Statistics Cumulative Exam 2 86%
The distribution of the number of words in text messages between employees at a large company is skewed right with a mean of 8.6 words and a standard deviation of 4.3 words. If a random sample of 39 messages is selected, what is the probability the sample mean is more than 10 words?
0.0210
Lucy recently asked the servers at her restaurant to only give straws to customers who request them. She thinks that about half of the customers will ask for straws but hopes that the rate will be less than half. She randomly selects 100 customers and finds that 43 of them ask for a straw. To determine if these data provide convincing evidence that the proportion of customers who will ask for a straw is less than 50%, 150 trials of a simulation are conducted. Lucy is testing the hypotheses: H0: p = 50% and Ha: p < 50%, where p = the true proportion of customers who will ask for a straw. Based on the results of the simulation, what is the estimate of the P-value of the test?
0.0733
A nontraditional deck of cards has 30 total cards: 5 hearts, 10 clubs, 8 spades, and 7 diamonds. The cards are shuffled, and the top card is noted. This process is repeated 100 times. What is the probability the top card is spades in more than 30% of the sample?
0.224
The probability that a mature hen will lay an egg on a given day is 0.80. Hannah has 12 hens. What is the probability that 10 of the 12 hens will lay eggs on a given day?
0.28
A snack company makes packages of grapes and honeydew slices. The weight of an individual package of grapes, G, is approximately Normally distributed with a mean of 3.25 ounces and a standard deviation of 0.91 ounces. The weight of an individual package of honeydew slices, H, is approximately Normally distributed with a mean of 4.51 ounces and a standard deviation of 2.02 ounces. Assume G and H are independent random variables. Let D = G - H. What is the probability that a randomly selected package of grapes weighs more than a randomly selected package of honeydew slices?
0.284
A local charity holds a carnival to raise money. In one activity, participants make a $3 donation for a chance to spin a wheel that has 10 spaces with the values, 0, 1, 2, 5, and 10. Whatever space it lands on, the participant wins that value. Let X represent the value of a random spin. The distribution is given in the table. What is the probability that the value is 0?
0.4
Two students are throwing water balloons at a target. Accuracy is measured as how close the balloon is from the center of the target. Tory's distances from the center of the target are approximately Normally distributed with a mean of 123 mm and a standard deviation of 31 mm. Adam's distances from the center is approximately Normally distributed with a mean of 108 mm and a standard deviation of 47 mm. If 5 attempts for Tory and 3 attempts for Adam are randomly selected, what is the probability that the mean distance from the center of the target for Tory is more than for Adam?
0.6887
At a carnival game, the chance of winning a prize is 0.45. Kylee plays the game 3 times. Using the table, what is the probability that she wins at least 1 prize?
0.83
A local charity holds a carnival to raise money. In one activity, participants make a $3 donation for a chance to spin a wheel that has 10 spaces marked with the values 0, 1, 2, 5, and 10. The participant wins the dollar amount marked on the space on which the wheel stops. Let X represent the value of a spin. The distribution of X is given in the table.
1
Based upon historical data, it is known that 8% of 12-egg cartons contain at least one broken egg. A grocery store manager would like to carry out a simulation to estimate the number of cartons, in a sample of 10, that would contain at least one broken egg. She assigns the digits to the outcomes. 01-08 = carton contains a broken egg 09-99, 00 = carton does not contain a broken egg Here is a portion of a random number table. In the first trial, line 1, 1 of the first 10 double-digit numbers is between 01 and 08, meaning that 1 of the 10 cartons of eggs contains at least one broken egg. Starting at line 2 and using a new line for each trial, carry out 4 more trials. Based on the 5 trials, how many cartons of eggs out of 10 cartons are expected to contain at least one broken egg, on average?
1
The final exam grade distribution for all students in the introductory statistics class at a local community college is displayed in the table, with A = 4, B = 3, C = 2, D = 1, and F = 0. Let X represent the grade for a randomly selected student from the class. What is the standard deviation of the distribution?
1.08
A shipping company claims that 95% of packages are delivered on time. A student wants to conduct a simulation to estimate the number of packages that would need to be randomly selected to find a package that was not delivered on time. The student assigns the digits to the outcomes. 00-04 = package not delivered on time 05-99 = package delivered on time Here is a portion of a random number table. Beginning at line 1, and starting each new trial right after the previous trial, carry out 5 trials of this simulation. Based on the 5 trials, what is the average number of packages that need to be selected in order to find a package that was not delivered on time?
2.25
A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is 11%. What is the mean of X, the number of patients seen until an injection-site reaction occurs?
9.0909
An avid golfer goes to the driving range to practice with her favorite golf club. In a random sample of 33 swings, she hits the ball, on average, 185 yards with a standard deviation of 16.7 yards. Which of the following correctly interprets the 99% confidence interval for the true mean number of yards she hits the ball with this club?
99% of all balls she hits with this club will lie between the values in the interval.
A teacher has a large container of blue, red, and green beads. She reports to the students that the proportion of blue beads in the container is 0.30. The students feel the proportion of blue beads is lower than 0.30. A student randomly selects 60 beads and finds that 12 of the beads are blue. The P-value for the test of the hypotheses, H0:p=0.30 and Ha:p<0.30, is 0.045. What is the correct interpretation of this value?
Assuming the true proportion of blue beads in the container is 0.30, there is a 4.5% probability that the null hypothesis is true.
In a statistics activity, students are asked to spin a penny and a dime and determine the proportion of times that each lands with tails up. The students believe that since a dime is lighter, it will have a lower proportion of times landing tails up compared to the penny. The students are instructed to spin the penny and the dime 30 times and record the number of times each lands tails up. For one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times. Let pD = the true proportion of times a dime will land tails up and pP = the true proportion of times a penny will land tails up. Which of the following is the correct standardized test statistic and P-value for the hypotheses, ?
B
At a local college, an admissions officer surveys the incoming class of 1,000 first-year students concerning their preference of major. The officer randomly selects 100 of them and asks if they intend to major in liberal arts. Of the 100 first-year students, 62 state they intend on majoring in liberal arts. Assuming the conditions for inference have been met, what is the 95% confidence interval for the true proportion of first-year students who intend on majoring in liberal arts?
B.
The proportion of all US adults who eat popcorn when they go to the movie theater is p = 0.87. A random sample of 20 US adults was selected and asked if they eat popcorn when they go to the movie theater. Which of the following is the shape of the sampling distribution of ?
Because , the sampling distribution of is not approximately Normal. Because p = 0.87 is closer to 1 than 0, the sampling distribution of is skewed to the right.
A carnival game is designed so that approximately 10% of players will win a large prize. If there is evidence that the percentage differs significantly from this target, then adjustments will be made to the game. To investigate, a random sample of 100 players is selected from the large population of all players. Of these players, 19 win a large prize. The question of interest is whether the data provide convincing evidence that the true proportion of players who win this game differs from 0.10. The computer output gives the results of a z-test for one proportion. Test and CI for One Proportion Test of p = 0.1 vs p ≠ 0.1 Sample1X19N100Sample p0.1995% CI(0.113, 0.267)Z-Value3.00P-value0.0027 What conclusion should be made at the a = 0.05 level?
Because the P-value < a = 0.05, there is convincing evidence that the true proportion of players who win this game differs from 0.10.
A student believes that a certain 6-sided number cube, with the numbers 1 to 6, is unfair and is more likely to land with a 6 facing up. The student rolls the cube 100 times and lands with 6 facing up 20 times. The P-value for the test of the hypotheses, H0:p=0.17 and Ha:p>0.17, is 0.19. What is the correct conclusion given a=0.05?
Because the P-value is greater than a=0.05, the student should fail to reject H0.
An avid traveler is investigating whether travel websites differ in their pricing. She chooses a random sample of 32 hotel rooms from across the state and notes the price of each room from site A and site B. The mean difference (site A - site B) in the prices for the rooms is $5.49 with a standard deviation of $18.65. Assuming the conditions for inference have been met, is there evidence that the prices of hotel rooms from site A and site B are different, on average? Use a significance level of a = 0.05.
Because the P-value is greater than α, there is not sufficient evidence that the prices of hotel rooms from site A and site B are different, on average.
A school guidance counselor is concerned that a greater proportion of high school students are working part-time jobs during the school year than a decade ago. A decade ago, 28% of high school students worked a part-time job during the school year. To investigate whether the proportion is greater today, a random sample of 80 high school students is selected. It is discovered that 37.5% of them work part-time jobs during the school year. The guidance counselor would like to know if the data provide convincing evidence that the true proportion of all high school students who work a part-time job during the school year is greater than 0.28. The guidance counselor tests the hypotheses H0: p = 0.28 versus Ha: p > 0.28, where p = the true proportion of all high school students who work a part-time job during the school year. The conditions for inference are met.
Because the P-value is less than α = 0.05, there is convincing evidence that the true proportion of all high school students who work a part-time job during the school year is greater than 0.28.
A conference consists of 5 sessions: A, B, C, D, and E. Here are the costs of the sessions. Session A: $50Session B: $50Session C: $100Session D: $150Session E: $200 A participant plans to attend 3 sessions. Here is a list of all possible samples of size 3 sessions from this population of 5 sessions: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, and CDE. Which of the following gives the sampling distribution of the sample minimum session price for samples of size 3, selected from the population of these 5 sessions without replacement?
D
A major car dealership has several stores in a big city. The owner wants to determine if there is a difference in the proportion of SUVs that are sold at stores A and B. The owner gathers the sales records for each store from the past year. A random sample of 55 receipts from store A shows that 30 of the sales were for SUVs. Another random sample of 60 receipts from store B shows that 42 of the sales were for SUVs. Assuming conditions for inference are met, what is the 99% confidence interval for the difference in proportions of sales that are SUVs?
D
A study is done to estimate the true mean satisfaction rating for all customers of a particular retail store. A random sample of 200 customers is selected and a 99% confidence interval for the true mean satisfaction rating is 7.8 to 8.4 where 1 represents very dissatisfied and 10 represents completely satisfied. Based upon this interval, what conclusion should be made about the hypotheses: = 8 versus 8 where μ = the true mean satisfaction rating for all customers of this store at α = 0.01?
Fail to reject H0. There is not convincing evidence that the true mean satisfaction rating of all customers in this store differs from 8.
A company executive claims that employees in his industry get 100 junk emails per day. To further investigate this claim, the tech department of the company conducts a study. The executive selects a random sample of 10 employees and records the number of junk emails they received that day. Here are the data: 125, 101, 109, 94, 122, 92, 119, 90, 118, 122. The tech department would like to determine if the data provide convincing evidence that the true mean number of junk emails received this day by employees of this company differs from 100. What are the appropriate hypotheses?
H0: μ = 100 versus Ha: μ ≠ 100, where μ = the true mean number of junk emails received this day by employees of this company
A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is 11%. A nurse will collect data by performing injections with this type of needle until five people experience injection-site reactions. Is it appropriate to use the geometric distribution to calculate probabilities in this situation?
No, because it is not looking for the first occurrence of success.
In a statistics activity, students are asked to determine the proportion of times that a spinning penny will land with tails up. The students are instructed to spin the penny 10 times and record the number of times the penny lands tails up. For one student, it lands tails side up six times. The student will construct a 90% confidence interval for the true proportion of tails up. Are the conditions for inference met?
No, the Large Counts Condition is not met.
Some stores print multiple coupons and advertisements on their receipts, making the receipts unusually long. A curious shopper makes the same purchase at a random sample of 10 stores and measures the length of each receipt (in inches). Here are the results: 8.25, 7.5, 5, 9.5, 12, 5.5, 9, 14, 11.5, 18 Are the conditions for constructing a t confidence interval met?
No, the Normal/large sample condition is not met.
A random sample of 30 professional tennis players is selected. For each tennis player, their first serve percentage and their second serve percentage are recorded. A 95% confidence interval for the mean difference (first - second) in serve percentages is found to be 0.10 to 0.18. A tennis coach claims that there is no difference in first and second serve percentages among professional tennis players. Is this claim supported by the confidence interval?
No, the confidence interval does not contain the value of 0.
A researcher is 95% confident that the interval from 19.7 posts to 27.3 posts captures the true mean amount of posts high school students make daily on social media. Is there evidence that the true mean number of posts high school students make is less than 27?
No. There is not evidence for the population mean to be less than 27, because there are values greater than 27 within the 95% confidence interval.
In early 2019, the US rate of recycling plastic water bottles was only 23%. A government agency designs an expensive program to increase the recycling rate. The program will be tested in Texas and, if successful, it will be used nationally. A hypothesis test is conducted with H0: The proportion of water bottles that are recycled is still 23% after the program, and Ha: The proportion of water bottles that are recycled is more than 23% after the program. What is a Type II error and its consequence in this context?
The agency concludes that the program does not increase the recycling rate, when in fact it does increase the rate. The agency will not implement a program that could have increased the water bottle recycling rate.
One professional basketball player typically attempts eight free throws per game. Let X represent the number of free throws made out of eight. The distribution for X is shown in the table. Which of the following is the correct interpretation of P(X ≤ 3)?
The probability of the number of free throws made out of 8 is at most 3.
A statistics class weighed 20 bags of grapes purchased from the store. The bags are advertised to contain 16 ounces, on average. The class calculated the 90% confidence interval for the true mean weight of bags of grapes from this store to be (15.875, 16.595) ounces. What is the sample mean weight of grapes, and what is the margin of error?
The sample mean weight is 16.235 ounces, and the margin of error is 0.360 ounces.
To estimate the benefits of an SAT prep course, a random sample of 10 students enrolled in the course is selected. For each of these students, their entrance score on the exam taken at the beginning of the course is recorded. Their exit score on the exam they take at the end of the course is recorded as well. The table displays the scores. The mean of the differences is 193 points, and the standard deviation of the differences is 62.73 points. The conditions for inference are met. A 98% confidence interval for the mean difference (after - before) in score is 137.04 points to 248.96 points. What is the correct interpretation of this interval?
The school can be 98% confident that students who take this course increase their SAT score, on average, by 137.04 to 248.96 points.
A large home improvement store is considering expanding its selection of moving products, such as cardboard boxes and packing tape. The store constructs a 90% confidence interval to estimate the proportion of all customers who have moved houses at least once in the last five years. In the random sample of 620 customers, 201 (32.4%) replied that they had moved at least once in the last five years. The sample yielded the 90% confidence interval (0.293, 0.355) for the proportion of all customers who have moved in the past five years. What is the correct interpretation of this confidence interval?
The store can be 90% confident that the interval from 0.293 to 0.355 captures the proportion of all customers of this store who have moved in the past five years.
In a statistics activity, students are asked to spin a penny and a dime and determine the proportion of times that each lands with tails up. The students believe that since a dime is lighter, it will have a lower proportion of times landing tails up compared with the penny. The students are instructed to spin the penny and the dime 30 times and record the number of times each lands tails up. For one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times. Let pD = the true proportion of times a dime will land tails up and pP = the true proportion of times a penny will land tails up. The P-value for this significance test is 0.296. Which of the following is the correct conclusion for this test of the hypotheses at the level?
The student should fail to reject the null hypothesis since 0.296 > 0.05. There is insufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.
From previous experience, the owner of an apple orchard knows that the mean weight of Gala apples is 140 grams. There has been more precipitation than usual this year, and the owner believes the weights of the apples will be heavier than usual. The owner takes a random sample of 30 apples and records their weights. The mean weight of the sample is 144 grams with a standard deviation of 13.2 grams. A significance test at an alpha level of produces a P-value of 0.054. What is the correct interpretation of the P-value?
There is a 5.4% chance that the null hypothesis is incorrect if the true mean weight is 140 grams.
A laundry detergent company wants to determine if a new formula of detergent, A, cleans better than the original formula, B. Researchers randomly assign 500 pieces of similarly soiled clothes to the two detergents, putting 250 pieces in each group. After washing the clothes, independent reviewers determine the cleanliness of the clothes on a scale of 1-10, with 10 being the cleanest. The researchers calculate the proportion of clothes in each group that receive a rating of 7 or higher. For detergent A, 228 pieces of clothing received a 7 or higher. For detergent B, 210 pieces of clothing received a rating of 7 or higher. Based on the 90% confidence interval, (0.02, 0.12), is there convincing evidence that the new formula of laundry detergent is better?
There is convincing evidence because the interval is entirely above 0.
A ski resort claims that there is a 75% chance of snow on any given day in January and that snowfall happens independently from one day to the next. A family plans a four-day trip to this ski resort in January. Let X represent the number of days it snows while the family is there. What are the mean and standard deviation of X?
UX=3,Ox=0.87
An athletic trainer would like to estimate how many additional calories are burned when completing a high-intensity interval training (HIIT) workout for 30 minutes rather than doing yoga for 30 minutes. A group of 30 volunteers is randomly assigned to take a 30-minute HIIT class or a 30-minute yoga class, and each volunteer wears a calorie-counting armband. Here are summary statistics and dotplots of the results. Are the conditions for inference met?
Yes, all three conditions for inference are met.
A manufacturing manager reports to the CEO that 5% of products that come off the assembly line are defective. A study is performed to investigate this claim. In a random sample of 250 products that come off the assembly line, 14 are found to be defective. The CEO wants to know if the data provide convincing evidence that the true proportion of defective products differs from 0.05. Are the conditions for inference met?
Yes, the conditions for inference are met.
The times to pop a 3.4-ounce bag of microwave popcorn without burning it are Normally distributed with a mean time of 140 seconds and a standard deviation of 20 seconds. A random sample of four bags is selected and the mean time to pop the bags is recorded. Which of the following describes the sampling distribution of all possible samples of size four?
approximately Normal with a mean of 140 seconds and a standard deviation of 10 seconds
A biologist wants to determine if different temperatures (15oC, 25oC, or 35oC) and amounts of sunlight (partial or full) will affect the growth of plants. He will test each combination of temperature and sunlight by randomly assigning 15 plants to each of the combinations. What type of sampling is described in this study?
more than two samples
A shoe company wants to determine if the new tread on its top line of running shoes lasts longer than the original tread. The company recruits 50 runners for a study. Each runner will perform their typical workout wearing one shoe with the original tread on one foot and another shoe with the new tread on the other foot. The foot that wears the new type of tread will be decided by flipping a coin. After one month, the runner will wear the new type of tread on the opposite foot. At the end of the second month, the difference in tread wear (New - Original) will be calculated. The company will then determine whether the new tread lasts longer than the original tread. What is the appropriate inference procedure?
one-sample t-test for UDIFF
A student would like to estimate the proportion of students at his school who have lunch during period 4. To do so, he selects a random sample of 40 students and finds that 30% of them have period 4 lunch. Later, he goes to the guidance office and finds that of the 380 students at this school, 25% of them have period 4 lunch. Which of the following properly describes the number, 25%?
sample
An English test has 50 multiple-choice questions, each with five answer choices and one correct answer. A student guesses each answer choice randomly. Let X represent the number of questions answered correctly. What is the shape of the probability histogram of X?
skewed right
What critical value of t* should be used for a 95% confidence interval for the population mean based on a random sample of 21 observations?
t* = 2.086
A doctor claims that the mean number of hours of sleep that seniors in high school get per night differs from the mean number of hours of sleep college seniors get per night. To investigate, he selects a random sample of 50 high school seniors from all high schools in his county. He also selects a random sample of 50 seniors from the colleges in his county. He constructs a 95% confidence interval for the true mean difference in the number of hours of sleep for seniors in high school and seniors in college. The resulting interval is (0.57, 1.25). Based upon the interval, can the doctor conclude that mean number of hours of sleep that seniors in high school get per night differs from the mean number of hours of sleep college seniors get per night?
yes because 0 is not in the confidence interval
A school administrator claims that 85% of the students at his large school plan to attend college after graduation. The statistics teacher at this school selects a random sample of 50 students from this school and finds that 76% of them plan to attend college after graduation. The administrator would like to know if the data provide convincing evidence that the true proportion of all students from this school who plan to attend college after graduation is less than 85%. What are the values of the test statistic and P-value for this test?
z = -1.78, P-value = 0. 0375
