Statistics
A two-sample t-test of the hypotheses H0 : μ1 - μ2 = 0 versus Ha : μ1 - μ2 > 0 produces a p-value of 0.03. Which of the following must be true? I. A 90 percent confidence interval for the difference in means will contain the value 0. II. A 95 percent confidence interval for the difference in means will contain the value 0. III. A 99 percent confidence interval for the difference in means will contain the value 0.
II and III only
A test of the hypotheses H0 : µ = 0 versus Ha : µ > 0 was conducted using a sample of size 7. The test statistic was t = 1.935. Which of the following is closest to the p-value of the test?
0.0506
In a test of H0 : μ = 8 versus Ha : μ ≠ 8, a sample of size of 220 leads to a p-value of 0.034. Which of the following must be true?
A 95% confidence interval for μ calculated from these data will not include μ = 8
Researchers for a company that manufactures batteries want to test the hypothesis that the mean battery life of their new battery is greater than the known mean battery life of their older version. The researchers selected random samples of 32 of the new batteries, subjected the batteries to continuous use, and determined the mean and standard deviation of the battery lives in the sample. Which of the following is an appropriate test for the researchers' hypothesis?
A one-sample -test for a population mean
Two 99 percent confidence intervals will be constructed to estimate the difference in means of two populations, R and W. One confidence interval, , will be constructed using samples of size 9 from each of R and W, and the other confidence interval, , will be constructed using samples of size 81 from each of R and W. When all other things remain the same, which of the following describes the relationship between the two confidence intervals?
The width of will be the width of .
A recent study reported the mean body mass index for adults in the United States was 26.8. A researcher believes the mean of marathon runners is less than 26.8. A random sample of 35 marathon runners had a mean of 22.7 with a standard deviation of 3.1. The researcher will conduct a one-sample -test for a population mean. Have the conditions for inference been met?
Yes, all conditions have been met.
Researchers are investigating the effectiveness of leg-strength training on cycling performance. A sample of 7 men will be selected to participate in a training program that lasts for one month. Peak power during cycling will be recorded for each man both before training and after training. The mean difference in times will be used to construct a 95 percent confidence interval for the mean difference in the population. When all other things remain the same, which of the following statements about the width of the interval is correct?
The interval will be narrower if 15 men are used in the sample.
A sports equipment researcher investigated how different types of wood used to make baseball bats might affect batting. The researcher selected a sample of 80 batters from summer baseball leagues and randomly assigned the batters to one of two groups: the ash bat group or the maple bat group. The mean number of hits for each group was recorded at the end of the season, and the difference in the sample means was calculated. Which of the following is the appropriate inference procedure for analyzing the results of the investigation?
A two-sample -interval for a difference between population means
The local ranger station tracked and tagged 2,844 adult female black bears in a national park. A random sample of 9 adult female black bears from those tagged had an average body weight of 203 pounds with standard deviation 25 pounds. Which of the following is a point estimate for the population mean weight of all female black bears that are tagged?
203
A study will be conducted to investigate whether there is a difference in pain relief for two brands of headache pills, N and P. Participants will be randomly assigned to one of two groups. One group will take pill N when they experience a headache, and the other group will take pill P when they experience a headache. Each participant will record the number of minutes it takes until relief from the headache is felt. The mean number of minutes will be calculated for each group. Which of the following is the appropriate test for the study?
A two-sample -test for a difference between population means
A random sample of 50 students at a large high school resulted in a 95 percent confidence interval for the mean number of hours of sleep per day of (6.73, 7.67). Which of the following statements best summarizes the meaning of this confidence interval?
About 95% of all random samples of 50 students from this population would result in a 95% confidence interval that covered the population mean number of hours of sleep per day.
A machine is designed to dispense at least 12 ounces of a beverage into a bottle. To test whether the machine is working properly, a random sample of 50 bottles was selected and the mean number of ounces for the 50 bottles was computed. A test of the hypotheses H0 : µ = 12 versus Ha : µ < 12 was conducted, where µ represents the population mean number of ounces of the beverage dispensed per bottle by the machine. The p-value for the test was 0.08. Which of the following is the most appropriate conclusion to draw at the significance level of α = 0.05?
Because the p-value is greater than the significance level, there is not convincing evidence that the population mean number of ounces dispensed into a bottle is less than 12 ounces.
A two-sample -test for a difference in means was conducted to investigate whether defensive players on a football team can bench-press more weight, on average, than offensive players. The conditions for inference were met, and the test produced a test statistic of and a -value of 0.15. Based on the -value and a significance level of , which of the following is the correct conclusion?
Fail to reject the null hypothesis because . There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players
A reporter responsible for the food section of a magazine investigated the belief that grocery stores sell beef at a higher price in the fall than in the spring. The reporter selected independent random samples of grocerystore beef prices in November and April and computed the mean and standard deviation for the samples. Which of the following are the correct null and alternative hypotheses for the reporter's investigation, where represents the mean price of beef in the fall and represents the mean price of beef in the spring?
Ho: mean of F- mean of S =0 Ha: mean of F- mean of S >0
In a random sample of 60 shoppers chosen from the shoppers at a large suburban mall, 36 indicated that they had been to a movie in the past month. In an independent random sample of 50 shoppers chosen from the shoppers in a large downtown shopping area, 31 indicated that they had been to a movie in the past month. What significance test should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of shoppers at the suburban mall who had been to a movie in the past month is the same as the proportion of shoppers in the large downtown shopping area who had been to a movie in the past month?
Two proportion z-test
Alicia would like to know if there is a difference in the average price between two brands of shoes. She selected and analyzed a random sample of 40 different types of Brand A shoes and 33 different types of Brand B shoes. Alicia observes that the boxplot of the sample of Brand A shoe prices shows two outliers. Alicia wants to construct a confidence interval to estimate the difference in population means. Is the sampling distribution of the difference in sample means approximately normal?
Yes, because the size of each sample is at least 30.
For a certain brand of canned corn, the company claims that the mean weight of the contents of the cans is 15.25 ounces. A random sample of 36 cans were selected. The sample was found to have mean 15.18 ounces and standard deviation 0.12 ounce. A hypothesis test will be conducted to investigate whether there is evidence to support the belief that the mean is less than 15.25 ounces. Which of the following is the correct test statistic for the hypothesis test?
t= 15.8-15.25/0.12/6
Last year the mean cost for a one-bedroom rental in a certain city was $1,200 per month. Eli is looking for a one-bedroom apartment and is investigating whether the mean cost is less now than what it was last year. A random sample of apartments had a sample mean of $1,180 per month. Assuming all conditions for inference are met, Eli will conduct a hypothesis test as part of his investigation. Which of the following is the correct set of hypotheses?
Ho: mean=1200 Ha: mean<1200
Which of the following correctly compares the -distribution and -distribution?
The density curve of the -distribution is more spread out than the density curve of the -distribution, especially for small sample sizes.
A consumer group studied two different manufacturers of cars, J and K, to investigate differences in gas mileage for cars made by the two manufacturers. For a similar type of car, a random sample of 15 cars from J and a random sample of 12 cars from K were selected, and the gas mileages, in miles per gallon , were recorded. The difference in the sample mean gas mileages was used to construct the 90 percent confidence interval . Assuming all conditions for inference were met, which of the following is a correct interpretation of the interval?
We are 90 percent confident that the population mean difference of gas mileage for the two car manufacturers is between 3.5 and 5.7 .
A national consumer agency selected independent random samples of 45 owners of newer cars (less than five years old) and 40 owners of older cars (more than five years old) to estimate the difference in mean dollar cost of yearly routine maintenance, such as oil changes, tire rotations, filters, and wiper blades. The agency found the mean dollar cost per year for newer cars was $195 with a standard deviation of $46. For older cars, the mean was $286 with a standard deviation of $58. Which of the following represents the 95 percent confidence interval to estimate the difference (newer minus older) in the mean dollar cost of routine maintenance between newer and older cars?
(196-286)+- 1.992 times square root (46)^2/45+(58)^2/40
Hannah wanted to investigate whether there was a difference in the time spent in the checkout line between two grocery stores in a large city. She went to Grocery Store J on a Monday morning and recorded the time, in minutes, it took 30 customers to go through a checkout line. Then she went to Grocery Store K on Monday afternoon and recorded the time it took 30 customers to go through a checkout line. Hannah calculated the mean number of minutes for the customers in each line. She intends to conduct a two-sample -test for a difference in means between the two stores. Have all conditions for inference been met?
No, the data were not collected using a random method.
A magazine article reported that college students spend an average of $100 on a first date. A university sociologist believed that number was too high for the students at the university. The sociologist surveyed 32 randomly selected students from the university and obtained a sample mean of $92.23 for the most recent first dates. A one-sample -test resulted in a -value of 0.026. Which of the following is a correct interpretation of the -value?
If the mean amount of money that students from the university spend on a first date is $100, the probability is 0.026 that a randomly selected group of 32 students from the university would spend a mean of $92.23 or less on their most recent first dates.
Makers of a new pain-relieving medication claim that it relieves chronic pain faster than the current topselling pain reliever on the market. A double-blind experiment was conducted in which 10 people who experience chronic pain were randomly selected to take either the new or the current medication. Each of the 10 people recorded the time, in minutes, from taking the medication until pain relief. After an appropriate time period, each of the 10 people took the other medication and recorded the time from taking the medication until pain relief. The medication each person took first was randomly determined, and because both medications look the same, the people in the study did not know which medication was taken first. The table below shows summary statistics for the results.
0.1802
A marine biologist wants to estimate the average weight of a population of dolphins living in a certain region of the ocean. The biologist will collect a random sample of dolphins and use the sample weights to create the estimate. Which of the following is an appropriate method for the biologist to use for inference to the population?
A one-sample -interval for a population mean
Animal researchers studying cows and horses conducted a two-sample -test for a difference in means to investigate whether grazing cows eat more grass, on average, than grazing horses. All conditions for inference were met, and the test produced a test statistic of and a -value of 0.0487. Which of the following is a correct interpretation of the -value?
Assuming that the mean amount of grass eaten by cows is equal to the mean amount of grass eaten by horses, the probability of observing a test statistic of at least 1.664 is 0.0487.
Researchers at a medical center studied the amount of caffeine, in milligrams (mg), contained in a 16-ounce cup of coffee made at one machine at the center's cafeteria. They selected a random sample of 40 16-ounce cups of coffee made at different times of the day during a one-month period. The mean and standard deviation of the amount of caffeine in the sample were 159.88 mg and 36.72 mg, respectively. A graph of the sample data revealed a right skew with one outlier. The researchers will construct a confidence interval to estimate the amount of caffeine for all 16-ounce cups made at the machine. Which of the following conditions is not needed for the inference?
The graph of the sample data is symmetric with no outliers.
A statistics student wants to compare the mean times needed to access flight information for two major airlines. Twenty randomly selected students accessed one airline's Web site, and the time required to locate the flight information using the Web site had a mean of 2.5 minutes and a standard deviation of 0.8 minute. Twenty different randomly selected students accessed the other airline's Web site, and the time required to locate the flight information using the Web site had a mean of 2.1 minutes and a standard deviation of 1.1 minutes. Assuming that the conditions for inference are met, which of the following statements about the pvalue obtained from the data and the conclusion of the significance test is true?
The p-value is greater than 0.10; therefore, there is no significant difference in mean search times on the two Web sites.
A sociologist will conduct a two-sample -test for a difference in means to investigate whether there is a significant difference, on average, between the salaries of people with bachelor's degrees and people with master's degrees. From a random sample of 32 people with a bachelor's degree, the average salary was $55,000 with standard deviation $3,500. From a random sample of 28 people with a master's degree, the average salary was $58,000 with a standard deviation of $4,000. With a null hypothesis of no difference in the means, which of the following is the test statistic for the appropriate test to investigate whether there is a difference in population means (master's degree minus bachelor's degree) ?
t=(58000-55000)/square root ((4000)^2/28)+((3500)^2/32
A biologist studied the frequency of croaks for frogs from two different regions. From a random sample of 32 frogs located in the northern region, the mean number of croaks per hour was 21.3, and from a random sample of 38 frogs located in the southern region, the mean number of croaks per hour was 28.9. To estimate the difference in the mean number of croaks (southern minus northern), a 95 percent confidence interval was constructed from the samples. The interval was reported as . Which of the following claims is supported by the interval?
The southern frogs are likely to have a greater mean number of croaks per hour than the northern frogs.
