxi drivers in the city that support such spending is less than the population proportion of bus drivers in the city that do. The p-value is greater than the level of significance, so fail to reject the null hypothesis. Conclude that there is not enough evidence to support the claim that the population proportion of taxi drivers in the city that support such spending is less than the population proportion of bus drivers in the city that do.

The proportion of mango trees on an island that produce only dwarf fruit is p1. The proportion of mango trees on a different island that produce only dwarf fruit is p2. What is the null hypothesis for a test to determine if the proportions are different between the two islands?

H0: p1−p2=0 The null hypothesis (H0) states that there is no difference between the proportion of successes for the populations, so p1=p2, or p1−p2=0.

Suppose the fraction of white sheep in Herd A is p1 and the fraction of white sheep in Herd B is p2. State the null hypothesis for a test to determine if Herd A has a lower proportion of white sheep.

H0: p1−p2=0 The null hypothesis (H0) states that there is no difference between the proportion of successes for the populations, so p1=p2, or p1−p2=0. H0:p1−p2=0

The null hypothesis (H0) states that there is no difference between the proportion of successes for the populations. In this case, the proportion of residents coping with depression in Country A is the same as the proportion of residents coping with depression in Country B. So, p1=p2, or p1−p2=0. H0:p1−p2=0

H0: p1−p2=0 The null hypothesis (H0) states that there is no difference between the proportion of successes for the populations, so p1=p2, or p1−p2=0. H0:p1−p2=0

Let p1 be the proportion of residents in Country A coping with depression, and let p2 be the proportion of residents in Country B coping with depression. What is the null hypothesis for a test to determine whether Country A has a greater proportion of residents coping with depression than Country B?

H0: p1−p2=0 The null hypothesis (H0) states that there is no difference between the proportion of successes for the populations. In this case, the proportion of residents coping with depression in Country A is the same as the proportion of residents coping with depression in Country B. So, p1=p2, or p1−p2=0. H0:p1−p2=0

In the search to determine if car 1 is slower to accelerate than car 2, the mean time it takes to accelerate to 30 miles per hour is recorded (Note: a car is slower to accelerate if it takes more time to accelerate). Twenty trials of the acceleration time for each car are recorded, and both populations have normal distributions with known standard deviations. What are the hypotheses used in this test?

H0:μ1≤μ2; Ha:μ1>μ2 If car 1 is slower to accelerate than car 2, then the time for car 1 to accelerate must be greater than the time for car 2: μ1>μ2. The null hypothesis must have some form of equality, so Ha is μ1>μ2 and H0 is μ1≤μ2.

To determine if loans from bank 1 are less affordable than loans from bank 2, the mean loan rate of the two competing banks is compared (Note: a higher rate means the loan is less affordable). Twenty-five loans are randomly sampled from each bank and the loan rates are recorded. Both populations have normal distributions with unknown standard deviations.

H0:μ1≤μ2; Ha:μ1>μ2 In order for loans from bank 1 to be less affordable than loans from bank 2, the mean loan rate of loans from bank 1 must be greater than the mean loan rate of loans from bank 2: μ1>μ2. The null hypothesis must have some form of equality, so Ha is μ1>μ2 and H0 is μ1≤μ2.

Two potato chip brands, 1 and 2, are being compared to see if brand 1 is healthier than brand 2. Eighteen bags of each brand are randomly sampled and the number of calories per serving are measured. Note: for this test, the brand with fewer calories is considered healthier. Both populations have normal distributions with known standard deviation.

H0:μ1≥μ2; Ha:μ1<μ2 Brand 1 being healthier than brand 2 means that brand 1 has fewer calories: μ1<μ2. The null hypothesis must have some form of equality, so Ha is μ1<μ2 and H0 is μ1≥μ2.

A physician wants to determine if a supplement is effective in helping men lose weight. She takes a random sample of overweight men and records their weight before the trial. She then prescribes the supplement and instructs them to take it for four weeks while making no other lifestyle changes. After the four-week period, she records the weight of the men again. Identify the null and alternative hypotheses for a hypothesis test where d=weight after trial−weight before trial.

H0:μd=0 Ha:μd<0 The physician wants to test if the hypothesized mean of the differences for the paired data is less than 0. Therefore, the null hypothesis is H0:μd=0, and the alternative hypothesis is Ha:μd<0.

A researcher wants to test to see if husbands are significantly older than their wives. To do this, he collects the ages of husbands and pairs them with the ages of their respective wives for a random set of married couples. Identify the null and alternative hypotheses for a hypothesis test, where each difference is calculated by subtracting the age of the wife from the age of the husband.

H0:μd=0 Ha:μd>0 The researcher wants to test if the hypothesized mean of the differences for the paired data is greater than 0. Therefore, the null hypothesis is H0:μd=0, and the alternative hypothesis is Ha:μd>0.

Suppose the proportion of students in School A diagnosed with ADHD is p1 and the proportion of students in School B diagnosed with ADHD is p2. State the alternative hypothesis for a test to determine if School A has the higher proportion of students diagnosed with ADHD.

Ha: p1−p2>0

The proportion of mango trees on an island that produce only dwarf fruit is p1. The proportion of mango trees on a different island that produce only dwarf fruit is p2. What is the alternative hypothesis for a test to determine if the proportion is the same on both islands?

Ha: p1−p2≠0

If the proportion of the population in City A that is under 25 years old is p1 and the proportion of the population in City B that is under 25 years old is p2, what is the alternative hypothesis for a test to determine if the proportion of the population that is under 25 years old is different in these cities?

Ha: p1−p2≠0 The alternative hypothesis (Ha) states that the difference between the proportion of successes in the populations from which the samples are taken is either greater than 0, less than 0, or different from 0. Here, the claim is that the proportion of the population that is under 25 years old is different in the cities, so p1−p2≠0. Ha:p1−p2≠0

A television station executive would like to determine whether there is more viewership during the 6 P.M. time slot or the 7 P.M. time slot. The executive takes a random sample of 25 days and collects the viewership numbers for the 6 P.M. time slot into one sample and the viewership numbers for the 7 P.M. time slot into a second sample. The executive claims that these samples are independent. Do you agree? Explain.

No, because the data points in the first sample can be linked to data points in the second sample. Two samples are independent if one sample has no effect on the other sample. In this case, the data for both samples are from the same day. It is possible that, on any given day, many of the people that like to watch the program at 6 P.M. also like to watch the program at 7 P.M. This means that a good viewership number at 6 P.M. could make it more likely that there is a good viewership number at 7 P.M. Therefore, these samples are dependent.

A librarian would like to determine whether a new book appeals more to adults or to children. The librarian asks 10 randomly selected adults and their children whether or not they will be interested in reading the book. The librarian then sorts the responses into two samples: one sample for the adults' answers and one sample for their children's answers. Are these samples independent? If not, how can these samples be made independent?

No, the samples are dependent. To make the samples independent, the librarian should take a random sample of adults and then separately take a random sample of children who are not related to those adults. An adult's response can affect the response of the child, and the opposite is also true. Thus, the samples are dependent. To fix this, the librarian can separately sample a random sample of adults and a random sample of children to make sure that the two samples are not related.

In a survey of 50 randomly selected taxi drivers in City A, 4 support increasing the gasoline tax. In a survey of 50 randomly selected bus drivers in City B, 3 support such spending. Check that the conditions are met for a hypothesis test to compare the population proportion of taxi drivers in City A that support increasing the gasoline tax to the proportion of bus drivers in City B.

Not all the values n1p-, n1q-, n2p-, and n2q- are at least 5. In this case, p-=0.07 and n1p⎯=3.5<5, so the third condition is not met.

Suppose 3% of a sample of 100 randomly selected people in City A and 6% of a sample of 100 randomly selected people in City B prefer eating beef that is well done. Check that the conditions are met for a hypothesis test to compare the population proportion of people that prefer eating beef well done in City A to the proportion in City B.

Not all the values n1p⎯⎯⎯, n1q⎯⎯⎯, n2p⎯⎯⎯, and n2q⎯⎯⎯ are at least 5.

A magazine would like to determine how long it takes people to solve the crossword puzzles included inside their magazine. The magazine creates 2 samples to conduct the study. Are the samples dependent or independent?

Not enough information is provided to determine independence. It is unknown how the samples were constructed. To determine whether the samples are dependent or independent, it is necessary to determine if one of the samples can be affected by the other. Are the same people solving crossword puzzles across multiple issues? Or are different groups of people solving one puzzle? Until these types of procedural questions are known, independence cannot be determined.

An executive believes that a new energy drink his company developed will increase an individual's stamina. In order to test this, he selects random individuals and times how long they can run without stopping. He then instructs the individuals to drink the energy drink for two weeks. After the two weeks, he times how long they can run again without stopping. If we were to perform a hypothesis test on the difference between running time after drinking the energy drink and running time before drinking the energy drink, what would be the null and alternative hypotheses?

Null: The population average difference in running time is equal to zero. Alternative: The population average difference in running time is greater than zero. The executive is testing whether the hypothesized population mean of the differences for the paired data is less than 0. Therefore, the null hypothesis is "the population average difference is equal to zero" and the alternative hypothesis is "the population average difference is greater than zero."

A therapist believes that the music a child listens to may have an impact on the amount of words on a list they can memorize. In order to test this, she selects a random sample of children for an experiment. She has the children look over a list of one hundred words for five minutes while listening to classic rock music. After the five minutes, she tests the children to see how many words they could remember in the order of the list. She then does the same experiment, this time with a different list of words, while listening to classical music. If we let d=classical−classic rock, identify the null and alternative hypotheses for an appropriate hypothesis test.

Null: The population average difference in words recited is equal to zero. Alternative: The population average difference in words recited is not equal to zero. The therapist wants to test if the hypothesized mean of the differences for the paired data is different from 0. Therefore, the null hypothesis is "the population average difference in words recited is equal to zero" and the alternative hypothesis is "the population average difference in words recited is not equal to zero."

Jeremy Remland is a civil engineering student who lives in a large city that has a vast public transportation network. He is writing a report for one of his classes that evaluates the city's public bus and subway services. Jeremy would like to make the claim in the report that the population mean commute time for city residents who take only the bus to commute to work is different than the population mean commute time for city residents who take only the subway to commute to work. Jeremy uses the results of similar studies that have already been completed to estimate that the population standard deviation is 8.07 minutes for commuters who take only the bus and 7.14 minutes for commuters who take only the subway. Jeremy conducts a survey using a random sample of city residents who take only the bus and who take only the subway to commute. The results of the survey are given in the table provided. Let μ1 be the population mean commute time, in minutes, for city residents who take only the bus and μ2 be the population mean commute time, in minutes, for city residents who take only the subway. The p-value rounded to three decimal places is 0.042, the significance level is α=0.05, the null hypothesis is H0:μ1−μ2=0, and the alternative hypothesis is Ha:μ1−μ2≠0. Bus x1=26.11 mins n1=171 Subway=24.46 mins n2=186

Reject the null hypothesis that the true difference between the population mean commute time for city residents who take only the bus to commute to work and the population mean commute time for city residents who take only the subway to commute to work is equal to zero. Based on the results of the hypothesis test, there is enough evidence at the α=0.05 level of significance to support the claim that the true difference between the population mean commute time for city residents who take only the bus to commute to work and the population mean commute time for city residents who take only the subway to commute to work is not equal to zero. Compare the p-value, 0.042, to α=0.05. Since the p-value is less than α, reject the null hypothesis H0. Therefore, there is enough evidence at the α=0.05 level of significance to support the claim that the true difference between the population mean commute time for city residents who take only the bus to commute to work and the population mean commute time for city residents who take only the subway to commute to work is not equal to zero.

Marianna is a website designer who wants to determine whether two different home-page formats lead visitors to click on different numbers of links. She writes two scripts: one to randomly display one of the home pages and another to record the page displayed and the number of links clicked. For home page A, the sample of 132 visitors clicked an average of 8.7 links with a standard deviation of 2.3. For home page B, the sample of 118 visitors clicked an average of 7.3 links with a standard deviation of 2.1. Let μ1 be the population mean number of clicks for home page A, and let μ2 be the population mean number of clicks for home page B. Marianna uses the alternative hypothesis Ha:μ1−μ2≠0 and assumes that the population standard deviation is the same for both home pages. If the p-value of the hypothesis test is less than 0.01 and the significance level is α=0.01, what conclusion could be made about the population mean number of clicks between the two home pages? Identify all of the appropriate conclusions to the hypothesis test below.

Reject the null hypothesis. The conclusion of the hypothesis test is that there is sufficient evidence to suggest that the population mean number of clicks for home page A is different than the population mean number of clicks for home page B. Compare the p-value that is less than 0.01 to α=0.01. Since the p-value is less than α, reject H0. Therefore, there is sufficient evidence at the α=0.01 level of significance to suggest that the population mean number of clicks for home page A is different than the population mean number of clicks for home page B.

Caden Carpenter is a professor of entomology who claims that for a certain species of dragonfly, the ones that live by a nearby river have a greater mean wingspan than those that live near a lake that is 10 kilometers away. To test this claim, Caden randomly captures, measures, and releases 45 dragonflies at each site. The sample statistics are provided in the table below. Let μ1 be the population mean wingspan of the dragonflies that live by the river, and let μ2 be the population mean wingspan of the dragonflies that live by the lake. Caden uses the alternative hypothesis Ha:μ1−μ2>0 and assumes that the population standard deviation for both groups of dragonflies is equal. If the p-value of the hypothesis test is less than 0.01 and the significance level is α=0.01, what conclusion could be made about the population mean wingspans of the dragonflies that live by the river and those that live by the lake? River x1=68.79mm s1=4.85mm Lake x2=65.18mm s2=4.24mm

Reject the null hypothesis. The conclusion of the hypothesis test is that there is sufficient evidence to suggest that the population mean wingspan of the dragonflies that live by the river is greater than the population mean wingspan of the dragonflies that live by the lake. Compare the p-value that is less than 0.01 to α=0.01. Since the p-value is less than α, reject H0. Therefore, there is sufficient evidence at the α=0.01 level of significance to suggest that the population mean wingspan of the dragonflies that live by the river is greater than the population mean wingspan of the dragonflies that live by the lake.

An analyst wants to determine if there is any difference in the amount of time teenagers play video games between two years. To do this, he takes a random sample of teenagers and gathers the average time they spent playing video games the previous year and compares it to the average time they spent playing video games this year. Suppose that data were collected for a random sample of 15 teenagers, where each difference is calculated by subtracting the time spent playing video games this year in hours per day from the time spent playing video games last year in hours per day. Assume that the times are normally distributed. Using a test statistic of t≈2.385, the significance level α=0.10, and the corresponding p-value between 0.01 and 0.05, draw a conclusion for the appropriate hypothesis test, where the null hypothesis is H0:μd=0 and the alternative hypothesis is Ha:μd≠0.

Reject the null hypothesis. The conclusion of the hypothesis test is that there is sufficient evidence to suggest that there is a difference in the amount of time teenagers spend playing video games. Compare the p-value to α=0.10. Since the p-value is between 0.01 and 0.05, it is less than α, so we reject H0. Therefore, there is sufficient evidence at the α=0.10 level of significance to suggest that there is a difference in the amount of time teenagers play video games between the two years.

In a survey of 100 randomly selected taxi drivers in a city, 72 support increased government spending on roads and bridges. In a survey of 100 randomly selected bus drivers in the same city, 89 support such spending. Test the alternative hypothesis that the population proportion of taxi drivers in the city that support such spending is different from the population proportion of bus drivers in the city. Use the level of significance α=0.01. The test statistic is z≈−3.03, and the p-value is approximately 0.002. Identify all of the appropriate conclusions to the hypothesis test below.

Reject the null hypothesis. The conclusion of the hypothesis test is that there is sufficient evidence to support the claim that the population proportion of taxi drivers in the city that support such spending is different from the population proportion of bus drivers in the city. The p-value is less than the level of significance, so reject the null hypothesis. Conclude that there is enough evidence to support the claim that the population proportion of taxi drivers in the city that support such spending is different from the population proportion of bus drivers in the city.

Assume that a researcher asked a random sample of middle school students what they thought they wanted to do when they finished school. Then, the researcher went to the same sample of students 10 years later and asked them what their plans were. Are these samples dependent or independent?

The samples are dependent. The same random sample of students is being used for both samples. While there is no doubt that people can change a lot over time, the answers from one of the samples definitely can have an effect on the answers in the other sample. Thus, the samples are dependent.

An internet researcher creates a poll with two related questions on an internet website. The researcher places each person's answer to the first question in one sample and each person's answer to the second question in the second sample. Are these samples independent or dependent?

The samples are dependent. The two samples consist of answers from the same group of people. Therefore, the answers can be linked by a person's answer to the first question in the first sample and same person's answer to the second question in the second sample. This makes the two samples dependent.

To test a drug intended to increase memory, a random sample of 80 subjects were asked to look at a poster with pictures of 20 common objects for five minutes. After waiting ten minutes, the subjects were then asked to list as many of the objects as they could recall. The subjects were then given the drug and asked to look at a different poster with pictures of 20 common objects for five minutes. After waiting ten minutes, the subjects were then asked to list as many of the objects as they could recall. Before taking the drug, 40% were able to list at least half of the objects. After taking the drug, 65% were able to list at least half of the objects. Check that the conditions are met for a hypothesis test to compare the population proportion of subjects that could list at least half of the objects before the drug to the proportion that could after taking the drug.

The samples are not independent.

In a survey of 50 randomly selected taxi drivers in City A, 27 support increased government spending on public transportation. In a survey of 50 randomly selected bus drivers in City B, 32 support such spending. Check that the conditions are met for a hypothesis test to compare the population proportion of people in City A that support such spending to the proportion of people in City B.

The samples are not random samples of the populations.

In a survey of 50 randomly selected taxi drivers in City A, 27 support increased government spending on public transportation. In a survey of 50 randomly selected bus drivers in City B, 32 support such spending. Check that the conditions are met for a hypothesis test to compare the population proportion of people in City A that support such spending to the proportion of people in City B.

The samples are not random samples of the populations. The conditions for a hypothesis test for the difference between two population proportions are shown below. 1) The samples are random. 2) The samples are independent. In this case, the samples are not random samples of the populations of the cities.

Suppose 10% of a sample of 75 randomly selected diners at a steakhouse in City A prefer eating beef over chicken and 8% of a sample of 75 randomly selected diners at a seafood restaurant in City B prefer eating beef over chicken. Check that the conditions are met for a hypothesis test to compare the population proportion of people that prefer eating beef over chicken in City A to the proportion in City B.

The samples are not random. The conditions for a hypothesis test for the difference between two population proportions are shown below. 1) The samples are random. 2) The samples are independent. In this case, the samples are not random samples of the populations of the cities in general.

Find the p-value for a right-tailed hypothesis test when the test statistic is z=0.3. Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=0.3 is approximately 0.618. This means the area to the right of z=0.3 is approximately 1−0.618=0.382. Therefore, the p-value is approximately 0.382.

Find the p-value for a two-tailed hypothesis test when the test statistic is z=1.3. Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=1.3 is approximately 0.903. This means the area to the right of z=1.3 is approximately 1−0.903=0.097. Twice this area is 2(0.097)=0.194. Therefore, the p-value is approximately 0.194.

In a survey of 100 randomly selected people in City A, 82 support increased government spending on roads and bridges. In a survey of 100 randomly selected people in City B, 72 support such spending. Test the alternative hypothesis that the population proportion of people in City A who support such spending is different from the population proportion of people in City B who do. The test statistic is z=1.68. What is the corresponding p-value? Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=1.68 is approximately 0.954. This means the area to the right of z=1.68 is approximately 1−0.954=0.046. Twice this area is 2(0.046)=0.092. Therefore, the p-value is approximately 0.092.

Researchers tested random samples of cattle from two towns for exposure to a particular virus. In Town A, of the 100 cattle selected, 22 tested positive for exposure, and in Town B, of the 100 cattle selected, 27 tested positive for exposure. To test the hypothesis that the population proportions of exposure is less in Town A at the α=0.05 level of significance, the test statistic is z=−0.82. What is the corresponding p-value, rounded to three decimal places? Compute your answer using a value from the table below.

Using a normal distribution table, the area to the left of z=−0.82 is approximately 0.206. Therefore, the p-value is approximately 0.206.

Consider the following samples. Which of these are clearly dependent?

a sample of 10 females and a sample of 10 males composed of wives and their husbands a sample of 100 freshmen who are sampled again a year later as sophomores

A local bank has two branches, one in Standish and the other in Limerick. To determine whether the wait time at the drive-through was different for the branches, the director of the bank had the manager at each branch use security camera footage to randomly select 40 customers who used the drive-through and to determine the wait time. The average wait time at the Standish branch was 93.12 seconds with a standard deviation of 14.65 seconds. The average wait time at the Limerick branch was 107.36 seconds with a standard deviation of 16.14 seconds. Let μ1 be the population mean wait time for drive-through customers at the Standish branch, and let μ2 be the population mean wait time for drive-through customers at the Limerick branch. The manager assumes the population standard deviations are equal and tests the alternative hypothesis Ha:μ1−μ2≠0. If the test statistic is t≈−4.13 and the number of degrees of freedom is 78, what is the p-value for this hypothesis test?

p-value <0.01

Ella Mayfield is researching the cost of preschool for a regional parenting magazine by comparing the hourly cost between two large metropolitan areas in the region, Waterboro and Sanford. Based on anecdotal evidence, she is testing the claim that the mean hourly cost of preschool in Sanford is lower than the mean hourly cost of preschool in Waterboro. She selects a random sample of preschools from both areas and records the rates. The sample statistics are shown in the table below. Let μ1 be the population mean hourly cost of preschool in Sanford, and let μ2 be the population mean hourly cost of preschool in Waterboro. Ella assumes the population standard deviations are not equal and tests the alternative hypothesis Ha:μ1−μ2<0. If the test statistic is t≈−0.99 and the number of degrees of freedom is 40, what is the p-value for this hypothesis test?

p-value <0.01

Lara Norris is a poultry farmer who claims that her Rhode Island Red hens on average lay more eggs than her Plymouth Rock hens. A random sample of 60 values of the daily egg production by the Rhode Island Red hens had a mean of 0.765 egg per hen per day and a standard deviation of 0.113. A random sample of 50 values of the daily egg production by the Plymouth Rock hens had a mean of 0.559 egg per hen per day and a standard deviation of 0.092. Let μ1 be the population mean number of eggs laid per hen per day by the Rhode Island Red hens and μ2 be the population mean number of eggs laid per hen per day by the Plymouth Rock hens. Lara assumes the population standard deviations are equal and uses the alternative hypothesis Ha:μ1−μ2>0. If the test statistic is t≈10.34 and the number of degrees of freedom is 108, what is the p-value for this hypothesis test?

p-value <0.01

A physician wants to test if temperature has an effect on heart rate. In order to do this, she compares the heart rates in beats per minute of several random volunteers after a period of time in a room with a temperature of 50∘F and after a period of time in a room with a temperature of 75∘F. Suppose that data were collected for a random sample of 11 volunteers, where each difference is calculated by subtracting the heart rate in beats per minute in the 50∘F room from the heart rate in beats per minute in the 75∘F room. Assume that the populations are normally distributed. The physician uses the alternative hypothesis Ha:μd≠0. Suppose the test statistic t is computed as t≈5.627, which has 10 degrees of freedom. What range contains the p-value?

p-value <0.01 Using the table of areas in the right tail for the t-distribution, in the row for 10 degrees of freedom, 5.627 is greater than 3.169. So, the area in the right tail is less than 0.005. Since this is a two-tailed test, multiply the area by 2. The p-value is less than 0.01.

Your friend believes that he has found a route to work that would make your commute faster than what it currently is under similar conditions. Suppose that data were collected for a random set of 7 days, where each difference is calculated by subtracting the time taken on the current route from the time taken on the new route. Assume that the populations are normally distributed. Your friend uses the alternative hypothesis Ha:μd<0. Suppose the test statistic t is computed as t≈−3.201, which has 6 degrees of freedom. What range contains the p-value?

p-value <0.01 this is a left-tailed test because the alternative hypothesis is Ha:μd<0. Using the table of areas in the right tail for the t-distribution, in the row for 6 degrees of freedom, 3.201 is greater than 3.143. So, the p-value is less than 0.01.

A sports analyst wants to determine if the number of free throws basketball players make in their second year is more than the number of free throws they make in their rookie year. To do this, he selects several random players and compares the average number of free throws they made per game in their second year to the average number of free throws they made per game in their rookie year. Suppose that data were collected for a random sample of 11 players, where each difference is calculated by subtracting the average number of free throws made per game in the player's rookie year from the average number of free throws made per game in the player's second year. Assume that the number of throws is normally distributed. The analyst uses the alternative hypothesis Ha:μd>0. Using a test statistic of t≈4.842, which has 10 degrees of freedom, determine the range that contains the p-value.

p-value <0.01 this is a right-tailed test because the alternative hypothesis is Ha:μd>0. The t-test statistic is 4.842 with 10 degrees of freedom. The t-distribution table gives the values for a right-tailed test. Using the table of areas in the right tail for the t-distribution, in the row for 10 degrees of freedom, 4.842 is greater than 2.764. So, the p-value is less than 0.01.

The partners at an investment firm want to know which of their two star financial planners, Brayden or Zoe, produced a higher mean rate of return last quarter for their clients. The partners reviewed last quarter's rates of return for random samples of clients who were managed by Brayden or Zoe. The mean rate of return for the sample of 30 of Brayden's clients was 3.54% with a standard deviation of 0.92%. The mean rate of return for a sample of 30 of Zoe's clients was 3.87% with a standard deviation of 2.08%. Let μ1 be the population mean rate of return for Brayden's clients and μ2 be the population rate of return for Zoe's clients. The partners assume the population standard deviations are not equal and, since Zoe's mean is higher, test the alternative hypothesis Ha:μ1−μ2<0. If the test statistic is t≈−0.80 with 29 degrees of freedom, what is the p-value for this hypothesis test?

p-value >0.10

Austin Clemens is writing a report about the city's climate for his high school environmental science class. To impress his teacher, Austin would like to show evidence that the population mean daily low temperature was lower during the five-year period from 1998-2002 than it was for the five-year period from 2013-2017. He researches the claim using a website that records all of the weather data observed at a local airport. After conducting the research, Austin assumes that the population standard deviation is 10.48∘F degrees for 1998-2002 and 11.29∘F for 2013-2017. Due to the large amount of data in each five-year period, Austin randomly selects the daily low temperatures for each group. The results are shown in the table below. Let μ1 be the population mean daily low temperature during the five-year period from 1998-2002 and μ2 be the population mean daily low temperature during the five-year period from 2013-2017. If the test statistic is z=−0.87, what is the p-value for this hypothesis test? 1998-2002 x1=44.31 n1=59 2013-2017 x2=46.02 n2=63

p-value= 0.192

Marketers at PaperClips, a regional office supply store, are researching whether the population mean amount spent on back-to-school shopping for students in grades K-12 has changed from 2016 to 2018. Based on market research, the marketers assume the population standard deviation is $101.52 for 2016 and $96.47 for 2018. Households across the region with at least one student in grades K-12 were randomly selected in 2016 and again in 2018. The results are shown in the table below. Let μ1 be the population mean amount spent on back-to-school shopping per household in 2016 and μ2 be the population mean amount spent on back-to-school shopping per household in 2018. If the test statistic is z=2.39, what is the p-value for this hypothesis test? 2016 x1=$521.54 n1=534 2018 x2=$506.92 n2=511

p-value=.016

A cell phone manufacturer claims that the population mean battery life of its flagship smartphone model, the Black Bear, is greater than the population mean battery life of the largest competitor, the Grizzly. A consumer advocacy publication tests this claim by purchasing a random sample of Black Bear smartphones and a random sample of Grizzly smartphones. Members of the publication charged each smartphone to full capacity and then had the smartphones play back the same videos until the batteries were completely depleted. The publication researched the population standard deviation of the battery life from the manufacturers. The population standard deviation for the Black Bear is assumed to be 0.71 hour, and the population standard deviation for the Grizzly is assumed to be 0.63 hour. The results of the battery life test are shown below. Let μ1 be the population mean battery life for the Black Bear and μ2 be the population mean battery life for the Grizzly. If the test statistic is z=1.38, what is the p-value for this hypothesis test? Black Bear x1=11.24 hours n1=54 Grizzly x2=11.06 hours n2=51

p-value=.084

Brandon Blake is a real estate agent working in several towns in a local area, mainly Littleton and Whitefield. Brandon has worked in the real estate business for several years and believes that the population mean selling price of a single-family home in Littleton during 2017 was greater than the population mean selling price of a single-family home in Whitefield during 2017. After consulting with a local realtors' association, he assumes the population standard deviation to be $14,496 for Littleton and $13,942 for Whitefield. Brandon randomly selects a sample of selling prices for single-family homes for each of the two towns during 2017. The results of the samples are shown in the table below. Let μ1 be the population mean selling price of a single-family home in Littleton during 2017 and μ2 be the population mean selling price of a single-family home in Whitefield during 2017. If the test statistic is z=1.27, what is the p-value for this hypothesis test? Littleton: x1=$214,601 n1=84 Whitefield x2=$211,862 n2=91

p-value=.102

A team of archeologists is reexamining a site where two additional sets of artifacts have been discovered. The sets of artifacts are in two separate areas of the archeological site, where each set is adjacent to artifacts that were discovered in the past. Since both sets of artifacts were discovered at about the same time years after the first examination, the members of the team would like to find out whether the population mean age of the artifacts in Set A is the different than the population mean age of the artifacts in Set B. Using information from the first examination of the site, the members of the team assume that the population standard deviation of the age of artifacts found in Set A is 135 years and the population standard deviation of the age of the artifacts found in Set B is 119 years. The team takes a random sample of the artifacts and finds the ages of each artifact using radiocarbon dating. The results are provided in the table below. Let μ1 be the population mean age, in years, of the artifacts in Set A and μ2 be the population mean age, in years, of the artifacts in Set B. If the test statistic is z=1.56, what is the p-value for this hypothesis test? Set A x1=2,726 n1=71 Set B x2=2,692 n2=65

p-value=.118

Leah Peschel is the bottling department manager for a bottling company that produces various soft drinks and juices. The company uses two different machines from different manufacturers to fill the bottles of its popular cola. Leah periodically verifies that the mean population amount of cola in the bottles filled by Machine 1 is the same as the mean population amount in the bottles filled by Machine 2. The manufacturers calibrated the machines at the time of installation and provided that information to the bottling company. Using this information, Leah assumes that the population standard deviation for Machine 1 is 0.021 ounce and the population standard deviation for Machine 2 is 0.019 ounce. Leah randomly selects samples of bottles filled by Machine 1 and Machine 2. The amount of cola in each bottle is recorded for both samples, and the results are shown in the table. Let μ1 be the population mean of Machine 1 and μ2 be the population mean of Machine 2. If the test statistic is z=2.22, what is the p-value for this hypothesis test? Machine 1 x1=12.522 n1=244 Machine 2 x2=12.518 n2=251

p-value=0.026

Simone Lahey is a district vice president of a bank. She is receiving complaints from bank customers about the amount of time they have to wait in the Mineola branch. She is investigating whether the population mean wait time of the Mineola branch is greater than the population mean wait time of the Westbury branch. Simone carefully reviews studies of the mean wait times conducted in the past and assumes that the population standard deviation is 1.09 minutes in the Mineola branch and 0.96 minute in the Westbury branch. Simone conducts a survey at each branch over a period of time by randomly selecting customers who wait in line and then recording each customer's wait time in minutes. The results of the survey are displayed in the table shown below. Let μ1 be the population mean wait time for customers in the Mineola branch and μ2 be the population mean wait time for customers in the Westbury branch. If the test statistic is z=1.84, what is the p-value for this hypothesis test? Mineola x1=3.06 minutes n1=194 Westbury x2=2.87 minutes n2=205

p-value=0.033

A large drug store chain outsources the manufacturing of its pain relief tablets to two different independent manufacturers, Manufacturer A and Manufacturer B. The quality control department of the chain would like to ensure that the population mean amount of the active ingredient in the tablets is still the same between the two manufacturers. The department assumes the population standard deviation for the amount of the active ingredient in a tablet based on the specifications from the manufacturers. The quality control department selects a random sample of tablets from each manufacturer across dozens of lots. The tablets are then analyzed using chromatography to determine the amount of the active ingredient in each tablet. The results of the analysis and the population standard deviations that are assumed for this hypothesis test are provided in the table below. Let μ1 be the population mean amount of the active ingredient, in milligrams, in tablets produced by Manufacturer A and μ2 be the population mean amount of the active ingredient, in milligrams, in tablets produced by Manufacturer B. If the test statistic is z=−1.65, what is the p-value for this hypothesis test? Manufacturer A x1=196.145 mg n1=251 σ1=0.578 mg Manufacturer B x2=196.236 n2=228 σ2=0.622 mg

p-value=0.098

The composition of a quarter dollar coin changed from silver and copper to a copper-nickel alloy in 1964. Hunter Rowland collects coins as a hobby and has access to large collections of quarters produced before and after 1964. He would like to show that the population mean mass of the quarters produced after 1964 is less than the population mean mass of the quarters produced before 1964. Hunter researches several coin collector websites and assumes the population standard deviation for each type of quarter from the information he gathered. Hunter randomly selects each type of quarter from the large collection and finds the mass of each one using a precise scale. The results of the samples and the population standard deviations that are assumed for this hypothesis test are shown in the table below. Assuming the conditions needed for the hypothesis test have been met, what is the z test statistic for this hypothesis test, rounded to two decimal places? After 1964 x1= 5.648 n1=37 σ1=.070 Before 1964 x2= 6.191 n1=39 σ1=.086

z=-30.26