Unit 5: Sampling Distributions
A machine is used to fill bags with a popular brand of trail mix. The machine is calibrated so the distribution of the weights of the bags of trail mix is normal, with mean 240 grams and standard deviation 3 grams. Of the following, which is the least weight of a bag in the top 5 percent of the distribution?
A. 234 grams
B. 240 grams
C. 243 grams
D. 246 grams
E. 248 grams
What is D?
"246 grams"
The manager of a city recreation center wants to estimate the percent of city residents who favor a proposal to build a new dog park. To gather data, the manager will select a random sample of city residents.
Which of the following is the most appropriate interval for the manager to use for such an estimate?
A. A one-sample 𝑧-interval for a sample proportion
B. A one-sample 𝑧-interval for a population proportion
C. A one-sample 𝑧-interval for a difference between population proportions
D. A two-sample 𝑧-interval for a difference between sample proportions
E. A two-sample 𝑧-interval for a difference between population proportions
What is B?
"A one-sample 𝑧-interval for a population proportion."
To study the effectiveness of a certain adult reading program, researchers will select a random sample of adults who are eligible for the program. The selected adults will be given a pretest before beginning the program and a posttest after completing the program. The difference in the number of correct answers on the pretest and the number of correct answers on the posttest will be recorded for each adult in the sample.
Which of the following is the most appropriate inference procedure for the researchers to use to analyze the results?
A. A one-sample 𝑧-interval for a population proportion
B. A one-sample 𝑡-interval for a sample mean difference
C. A matched-pairs 𝑡-interval for a population mean difference
D. A matched-pairs 𝑡-interval for a sample mean difference
E. A two-sample 𝑡-interval for a difference between means
What is C. "A matched-pairs 𝑡-interval for a population mean difference?"
One of the conditions necessary to use a chi-square goodness of fit test to test the distribution of proportions for a categorical variable is that all expected counts should be greater than 5. The reason for the condition is to ensure that...
A. the p-value is accurate
B. the degree of dependency among observations is negligible
C. the frequency of each category is at least
D. the central limit theorem is applicable for the categorical variable\
E. the sampling method is not biased
What is A?
"the p-value is accurate."
A fishery raises sea cucumbers for export and sells them individually. The fishery manager planned to select a random sample of 100 dried sea cucumbers and use a 90% confidence interval to estimate the slope of the regression line to predict the market value (in dollars) from the length of the specimens (in centimeters). However, due to time constraints, the fishery manager will instead select a random sample of 40 dried sea cucumbers. If other factors do not change, which of the following best describes the effect on the confidence interval for the slope due to the manager decreasing the sample size to 40?
A. The width of the confidence interval will decrease.
B. The width of the confidence interval will increase.
C. The confidence level of the confidence interval will decrease.
D. The confidence level of the confidence interval will increase.
E. The standard error for the slope will decrease.
What is B?
"The width of the confidence will increase."
The life span of a battery is the amount of time the battery will last. The distribution of life span for a certain type of battery is approximately normal with mean 2.5 hours and standard deviation 0.25 hour. Suppose one battery will be selected at random. Which of the following is closest to the probability that the selected battery will have a life span of at most 2.1 hours?
A. 0.055
B. 0.110
C. 0.445
D. 0.890
E. 0.945
What is A?
"0.055"
A sample of manufactured items will be selected from a large population in which 8 percent of the items are defective. Of the following, which is the least value for a sample size that will allow for the sampling distribution of the sample proportion to be assumed approximately normal?
A. 11
B. 30
C. 100
D. 125
E. 250
What is D?
"125"
The mean and standard deviation of a random sample of 7 baby orca whales were calculated as 430 pounds and 26.9 pounds, respectively. Assuming all conditions for inference are met, which of the following is a 90 percent confidence interval for the mean weight of all baby orca whales?
A. 26.9±1.895(430/√7)
B. 26.9±1.943(430/√7)
C. 430±1.440(26.9/√7)
D. 430±1.895(26.9/√7)
E. 430±1.943(26.9/√7)
What is E. "430±1.943(26.9/√7)?"
A 𝜒2 goodness-of-fit test was used to test the hypothesis that students at a local university select majors in the same proportions as other universities in the state. A chi-square test statistic of 𝜒2=45.6 was calculated with a corresponding 𝑝-value of 0.005. Which of the following is correct?
A. There is sufficient evidence to conclude that students at the local university do not select majors in the same proportions as do students in the rest of the state.
B. There is sufficient evidence to conclude that students at the local university select majors in the same proportions as do students in the rest of the state.
C. There is insufficient evidence to conclude that students at the local university do not select majors in the same proportions as do students in the rest of the state.
D. There is insufficient evidence to conclude that students at the local university select majors in the same proportions as do students in the rest of the state.
E. Students at the local university select majors in the same proportions as do students in the rest of the state.
What is A?
"There is sufficient evidence to conclude that students at the local university do not select majors in the same proportions as do students in the rest of the state."
A study is conducted to examine the relationship between time spent watching news at night and perceived stress level the following morning using a self-assessment test for stress. A 90% confidence interval for the slope of the regression of stress level and time spent watching news at night is determined to be (-4.32, 3.58). Based on this information, which of the following statements must be true about the point estimate of the confidence interval?
A. The point estimate is 0
B. The point estimate is negative and greater than -4.32
C. The point estimate is negative and less than -4.32
D. The point estimate is positive and greater than 3.58
E. The point estimate is positive and less than 3.58
What is B?
"The point estimate is negative and greater than -4.32."
Recipes for the same type of cookies can vary in terms of ingredients and baking times. From a collection of chocolate chip cookie recipes, a baker randomly selected 5 recipes. From a collection of oatmeal raisin cookie recipes, the baker randomly selected 4 recipes. The mean baking times, in minutes, for each sample were recorded as x̅C and x̅O, respectively.
What is the correct unit of measure for the standard deviation of the sampling distribution of x̅C and x̅O?
A. Minutes
B. Minutes squared
C. Recipes
D. Number of Raisins
E. Number of Chocolate Chips
What is C?
"Recipes"
A major credit card company is interested in the proportion of individuals who use a competitor’s credit card. Their null hypothesis is H0: 𝑝=0.65, and based on a sample they find a sample proportion of 0.70 and a 𝑝-value of 0.053. Is there convincing statistical evidence at the 0.05 level of significance that the true proportion of individuals who use the competitor’s card is actually greater than 0.65?
A. Yes, because the sample proportion 0.70 is greater than the hypothesized proportion 0.65.
B. Yes, because the 𝑝-value 0.053 is greater than the significance level 0.05.
C. No, because the sample proportion 0.70 is greater than the hypothesized proportion 0.65.
D. No, since the sample proportion 0.70 is exactly 0.05 away from the hypothesized proportion 0.65.
E. No, because the 𝑝-value 0.053 is greater than the significance level 0.05.
What is E?
"No, because the 𝑝-value 0.053 is greater than the significance level 0.05"
A recent report indicated that families in a certain country typically spend about $175 per week on groceries. To investigate whether families in a certain city typically spend more than $175 per week, an economist selected a random sample of 500 families in the city and found the sample mean to be $176.24. With all conditions for inference met, a hypothesis test resulted in a 𝑝-value of 0.0021.
At the significance level of 𝛼=0.05, which of the following is a correct conclusion?
A. The 𝑝-value is less than 0.05, and the null hypothesis is rejected. There is convincing statistical evidence that the mean is greater than $175.
B. The 𝑝-value is less than 0.05, and the null hypothesis is not rejected. There is convincing statistical evidence that the mean is greater than $175.
C. The 𝑝-value is less than 0.05, and the null hypothesis is not rejected. There is not convincing statistical evidence that the mean is greater than $175.
D. The 𝑝-value is greater than 0.05, and the null hypothesis is rejected. There is convincing statistical evidence that the mean is greater than $175.
E. The 𝑝-value is greater than 0.05, and the null hypothesis is not rejected. There is not convincing statistical evidence that the mean is greater than $175.
What is A. "The 𝑝-value is less than 0.05, and the null hypothesis is rejected. There is convincing statistical evidence that the mean is greater than $175?"
A city has completed a new initiative to encourage alternate forms of transportation to the airport. Before the initiative, 30 percent of people traveling to the airport were dropped off by family or friends, 30 percent drove to the airport, 20 percent used ride-sharing services, 10 percent took taxis, and 10 percent used public transportation. To determine whether the distribution has changed, the form of transportation was recorded for each person in a random sample of 200 people traveling to the airport. The test statistic for the chi-square goodness-of- fit test was 6.55 with a p-value of 0.161. Assume that the distribution has not changed. If the airport selected another random sample of the same size, which of the following is closest to the probability that the test statistic for a chi-square goodness-of-fit of the new data is less than the test statistic of the previous chi-square goodness-of-fit test?
A. 0.081
B. 0.161
C. 0.322
D. 0.500
E. 0.839
What is E?
"0.839"
A marketing manager believes that linear regression can be used to predict the global sales (in millions of dollars) for a video game from the average amount of time (in hours) for players to complete the game. The manager collected data on 100 recent video games and created the following regression equation:
predicted global sales=0.707+0.534(completion time)
Which of the following is the best interpretation of the slope of the regression line?
A. The global sales increase, on average, by 0.5034 million dollars for each additional 0.707 hours of completion time.
B. The global sales increase, on average, by 0.534 million dollars for each addition hour of completion.
C. The global sales increase, on average, by 1 million dollars for each additional 0.534 hours of completion time.
D. The global sales increase, on average, by 0.707 million dollars for each additional 0.534 hours of completion time.
E. The global sales increase, on average, by 0.707 million dollars for each additional hour of completion time.
What is B?
"The global sales increase, on average, by 0.534 million dollars for each addition hour of completion."
City officials estimate that 46 percent of all city residents are in favor of building a new city park. A random sample of 150 city residents will be selected. Suppose that 51 percent of the sample are in favor of building a new city park. Which of the following is true about the sampling distribution of the sample proportion for samples of size 150?
A. The distribution is not normal, and the mean is 0.46.
B. The distribution is not normal, and the mean is 0.51.
C. The distribution is not normal, and the mean is the average of 0.46 and 0.51.
D. The distribution is approximately normal, and the mean is 0.46.
E. The distribution is approximately normal, and the mean is 0.51.
What is D?
"The distribution is approximately normal, and the mean is 0.46."
A significance test is conducted for which the alternative hypothesis states that more than 83 percent of all adult sea turtles on a certain beach are female. A random sample of adult sea turtles from the beach found that 85 percent of the sea turtles were female. The 𝑝-value for the test is 0.4058. If the null hypothesis is true, which of the following statements is a correct interpretation of the 𝑝-value?
A. Of all possible samples of the same size, 40.58 percent will result in 85 percent of adult sea turtles on the beach being female.
B. Of all possible samples of the same size, 40.58 percent will result in 85 percent or more of adult sea turtles on the beach being female.
C. Of all possible samples of the same size, 40.58 percent will result in 85 percent or less of adult sea turtles on the beach being female.
D. Of all possible samples of the same size, 20.29 percent will result in 85 percent or more of adult sea turtles on the beach being female.
E. Of all possible samples of the same size, 20.29 percent will result in 85 percent of adult sea turtles on the beach being female.
What is B?
"Of all possible samples of the same size, 40.58 percent will result in 85 percent or more of adult sea turtles on the beach being female"
An occupational safety officer for a large company is conducting a study to investigate back problems in office workers who use a computer for most of the workday. The study will investigate the difference in back problems for workers who stand and workers who sit. A group of 68 volunteers have agreed to participate in the nine-month study. Half the group is randomly assigned to work while standing, and the other half is assigned to work while sitting. At the end of the study, the mean number of back problems between the two groups will be calculated. The officer will use the results to estimate the difference in the mean number of back problems between those who work while standing and those who work while sitting.
Which of the following is an appropriate inference procedure for the study?
A. A one-sample 𝑡-interval for a population mean
B. A one-sample 𝑡-interval for a sample mean
C. A matched pairs 𝑡-interval for a mean difference
D. A two-sample 𝑡-interval for a difference between sample means
E. A two-sample 𝑡-interval for a difference between population means
What is E. "A two-sample 𝑡-interval for a difference between population means?"
A polling organization uses random digit dialing of registered voters in a county to gauge the voters’ opinions about a ballot initiative to increase taxes that would pay for a new county park. The polling organization will investigate whether there are differences in how voters respond based on the highest level of education completed. The phone survey asked the voters contacted whether they approve or disapprove of the tax increase. The voters were also asked to indicate their highest level of education completed: high school, bachelor’s degree, or master’s degree.
Which of the following is the appropriate test to investigate whether there is an association between opinion about the ballot initiative and highest level of education completed?
A. A two-sample 𝑡-test for a difference between means
B. A two-sample 𝑧-test for a difference between proportions
C. A chi-square test of homogeneity
D. A chi-square test of independence
E. A chi-square goodness-of-fit test
What is D?
"A chi-square test of independence."
A group of marine biologists investigated the linear relationship between chlorophyll concentration and temperature on the ocean surface. The biologists plan to conduct a test of the slope of the population regression line to determine if the chlorophyll concentration decreases as the temperature increases. If all conditions for inference are met, which of the following is the most appropriate set of hypotheses to evaluate the biologists' claim?
A. Ho: The slope of the population regression line is less than 0.
Ha: The slope of the population regression line is equal to 0.
B. Ho: The slope of the population regression line is greater than 0.
Ha: The slope of the population regression line is equal to 0.
C. Ho: The slope of the population regression line is less than 0.
Ha: The slope of the population regression line is greater than 0.
D. Ho: The slope of the population regression line is equal to 0.
Ha: The slope of the population regression line is less than 0.
E. Ho: The slope of the population regression line is equal to 0.
Ha: The slope of the population regression line is greater than 0.
What is D?
"Ho: The slope of the population regression line is equal to 0.
Ha: The slope of the population regression line is less than 0."
A manufacturer of computer monitors estimates that 4 percent of all the monitors manufactured have a screen defect. Let 𝑝𝑑 represent the population proportion of all monitors manufactured that have a screen defect. For the sampling distribution of the sample proportion for samples of size 100, 𝜇𝑃^𝑑=0.04. Which of the following is the best interpretation of 𝜇𝑃^𝑑=0.04?
A. For all samples of size 100, the mean of all possible sample proportions of monitors manufactured that have a screen defect is 0.04.
B. For all samples of size 100, the standard deviation of all possible sample proportions of monitors manufactured that have a screen defect is 0.04.
C. For a randomly selected screen monitor, the probability that the selected monitor will have a screen defect is 0.04.
D. For each sample of size 100, the proportion of monitors manufactured that will have a screen defect is 0.04.
E. For a randomly selected screen monitor, the mean number of defects for the monitor will be equal to 4.
What is A?
"For all samples of size 100, the mean of all possible sample proportions of monitors manufactured that have a screen defect is 0.04."
At many college bookstores, students can decide whether to purchase or to rent a textbook for a class. A study was conducted to investigate whether the percent of rented textbooks for all science classes in the state was greater than the percent of rented textbooks for all literature classes in the state. The following hypothesis test was done at the significance level of 𝛼=0.05.
H0: 𝑝S=𝑝L
Ha: 𝑝S>𝑝L
In the hypotheses, 𝑝S represents the proportion of all science textbooks that are rented, and 𝑝L represents the proportion of all literature textbooks that are rented.
All conditions for inference were met, and the resulting 𝑝-value was 0.035. Which of the following is the correct decision for the test?
A. The 𝑝-value is less than 𝛼. Since 0.035<0.05, the null hypothesis is rejected, and the claim is supported. There is convincing statistical evidence that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.
B. The 𝑝-value is less than 𝛼, and the null hypothesis is rejected. There is not convincing evidence to support the claim that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.
C. The 𝑝-value is less than 𝛼, and the null hypothesis is not rejected. There is not convincing evidence to support the claim that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.
D. The 𝑝-value is greater than 𝛼, and the null hypothesis is rejected. There is convincing evidence to support the claim that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.
E. The 𝑝-value is greater than 𝛼, and the null hypothesis is not rejected. There is not convincing evidence to support the claim that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.
What is A?
"The 𝑝-value is less than 𝛼. Since 0.035<0.05, the null hypothesis is rejected, and the claim is supported. There is convincing statistical evidence that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented"
Researchers studying two populations of wolves conducted a two-sample 𝑡-test for the difference in means to investigate whether the mean weight of the wolves in one population was different from the mean weight of the wolves in the other population. All conditions for inference were met, and the test produced a test statistic of 𝑡=2.771 and a 𝑝-value of 0.01.
Which of the following is a correct interpretation of the 𝑝-value?
A. Assuming that the mean weights of wolves in the populations are equal, the probability of obtaining a test statistic that is greater than 2.771 or less than −2.771 is 0.01.
B. Assuming that the mean weights of wolves in the populations are equal, the probability of obtaining a test statistic that is greater than 2.771 is 0.01.
C. Assuming that the mean weights of wolves in the populations are different, the probability of obtaining a test statistic that is greater than 2.771 or less than −2.771 is 0.01.
D. Assuming that the mean weights of wolves in the populations are different, the probability of obtaining a test statistic that is greater than 2.771 is 0.01.
E. Assuming that the mean weights of wolves in the populations are different, the probability of obtaining a test statistic that is less than 2.771 is 0.01.
What is A. "Assuming that the mean weights of wolves in the populations are equal, the probability of obtaining a test statistic that is greater than 2.771 or less than −2.771 is 0.01?"
A sports medicine clinic will conduct a study to investigate whether a swimming stroke specialty is related to the occurrence of shoulder impingement syndrome. Data will be collected from a random sample of 75 competitive swimmers. Each swimmer will be categorized based on stroke specialty (freestyle, backstroke, breaststroke, butterfly) and whether they have ever experienced shoulder impingement. Which of the following are appropriate hypotheses for the study?
A. Ho: The distribution of swimming stroke specialties is uniform in the population of swimmers.
Ha: The distribution of swimming stroke specialties is not uniform in the population of swimmers.
B. Ho: The mean number of shoulder impingement episodes is the same across all swimming stroke specialties.
Ha: The mean number of shoulder impingement episodes is not the same across all swimming stroke specialties.
C. Ho: The population proportion of shoulder impingement is equal across all swimming stroke specialties
Ha: The population proportion of shoulder impingement is not equal across all swimming stroke specialties
D. Ho: There is no linear relationship between swimming stroke specialty and shoulder impingement
Ha: There is a linear relationship between swimming stroke specialty and shoulder impingement
E. Ho: There is no association between swimming stroke specialty and shoulder impingement
Ha: There is an association between swimming stroke specialty and shoulder impingement
What is E?
"Ho: There is no association between swimming stroke specialty and shoulder impingement
Ha: There is an association between swimming stroke specialty and shoulder impingement"
Sheronna, a licensed dietician, wants to demonstrate the relationship between the number of tortilla chips eaten and the total amount of sodium intake (milligrams) in a single day. She selected a random sample of 35 adults at a particular restaurant and recorded the number of tortilla chips eaten and the total amount of sodium intake at the end of the day. A t-test for the slope of a regression line test was conducted for Ho: β=0 versus Ha: β>0, where β represents the slope of the population regression line to predict milligrams of sodium intake at the end of the day based on the number of tortilla chips eaten at the restaurant. A test statistic of 3.55 and a p-value of 0.006 were calculated. Assuming all conditions for the inference are met, which of the following can be conducted about the test at a significance level of 𝛼=0.01?
A. The p-value is less than 0.01, and Ho should be rejected.
B. The p-value is less than 0.01, and Ho should not be rejected.
C. The test statistic is greater than 0.01, and Ho should be rejected.
D. The test statistic is greater than 0.01, and Ho should not be rejected.
E. The test statistic is greater than the p-value, and Ho should be rejected.
What is A?
"The p-value is less than 0.01, and Ho should be rejected."