Exam 10: Statistical Inference Concerning Two Populations

The difference between the two sample means is an interval estimator of the difference between two population means .

A bank is trying to determine which model of safe to install.The bank manager believes that each model is equally resistant to safe crackers but sets up a test to be sure.He hires nine safe experts to break into each of the models,timing each endeavor.The results (in seconds)are given next,paired by expert.Let D be the difference: Time to break Safe 1 minus Time to break Safe 2. Assuming the difference D is normally distributed,which of the following is the 95% confidence interval for μ_D?

Two random samples are considered independent if the observations in the first sample are different from the observations of the second sample.

If the underlying populations cannot be assumed to be normal,then by the central limit theorem,the sampling distribution of is approximately normal only if both sample sizes are sufficiently large-that is,when _________ .

Weather forecasters would like to report on temperatures in Washington,DC.In particular,they would like to compare the temperatures in December to temperatures in January to help potential tourists understand whether there is a difference.They believe it is colder in January but want to conduct a test to support their assertion.The average December temperature in a random sample of 16 days is 35 and the average January temperature in a random sample of 20 days is 33.The standard deviations based on historic information are 4 for December and 5 for January. a.Specify the competing hypotheses to determine if it is colder in January than in December. b.Find the critical value(s)of the appropriate test at the 5% significance level. c.Calculate the value of the test statistic. d.Make a conclusion at the 5% significance level.

A statistics professor at a large university hypothesizes that students who take statistics in the morning typically do better than those who take it in the afternoon.He takes independently random samples,each of size 36,consisting of students who took a morning and an afternoon class,and compares the scores of each group on a common final exam.He finds that the morning group scored an average of 74 with a standard deviation of 8,while the evening group scored an average of 68 with a standard deviation of 10.The population standard deviation of scores is unknown but is assumed to be equal for morning and evening classes.Let µ₁ andµ₂ represent the population mean final exam scores of statistics' courses offered in the morning and the afternoon,respectively.Compute the appropriate test statistic to analyze the claim at the 1% significance level.

A new sales training program has been instituted at a rent-to-own company.Prior to the training,10 employees were tested on their knowledge of products offered by the company.Once the training was completed,the employees were tested again in an effort to determine whether the training program was effective.Scores are known to be normally distributed.The sample scores on the tests are listed next.Use pretest score as µ₁ for population 1 and posttest score as µ₂ for population 2,orµ_D as the difference calculated as pretest score minus posttest score. Which of the following is the value of the appropriate test statistic?

A recent study from the University of California,Berkeley,suggests that compared to black Americans,white Americans are less likely to cross the race barrier in online dating.In a sample of 120 white men,24 initiated contact with someone outside of their race,whereas in a similar sample of 90 black men,36 initiated contact with someone outside of their race. a.Specify the competing hypotheses to determine if the proportion of white Americans initiating a contact outside their race is exactly 10 percentage points lower than that of black Americans. b.Find the critical value(s)of the test. c.Calculate the value of the test statistic. d.Make a conclusion at the 5% significance level.

A restaurant chain has two locations in a medium-sized town and,believing that it has oversaturated the market for its food,is considering closing one of the restaurants.The manager of the restaurant with a downtown location claims that his restaurant generates more revenue than the sister restaurant by the freeway.The CEO of this company,wishing to test this claim,randomly selects 36 monthly revenue totals for each restaurant.The revenue data from the downtown restaurant have a mean of $360,000 and a standard deviation of $50,000,while the data from the restaurant by the freeway have a mean of $340,000 and a standard deviation of $40,000.Assume there is no reason to believe the population standard deviations are equal,and let μ₁ and μ₂ denote the mean monthly revenue of the downtown restaurant and the restaurant by the freeway,respectively.Which of the following is the correct value of the test statistic to analyze the claim?

A veterinarian wants to know if pit bulls or golden retrievers have a higher incidence of tooth decay at the age of three.The vet surveys 120 three-year-old pit bulls and finds 30 of them have tooth decay.The vet then surveys 160 three-year-old golden retrievers and finds 32 of them have tooth decay.Number the population of pit bulls and golden retrievers by 1 and 2,respectively.Calculate a 90% confidence interval for the difference in the population proportion of pit bulls and golden retrievers that have tooth decay.Which of the following is correct?

In the test for comparing two population means when population variances are unknown and unequal,a student calculates the degrees of freedom using the proper formula as 34.7.How many degrees of freedom should the student assume to find the p-value of the test?

In August 2010,Massachusetts enacted a 150-day right-to-cure period that mandates that lenders give homeowners who fall behind on their mortgage an extra five months to become current before beginning foreclosure proceedings.Policymakers claimed that the policy would result in a higher proportion of delinquent borrowers becoming current on their mortgages.To test this claim,researchers took a sample of 244 homeowners in danger of foreclosure in the time period surrounding the enactment of this law.Of the 100 who fell behind just before the law was enacted,30 were able to avoid foreclosure,and of 144 who fell behind just after the law was enacted,48 were able to avoid foreclosure.Let p₁andp₂ represent the proportion of delinquent borrowers who avoid foreclosure just before and just after the right-to-cure law is enacted,respectively.Which of the following is the appropriate test statistic value to analyze the claim at the 10% significance level?

A restaurant chain has two locations in a medium-sized town and,believing that it has oversaturated the market for its food,is considering closing one of the restaurants.The manager of the restaurant with a downtown location claims that his restaurant generates more revenue than the sister restaurant by the freeway.The CEO of this company,wishing to test this claim,randomly selects 36 monthly revenue totals for each restaurant.The revenue data from the downtown restaurant have a mean of $360,000 and a standard deviation of $50,000,while the data from the restaurant by the freeway have a mean of $340,000 and a standard deviation of $40,000.Assume there is no reason to believe the population standard deviations are equal,and let μ₁ and μ₂ denote the mean monthly revenue of the downtown restaurant and the restaurant by the freeway,respectively.At the 5% significance level,does the evidence support the manager's claim?

The choice of an appropriate test for comparing two population means depends on whether we deal with _________ .

The margin of error in the confidence interval for the difference equals the standard error multiplied by either or depending on whether or not the population variances are known.

A 7,000-seat theater is interested in determining whether there is a difference in attendance between shows on Tuesday evening and those on Wednesday evening.Two independent samples of 25 weeks are collected for Tuesday and Wednesday.The mean attendance on Tuesday evening is calculated as 5,500,while the mean attendance on Wednesday evening is calculated as 5,850.The known population standard deviation for attendance on Tuesday evening is 550 and the known population standard deviation for attendance on Wednesday evening is 445.Which of the following is the value of the appropriate test statistic?

A particular personal trainer works primarily with track and field athletes.She believes that her clients run faster after going through her program for six weeks.How might she test that claim?

A statistics professor at a large university hypothesizes that students who take statistics in the morning typically do better than those who take it in the afternoon.He takes independently random samples,each of size 36,consisting of students who took a morning and an afternoon class,and compares the scores of each group on a common final exam.He finds that the morning group scored an average of 74 with a standard deviation of 8,while the evening group scored an average of 68 with a standard deviation of 10.The population standard deviation of scores is unknown but is assumed to be equal for morning and evening classes.Let µ₁ andµ₂ represent the population mean final exam scores of statistics' courses offered in the morning and the afternoon,respectively.At the 1% significance level,does the evidence support the professor's claim?

A particular bank has two loan modification programs for distressed borrowers: Home Affordable Modification Program (HAMP)modifications,where the federal government pays the bank $1,000 for each successful modification,and non-HAMP modifications,where the bank does not receive a bonus from the federal government.To qualify for a HAMP modification,borrowers must meet a set of financial suitability criteria.What type of hypothesis test should we use to test whether borrowers from this particular bank who receive HAMP modifications are more likely to re-default than those who receive non-HAMP modifications?

Suppose you want to perform a test to compare the mean GPA of all freshmen with the mean GPA of all sophomores in a college? What type of sampling is required for this test?