# Prob & Stats. Chapter 9 Review

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1 Chapter 9 Review Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Also assume that the population standard deviations are equal (σ1 = σ2), so that the standard error of the difference between means is obtained by pooling the sample variances. 1) A researcher was interested in comparing the resting pulse rates of people who exercise regularly and people who do not exercise regularly. Independent simple random samples were obtained of 16 people who do not exercise regularly and 12 people who do exercise regularly. The resting pulse rate (in beats per minute) of each person was recorded. The summary statistics are as follows. Do Not Exercise Do Exercise x 1 = 73.4 beats/min x 2 = 69.7 beats/min s1 = 10.3 beats/min n1 = 16 n2 = 12 s2 = 8.6 beats/min Construct a 90% confidence interval for the difference between the mean pulse rate of people who do not exercise regularly and the mean pulse rate of people who exercise regularly. 2) A paint manufacturer wanted to compare the drying times of two different types of paint. Independent simple random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying times, in hours, were recorded. The summary statistics are as follows. Type A Type B x 1 = 70.9 hr x 2 = 68.4 hr s1 = 3.7 hr n1 = 11 n2 = 9 s2 = 3.2 hr Construct a 99% confidence interval for µ1 - µ2, the difference between the mean drying time for paint type A and the mean drying time for paint type B. Test the indicated claim about the variances or standard deviations of two populations. Assume that both samples are independent simple random samples from populations having normal distributions. 3) Random samples of 13 women and 11 men yielded the following scores on a test: Women: 70, 78, 62, 96, 75, 68, 41, 74, 80, 47, 73, 94, 65 Men: 72, 60, 52, 87, 66, 74, 95, 50, 81, 70, 72 Use a 0.05 significance level to test the claim that test scores for women have a larger standard deviation than test scores for men. (Note: s1 = and s2 = ). 4) When 25 randomly selected customers enter any one of several waiting lines, their waiting times have a standard deviation of 5.07 minutes. When 16 randomly selected customers enter a single main waiting line, their waiting times have a standard deviation of 2.02 minutes. Use a 0.05 significance level to test the claim that there is more variation in the waiting times when several lines are used. Chapter 9 Review - pg.1

2 5) The manager of a juice bottling factory is considering installing a new juice bottling machine which she hopes will reduce the amount of variation in the volumes of juice dispensed into 8-fluid-ounce bottles. Random samples of 10 bottles filled by the old machine and 9 bottles filled by the new machine yielded the following volumes of juice (in fluid ounces). Old machine: 8.1, 8.0, 7.9, 8.2, 8.5, 8.1, 8.1,8.2, 7.8, 7.9 New machine: 8.0, 8.1, 8.0, 8.1, 7.9, 8.0, 7.9, 8.0, 8.1 Use a 0.05 significance level to test the claim that the volumes of juice filled by the old machine vary more than the volumes of juice filled by the new machine. Solve the problem. 6) A researcher wishes to compare how students at two different schools perform on a math test. He randomly selects 40 students from each school and obtains their test scores. He pairs the first score from school A with the first school from school B, the second score from school A with the second school from school B and so on. He then performs a hypothesis test for matched pairs. Is this approach valid? Why or why not? If it is not valid, how should the researcher have proceeded? 7) A manager at a bank is interested in the standard deviation of the waiting times when a single waiting line is used and when individual lines are used. He wishes to test the claim that the population standard deviation for waiting times when multiple lines are used is greater than the population standard deviation for waiting times when a single line is used. Find the P-value (or range of p-values) for a test of this claim given the following sample data. Sample 1: multiple waiting lines: n1 = 13, s1 = 2.1 minutes Sample 2: single waiting line: n2 = 16, s2 = 1.1 minutes 8) A researcher wishes to determine whether listening to music affects students' performance on memory test. He randomly selects 50 students and has each student perform a memory test once while listening to music and once without listening to music. He obtains the mean and standard deviation of the 50 "with music" scores and obtains the mean and standard deviation of the 50 "without music scores". He then performs a hypothesis test for two means assuming large and independent samples. Is this approach appropriate? If not, how would you proceed. 9) A researcher wished to perform a hypothesis test to test the claim that the rate of defectives among the computers of two different manufacturers are the same. She selects two independent random samples and obtains the following sample data. Manufacturer A: n1 = 400, rate of defectives: 1.5% Manufacturer B: n2 = 200, rate of defectives: 3.5% Can the methods of this section be used to perform a hypothesis test to test for the equality of the two population proportions? Go through the steps of checking whether the conditions for the hypothesis test for two population proportions are satisfied. Show your calculations and state your conclusion. Chapter 9 Review - pg.2

3 10) A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. A 95% confidence interval for the difference between the population means is (1.4, 8.7). If the hypothesis test is based on the same samples, which of the following do you know for sure: A: The hypothesis µ1 = µ2 would be rejected at the 5% level of significance. B: The hypothesis µ1 = µ2 would be rejected at the 10% level of significance. C: The hypothesis µ1 = µ2 would be rejected at the 1% level of significance. Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis. 11) A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels) before and after using the additive. The results are shown below. Plot: Before After You wish to test the following hypothesis at the 10 percent level of significance. H 0 : µ d = 0 against H 1 : µ d 0. What decision rule would you use? Construct the indicated confidence interval for the difference between population proportions p 1 - p 2. Assume that the samples are independent and that they have been randomly selected. 12) A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. Construct a 99% confidence interval for the difference between the two population proportions. Construct a confidence interval for µ d, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed. 13) Ten different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. Construct a 90% confidence interval for the mean of the differences. Before After ) A test of writing ability is given to a random sample of students before and after they completed a formal writing course. The results are given below. Construct a 99% confidence interval for the mean difference between the before and after scores. Before After ) A coach uses a new technique in training middle distance runners. The times for 9 different athletes to run 800 meters before and after this training are shown below. Athlete A B C D E F G H I Time before training (seconds) Time after training (seconds) Construct a 99% confidence interval for the mean difference of the "before" minus "after" times. Chapter 9 Review - pg.3

4 Assume that you plan to use a significance level of α = 0.05 to test the claim that p 1 = p 2. Use the given sample sizes and numbers of successes to find the z test statistic for the hypothesis test. 16) A random sampling of sixty pitchers from the National League and fifty-two pitchers from the American League showed that 19 National and 11 American League pitchers had E.R.A's below ) A report on the nightly news broadcast stated that 10 out of 108 households with pet dogs were burglarized and 20 out of 208 without pet dogs were burglarized. 18) In a vote on the Clean Water bill, 46% of the 205 Democrats voted for the bill while 47% of the 230 Republicans voted for it. Test the indicated claim about the means of two populations. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Use the traditional method or P-value method as indicated. 19) A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure, measured in mm Hg, by following a particular diet. Use a significance level of 0.01 to test the claim that the treatment group is from a population with a smaller mean than the control group. Use the traditional method of hypothesis testing. Treatment Group Control Group n 1 = 35 n 2 = 28 x1 = x2 = s 1 = 38.7 s 2 = ) A researcher was interested in comparing the amount of time (in hours) spent watching television by women and by men. Independent simple random samples of 14 women and 17 men were selected, and each person was asked how many hours he or she had watched television during the previous week. The summary statistics are as follows. Women Men x 1 = 11.8 hr x 2 = 13.5 hr s1 = 3.9 hr s2 = 5.2 hr n1 = 14 n2 = 17 Use a 0.05 significance level to test the claim that the mean amount of time spent watching television by women is smaller than the mean amount of time spent watching television by men. Use the traditional method of hypothesis testing. Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is µ d = 0. Compute the value of the t test statistic. Round intermediate calculations to four decimal places as needed and final answers to three decimal places as needed. 21) The following table shows the weights of nine subjects before and after following a particular diet for two months. You wish to test the claim that the diet is effective in helping people lose weight. What is the value of the appropriate test statistic? Subject A B C D E F G H I Before After Chapter 9 Review - pg.4

5 Answer Key Testname: CHAPTER 9 REVIEW 1) beats/min < µ 1 - µ 2 < 9.97 beats/min 2) hrs < µ 1 - µ 2 < 7.01 hrs 3) H0: σ1 = σ2 H1: σ1 > σ2 Test statistic: F = 1.28 Upper critical F value: Fail to reject the null hypothesis. At the 5% significance level, there is not sufficient evidence to the support the claim that test scores for women have a larger standard deviation than test scores for men. 4) H 0 : σ 1 = σ 2 H1: σ 1 > σ 2 Test statistic: F = 6.3. Upper critical F value: Reject H 0. There is sufficient evidence to support the claim that there is more variation in the waiting times when several lines are used. 5) H0: σ1 = σ2 H1: σ1 > σ2 Test statistic: F = 6.47 Upper critical F value: Reject the null hypothesis. There is sufficient evidence to support the claim that the volumes of juice filled by the old machine vary more than the volumes of juice filled by the new machine. 6) There is no natural pairing here; hence, it is not appropriate to perform a test for matched pairs. The two samples are independent and a hypothesis test for large, independent samples should have been performed. 7) 0.01 < P-value < ) The data consists of matched pairs since for each person there is a "with music" score and a "without music" score. The two samples are dependent not independent. The researcher should calculate, for each person, the difference d between their "with music" score and their "without music" score and then use these differences to perform a hypothesis test for matched pairs. 9) x1 = 400(0.015) = 6 x2 = 200(0.035) = 7 p = = n1p = 400(0.0217) = 8.7 n2p = 200(0.0217) = 4.3 Because n2p < 5, the conditions for the hypothesis test are not satisfied. 10) A and B 11) Reject H 0 if test statistic is less than or greater than ) < p 1 - p 2 < ) 1.8 < µ d < ) -0.5 < µ d < ) < µ d < ) z = ) z = ) z = Chapter 9 Review - pg.5

6 Answer Key Testname: CHAPTER 9 REVIEW 19) H 0 : µ 1 = µ 2. H 1 : µ 1 < µ 2. Test statistic: t = Critical value: Do not reject the null hypothesis. There is not sufficient evidence to support the claim that the treatment group is from a population with a smaller mean than the control group. 20) H0: µ1 = µ2 H1: µ1 < µ2 Test statistic: t = Critical value: t = Do not reject H0. At the 5% significance level, there is not sufficient evidence to support the claim that the mean amount of time spent watching television by women is smaller than the mean amount of time spent watching television by men. 21) Chapter 9 Review - pg.6

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