Exam 10: Statistical Inference Concerning Two Populations

The test statistic for testing the difference between two means are valid only when the sampling distribution of is_________ .

When the hypothesized difference of the population proportions is equal to 0,we _________ .

The student senate at a local university is about to hold elections.A representative from the women's sports program and a representative from the men's sports program must both be elected.Two candidates,an incumbent and a challenger,are vying for each position and early polling results are presented next.A hypothesis test must be performed to determine whether the percentages of supporting votes are different between the two incumbent candidates.In a sample of 100 voters,67 said that they would vote for the women's incumbent candidate.In a separate sample of 100 voters,55 said they would vote for the men's incumbent candidate.Let p₁ and p₂ be the proportions of supporting votes for the incumbent candidates representing women's and men's sports programs,respectively.Which of the following hypothesis tests must be performed to determine whether the percentages of supporting votes are different between the two incumbent candidates?

The statistical inference concerning the difference between two population proportions is used for qualitative data.

In the case when and are unknown and can be assumed equal,we can calculate a pooled estimate of the population variance.

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.Which of the following is(are)the appropriate critical value(s)to test the professor's claim at the 1% significance level?

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 appropriate decision given a 5% level of significance?

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.Which of the following hypotheses will test the professor's claim?

A coach is examining two basketball players' free-throw percentages over the last few games.Bob made 34 out of 85 shots and Joe made 72 out of 125 shots.The coach wants to know if Bob is a 10% worse free-throw shooter than Joe.Which of the following would be the p-value of the corresponding test?

The confidence interval for the difference is based on the same approach used in the case of one sample: Point Estimate ± Standard Error.

A company claims that you can expect your car to get one mpg better gas mileage while using their gasoline additive.A magazine did a study to find out how much a car's gas mileage improved while using the gasoline additive.The study used 36 cars and recorded the average mpg with and without the additive for each car in the study.The cars with the additive averaged 1.20 mpg better than without and had a variance of 0.36 (mpg)². a.Specify the competing hypotheses to determine if the gasoline additive improved gas mileage by at least one mpg.Use the matched-pairs sampling. b.Calculate the value of the test statistic and the criticalvalue. c.Make a conclusion at the 5% significance level.

Assume the competing hypotheses take the following form H₀: µ₁-µ₂ = 0,H_A: µ₁-µ₂ ≠ 0,where µ₁ is the population mean for population 1 and µ₂ is the population mean for population 2.Also assume that the populations are normally distributed with known variances and we use independent sampling.Which of the following expressions is appropriate test statistic?

What type of test for population means should be performed when examining a situation in which employees are first tested,then trained,and finally retested?

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.Assuming α = 0.10,does the evidence support the policymakers' claim?

We estimate the unknown standard deviation for by the _________ .

Calcium is an essential nutrient for strong bones and for controlling blood pressure and heart beat.Because most of the body's calcium is stored in bones and teeth,the body withdraws the calcium it needs from the bones.Over time,if more calcium is taken out of the bones than is put in,the result may be thin,weak bones.This is especially important for women who are often recommended a calcium supplement.A consumer group activist assumes that calcium content in two popular supplements are normally distributed with the same unknown population variance,and uses the following information obtained under independent sampling: Let μ₁ and μ₂ denote the corresponding population means.Construct a 95% confidence interval for the difference μ₁ - μ₂.

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. Should we conclude that the average time it takes experts to crack the safes does not differ by model at the 5% significance level?

The student senate at a local university is about to hold elections.A representative from the women's sports program and a representative from the men's sports program must both be elected.Two candidates,an incumbent and a challenger,are vying for each position and early polling results are presented next.A hypothesis test must be performed to determine whether the percentages of supporting votes are different between the two incumbent candidates.In a sample of 100 voters,67 said that they would vote for the women's incumbent candidate.In a separate sample of 100 voters,55 said they would vote for the men's incumbent candidate.Let p₁ and p₂ be the proportions of supporting votes for the incumbent candidates representing women's and men's sports programs,respectively.Which of the following are the competing hypotheses to determine whether the percentages of supporting votes are different between the two incumbent candidates?

You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood.Let men and women be represented by populations 1 and 2,respectively.Which of the following hypotheses is relevant to this claim?

When calculating the standard error of ,under what assumption do you pool the sample variances and ?