Stats Review

A fair six-sided die, with sides numbered 1 through 6, will be rolled a total of 15 times. Let x¯1 represent the average of the first ten rolls, and let x¯2represent the average of the remaining five rolls. What is the mean μ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2 ?

A certain statistic dˆ is being used to estimate a population parameter D. The expected value of dˆ is not equal to D. What property does dˆexhibit?

At a large university, the division of computing services surveyed a random sample of 45 biology majors and 55 business majors from populations of over 1,000 biology and 1,000 business majors. The sampled students were asked how many hours they spend per week using any university computer lab. Let x¯1 represent the average hours per week spent in any university computer lab by the 45 biology majors, and let x¯2 represent the average hours per week spent in any university computer lab by the 55 business majors.

Suppose the variance in trunk diameter of the giant sequoia tree species is 15.7m2, while the variance in trunk diameter of the California redwood tree species is 10.6m2. Let x¯1 represent the average trunk diameter of four randomly sampled giant sequoia trees, and let x¯2represent the average trunk diameter of three randomly sampled California redwood trees. If the random sampling is done with replacement, what is the standard deviation σ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2 ?

Two non-profit organizations, L and M, accept donations from people. In a certain month, 140 people donated to organization L, with an average donation amount of x¯L=$113, and 42 people donated to organization M, with an average donation amount of x¯M=$390.

Samples G and H were selected from the same population of quantitative data and the mean of each sample was determined. The mean of sample G is equal to the mean of the population.

Suppose that 25 percent of women and 22 percent of men would answer yes to a particular question. In a simulation, a random sample of 100 women and a random sample of 100 men were selected, and the difference in sample proportions of those who answered yes, p̂women - p̂men, was calculated. The process was repeated 1,000 times. Which of the following is most likely to be a representation of the simulated sampling distribution of the difference between the two sample proportions?

The continuous random variable N has a normal distribution with mean 7.5 and standard deviation 2.5. For which of the following is the probability equal to 0 ?

Clara recorded 50 numerical observations on a certain variable and then calculated the mean x¯ and the standard deviation s for the observations. To help decide whether a normal model is appropriate, she created the following chart.

The mean number of pets owned by the population of students at a large high school is 3.2 pets per student with a standard deviation of 1.7 pets. A random sample of 16 students will be selected and the mean number of pets for the sample will be calculated.

The distribution of time needed to complete a certain programming task is approximately normal, with mean 47 minutes and standard deviation 6 minutes. Which of the following is closest to the probability that a randomly chosen task will take less than 34 minutes or more than 60 minutes to complete?

Data will be collected on the following variables. Which variable is most likely to be approximated by a normal model?

City R is a large city with 4 million residents, and City S is a smaller city with 0.25 million residents. Researchers believe that the proportion of City S residents who regularly ride bicycles is between 10 percent and 25 percent and the proportion of City R residents who regularly ride bicycles is between 20 percent and 50 percent.

A sample of size n will be selected from a population with population proportion p. Which of the following must be true for the sampling distribution of the sample proportion to be approximately normal?

For which of the following is the shape of the sampling distribution of the sample mean approximately normal? A random sample of size 5 from a population that is approximately normal A random sample of size 10 from a population that is strongly skewed to the right A random sample of size 60 from a population that is strongly skewed to the left

Approximately 52 percent of all recent births were boys. In a simple random sample of 100 recent births, 49 were boys and 51 were girls. The most likely explanation for the difference between the observed results and the expected results in this case is

A certain skin cream is 80 percent effective in curing a common rash. A random sample of 100 people with the rash will use the cream. Which of the following is the best description of the shape of the sampling distribution of the sample proportion of those who will be cured?

Two different drugs, X and Y, are currently in use to treat a certain condition. About 7 percent of the people using either drug experience side effects. A random sample of 75 people using drug X and a random sample of 150 people using drug Y are selected. The proportion of people in each sample who experience side effects is recorded.

For a certain population of sea turtles, 18 percent are longer than 6.5 feet. A random sample of 90 sea turtles will be selected. What is the standard deviation of the sampling distribution of the sample proportion of sea turtles longer than 6.5 feet for samples of size 90 ?

A certain statistic will be used as an unbiased estimator of a parameter. Let J represent the sampling distribution of the estimator for samples of size 40, and let K represent the sampling distribution of the estimator for samples of size 100.

A local arts council has 200 members. The council president wanted to estimate the percent of its members who have had experience in writing grants. The president randomly selected 30 members and surveyed the selected members on their grant-writing experience. Of the 30 selected members, 12 indicated that they did have the experience. Have the conditions for inference with a one-sample z-interval been met? A Yes, all conditions for inference have been met. B No, because the sample size is not large enough to satisfy the conditions for normality. C No, because the sample was not selected at random. D No, because the sample size is not less than 10 percent of the population size. E No, because the sample is not representative of the population.

From a random sample of potential voters in an upcoming election, 47% indicated they intended to vote for Candidate R. A 95 percent confidence interval was constructed from the sample, and the margin of error for the estimate was 5%. Which of the following is the best interpretation of the interval? A We are 95% confident that the proportion who intend to vote for Candidate R from the random sample is between 42% and 52%. B We are 95% confident that the proportion who intend to vote for Candidate R from the population is between 42% and 52%. C We are 95% confident that the proportion who intend to vote for Candidate R from the random sample is 47%. D We are 95% confident that the proportion who intend to vote for Candidate R from the population is 47%. E We are confident that 95% of the population intend to vote for Candidate R.

A random sample of residents in city J were surveyed about whether they supported raising taxes to increase bus service for the city. From the results, a 95 percent confidence interval was constructed to estimate the proportion of people in the city who support the increase. The interval was (0.46,0.52). Based on the confidence interval, which of the following claims is supported? A More than 90 percent of the residents support the increase. B More than 60 percent of the residents support the increase. C More than 40 percent of the residents support the increase. D Fewer than 10 percent of the residents support the increase. E Fewer than 25 percent of the residents support the increase.

A survey was conducted to determine what percentage of college seniors would have chosen to attend a different college if they had known then what they know now. In a random sample of 100 seniors, 34 percent indicated that they would have attended a different college. A 90 percent confidence interval for the percentage of all seniors who would have attended a different college is A 24.7% to 43.3% B 25.8% to 42.2% C 26.2% to 41.8% D 30.6% to 37.4% E 31.2% to 36.8%

A city planner wants to estimate the proportion of city residents who commute to work by subway each day. A random sample of 30 city residents was selected, and 28 of those selected indicated that they rode the subway to work. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal? A No, because the size of the population is not known. B No, because the sample is not large enough to satisfy the normality conditions. C Yes, because the sample is large enough to satisfy the normality conditions. D Yes, because the sample was selected at random. E Yes, because sampling distributions of proportions are modeled with a normal model.

The management team of a company with 10,000 employees is considering installing charging stations for electric cars in the company parking lots. In a random sample of 500 employees, 15 reported owning an electric car. Which of the following is a 99 percent confidence interval for the proportion of all employees at the company who own an electric car? A B C D E

In 2009 a survey of Internet usage found that 79 percent of adults age 18 years and older in the United States use the Internet. A broadband company believes that the percent is greater now than it was in 2009 and will conduct a survey. The company plans to construct a 98 percent confidence interval to estimate the current percent and wants the margin of error to be no more than 2.5 percentage points. Assuming that at least 79 percent of adults use the Internet, which of the following should be used to find the sample size (n) needed? A 1.96(0.5)n−−−−√≤0.025 B 1.96(0.5)(0.5)n−−−−−−√≤0.025 C 2.33(0.5)(0.5)n−−−−−−√≤0.05 D 2.33(0.79)(0.21)n−−−−−−−−√≤0.025 E 2.33(0.79)(0.21)n−−−−−−−−√≤0.05

Jessica wanted to determine if the proportion of males for a certain species of laboratory animal is less than 0.5. She was given access to appropriate records that contained information on 12,000 live births for the species. To construct a 95 percent confidence interval, she selected a simple random sample of 100 births from the records and found that 31 births were male. Based on the study, which of the following expressions is an approximate 95 percent confidence interval estimate for p, the proportion of males in the 12,000 live births? A B C D E

A random sample of 300 students is selected from a large group of students who use a computer-equipped classroom on a regular basis. Occasionally, students leave their USB drive in a computer. Of the 300 students questioned, 180 said that they write their name on their USB drive. Which of the following is a 98 percent confidence interval for the proportion of all students using the classroom who write their name on their USB drive? A 0.4±2.33(0.4)(0.6)300−−−−−−√ B 0.4±1.96(0.4)(0.6)300−−−−−−√ C 0.6±2.33(0.6)(0.4)300−−−−−−√ D 0.6±1.96(0.6)(0.4)300−−−−−−√ E 0.6±2.05(0.6)(0.4)300−−−−−−√

Based on a survey of a random sample of 900 adults in the United States, a journalist reports that 60 percent of adults in the United States are in favor of increasing the minimum hourly wage. If the reported percent has a margin of error of 2.7 percentage points, which of the following is closest to the level of confidence? A 80.0% B 90.0% C 95.0% D 95.5% E 99.0%

Elly and Drew work together to collect data to estimate the percentage of their classmates who own a particular brand of shoe. Using the same data, Elly will construct a 90 percent confidence interval and Drew will construct a 99 percent confidence interval. Which of the following statements is true? A The midpoint of Elly's interval will be greater than the midpoint of Drew's interval. B The midpoint of Elly's interval will be less than the midpoint of Drew's interval. C The width of Elly's interval will be greater than the width of Drew's interval. D The width of Elly's interval will be less than the width of Drew's interval. E The width of Elly's interval will be equal to the width of Drew's interval.

On the day before an election in a large city, each person in a random sample of 1,000 likely voters is asked which candidate he or she plans to vote for. Of the people in the sample, 55 percent say they will vote for candidate Taylor. A margin of error of 3 percentage points is calculated. Which of the following statements is appropriate? A The proportion of all likely voters who plan to vote for candidate Taylor must be the same as the proportion of voters in the sample who plan to vote for candidate Taylor (55 percent), because the data were collected from a random sample. B The sample proportion minus the margin of error is greater than 0.50, which provides evidence that more than half of all likely voters plan to vote for candidate Taylor. C It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because the 1,000 likely voters in the sample represent only a small fraction of all likely voters in a large city. D It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because this is not an experiment. E It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because this is a random sample and not a census.

A marketing company wants to estimate the proportion of consumers in a certain region of the country who would react favorably to a new marketing campaign. Further, the company wants the estimate to have a margin of error of no more than 5 percent with 90 percent confidence. Of the following, which is closest to the minimum number of consumers needed to obtain the estimate with the desired precision? A 136 B 271 C 385 D 542 E 769

The manager of a magazine wants to estimate the percent of magazine subscribers who approve of a new cover format. To gather data, the manager will select a random sample of subscribers. Which of the following is the most appropriate interval for the manager to use for such an estimate? A A two-sample z-interval for a difference between sample proportions B A two-sample z-interval for a difference between population proportions C A one-sample z-interval for a sample proportion D A one-sample z-interval for a population proportion E A one-sample z-interval for a difference between population proportions

A factory manager selected a random sample of parts produced on an old assembly line and a random sample of parts produced on a new assembly line. The difference between the sample proportion of defective parts made on the old assembly line and the sample proportion of defective parts made on the new assembly line (old minus new) was 0.006. Under the assumption that all conditions for inference were met, a hypothesis test was conducted with the alternative hypothesis being the proportion of defective parts made on the old assembly line is greater than that of the new assembly line. The p-value of the test was 0.018. Which of the following is the correct interpretation of the p-value? A If there is a difference of 0.018 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.006. B If there is a difference of 0.006 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.018. C If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference equal to 0.006 is 0.018. D If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at least 0.006 is 0.018. E If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at most 0.006 is 0.018.

A company that ships glass for a glass manufacturer claimed that its shipping boxes are constructed so that no more than 8 percent of the boxes arrive with broken glass. The glass manufacturer believed the actual percent is greater than 8 percent. The manufacturer selected a random sample of boxes and recorded the proportion of boxes that arrived with broken glass. The manufacturer tested the hypotheses H0_p=0.08 versus Ha:p>0.08 at the significance level of α=0.01. The test yielded a p-value of 0.001. Assuming all conditions for inference were met, which of the following is the correct conclusion? A The p-value is greater than α, and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08. B The p-value is greater than α, and the null hypothesis is rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08. C The p-value is greater than α, and the null hypothesis is not rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08. D The p-value is less than α, and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08. E The p-value is less than α, and the null hypothesis is not rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.

A representative of a car manufacturer in the United States made the following claim in a news report. Ten years ago, only 53 percent of Americans owned American-made cars, but that figure is significantly higher today. A research group conducted a study to investigate whether the claim was true. The group found that 56 percent of a randomly selected sample of car owners in the United States owned American-made cars. A test of the appropriate hypotheses resulted in a p-value of 0.283. Assuming the conditions for inference were met, is there sufficient evidence to conclude, at the significance level of a = 0.05, that the proportion of all car owners in the United States who own American-made cars has increased from what it was ten years ago? A Yes, because 0.56 > 0.53. B Yes, because a reasonable interval for the proportion is 0.56 ± 0.283. C Yes, because 0.56 - 0.53 = 0.03 and 0.03 < 0.05. D No, because 0.283 < 0.53. E No, because 0.283 > 0.05.

In the states of Florida and Colorado, veterinarians investigating obesity in dogs obtained random samples of pet medical records and recorded the weights of the dogs in the samples. A test was conducted of H0 : p1= p2 versus Ha : p1 ≠ p2, where p1 represents the proportion of all overweight dogs in Florida and p2 represents the proportion of all overweight dogs in Colorado. The resulting test statistic for a two-sample z-test for a difference between proportions was 1.85. At the significance level α = 0.05, which of the following is a correct conclusion? A There is not sufficient statistical evidence to conclude that the proportion of all overweight dogs in Florida is different from the proportion of all overweight dogs in Colorado because the p-value is greater than 0.05. B There is not sufficient statistical evidence to conclude that the proportion of all overweight dogs in Florida is different from the proportion of all overweight dogs in Colorado because the z-test statistic is greater than 0.05. C There is sufficient statistical evidence to conclude that the proportion of all overweight dogs in Florida is different from the proportion of all overweight dogs in Colorado because the p-value is greater than 0.05. D There is sufficient statistical evidence to conclude that the proportion of all overweight dogs in Florida is different from the proportion of all overweight dogs in Colorado because the p-value is less than 0.05. E There is sufficient statistical evidence to conclude that the proportion of all overweight dogs in Florida is greater than the proportion of all overweight dogs in Colorado because the z-test statistic is positive.

In order to make statistical inferences when testing a population proportion p, which of the following conditions verify that inference procedures are appropriate? The data are collected using a random sample or random assignment. The sample size is less than 10 percent of the population size. np0≥10 and n(1−p0)≥10 for sample size n and hypothesized proportion p0. A I only B II only C III only D II and III only E I, II, and III

A medical doctor uses a diagnostic test to determine whether a patient has arthritis. A treatment will be prescribed only if the doctor thinks the patient has arthritis. The situation is similar to using a null and an alternative hypothesis to decide whether to prescribe the treatment. The hypotheses might be stated as follows. H0 : The patient does not have arthritis Ha : The patient has arthritis Which of the following represents a Type II error for the hypotheses? A Diagnosing arthritis in a patient who has arthritis B Failing to diagnose arthritis in a patient who has arthritis C Diagnosing arthritis in a patient who does not have arthritis D Failing to diagnose arthritis in a patient who does not have arthritis E Prescribing treatment to a patient regardless of the diagnosis

To investigate whether there is a difference in opinion on a certain proposal between two voting districts, A and B, two independent random samples were taken. From district A, 35 of the 50 voters selected were in favor of the proposal, and from district B, 36 of the 60 voters selected were in favor of the proposal. Which of the following is the test statistic for the appropriate test to investigate whether there is a difference in the proportion of voters who are in favor of the proposal between the two districts (district A minus district B)? A 35−363550+3660√ B 35−360.750+0.660√ C 0.7−0.6(0.65)(0.35)(150+160)√ D 0.7−0.6(0.7)(0.6)(150+60)√ E 0.7−0.6(0.7)(0.6)150+160√

A study was conducted to investigate whether a new drug could significantly reduce pain in people with arthritis. From a group of 500 people with arthritis, 250 were randomly assigned to receive the drug (group 1) and the remaining people were assigned a placebo (group 2). After one month of treatment, 225 people in group 1 reported pain relief and 150 people in group 2 reported pain relief. Let pˆC represent the combined (or pooled) sample proportion for the two samples. Have the conditions for inference for testing the difference in population proportions been met? A No. The people in the study were not selected at random. B No. The number of people in the study was too large compared with the size of the population. C No. The normality of the sampling distribution cannot be assumed because pˆC times each sample size is not sufficiently large. D No. The normality of the sampling distribution cannot be assumed because 1−pˆC times each sample size is not sufficiently large. E Yes. All conditions for inference have been met.

A one-sided hypothesis test is to be performed with a significance level of 0.05. Suppose that the null hypothesis is false. If a significance level of 0.01 were to be used instead of a significance level of 0.05, which of the following would be true? A Neither the probability of a Type II error nor the power of the test would change. B Both the probability of a Type II error and the power of the test would decrease. C Both the probability of a Type II error and the power of the test would increase. D The probability of a Type II error would decrease and the power of the test would increase. E The probability of a Type II error would increase and the power of the test would decrease.

Which of the following gives the probability of making a Type I error? A The sample size B The power C The significance level D The standard error E The p-value

Consider the results of a hypothesis test, which indicate there is not enough evidence to reject the null hypothesis. Which of the following statements about error is correct? A A Type I error could have been made, but not a Type IIerror. B A Type II error could have been made, but not a Type Ierror. C Both types of error could have been made, but the probability of a Type I error is greater than the probability of a Type II error. D Both types of error could have been made, but the probability of a Type I error is less than the probability of a Type II error. E The type of error that could have been made is not possible to determine without knowing the statement of the null hypothesis.

Dan selected a random sample of 100 students from the 1,200 at his school to investigate preferences for making up school days lost due to emergency closings. The results are shown in the table below. Dan incorrectly performed a large sample test of the difference in two proportions using 58/100 and 42/100 and calculated a p-value of 0.02. Consequently, he concluded that there was a significant difference in preference for the two options. Which of the following best describes his error in the analysis of these data? A No statistical test was necessary because 0.58 is clearly larger than 0.42. B The results of the test were invalid because less than 10% of the population was sampled. C Dan performed a two-tailed test and should have performed a one-tailed test. D A one-sample test for a proportion should have been performed because only one sample was used. E More options should have been included, and a chi-square test should have been performed.

At a manufacturing company for medical supplies, machines produce parts used in highly specialized lasers. Company researchers are testing a new machine designed to improve the precision of the parts. The null hypothesis is that the new machine does not improve the precision. For the researchers, the more consequential error would be that the new machine actually improves the precision, but the test does not detect the improvement. Which of the following should the researchers do to avoid the more consequential error? A Increase the significance level to increase the probability of a Type I error. B Increase the significance level to decrease the probability of a Type I error. C Decrease the significance level to increase the probability of a Type I error. D Decrease the significance level to decrease the probability of a Type I error. E Decrease the significance level to decrease the standard error.

Machines at a bottling plant are set to fill bottles to 12 ounces. The quality control officer at the plant periodically tests the machines to be sure that the bottles are filled to an appropriate amount. The null hypothesis of the test is that the mean is at least 12 ounces. The alternative hypothesis is that the mean is less than 12 ounces. Which of the following describes a Type I error that could result from the test? A The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces. B The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is less than 12 ounces. C The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is 12 ounces. D The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces. E The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is 11 ounces.

Researchers were investigating whether there is a significant difference between two medications, R and S, designed to reduce fleas found on cats. From a sample of 300 cat owners, the researchers randomly assigned 150 cat owners to use medication R on their cats and the remaining cat owners to use medication S. For the cats using medication R, 88 percent had no fleas. For the cats using medication S, 90 percent had no fleas. Which of the following is the most appropriate method for analyzing the results? A A two-sample z-test for a difference in population proportions B A two-sample z-test for a difference in sample proportions C A one-sample z-test for a sample proportion D A one-sample z-test for a population proportion E A one-sample z-test for a difference in sample proportions

The germination rate is the rate at which plants begin to grow after the seed is planted. A seed company claims that the germination rate for their seeds is 90 percent. Concerned that the germination rate is actually less than 90 percent, a botanist obtained a random sample of seeds, of which only 80 percent germinated. What are the correct hypotheses for a one-sample z-test for a population proportion p ? A H0_p=0.80Ha:p<0.80 B H0_p=0.80Ha:p>0.80 C H0_p=0.90Ha:p<0.90 D H0_p=0.90Ha:p>0.90 E H0_p=0.90Ha:p≠0.90

A one-sample z-test for a population proportion will be conducted using a simple random sample selected without replacement from a population. Which of the following is a check for independence? A np0≥10 and n(1−p0)≥10 for sample size n and population proportion p0. B Each sample proportion value is less than or equal to 0.5. C The sample size is more than 10 times the population size. D The population size is more than 10 times the sample size. E The population distribution is approximately normal.