Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations. True or false? The standard deviation of all possible s between the two sample means equals the square root of the of variances each divided by the corresponding sample size. 2) Suppose that x is a variable on each of two populations. Independent samples of sizes 5 and 8, respectively, are selected from the two populations. True or false? x 1 - x 2 is normally distributed, regardless of the distributions of the variable on the two populations. 3) Suppose a researcher wants to perform a hypothesis test to compare systolic blood pressure for men with systolic blood pressure for women, based on independent samples. She obtains a random sample of 30 married couples and determines systolic blood pressure for the men, x 1, and systolic blood pressure for the women, x 2. She formulates the hypotheses as follows H0: μ1 = μ2 Ha: μ1 μ2 and will reject the null hypothesis if x 1 - x 2, the observed sample means is too large. She will use a two-means z-test. Identify the flaw in her methodology. Determine the null and alternative hypotheses for the proposed hypothesis test. 4) A researcher wants to perform a hypothesis test to determine whether length of marriages in California differs from length of marriages in Texas. 5) A researcher is interested in determining whether men who have completed a postgraduate degree (master's or Phd) have greater earning potential than those who have completed a Bachelor's degree only. She will perform a hypothesis test to determine whether salary of men who have completed a postgraduate degree is greater than salary of men with a Bachelor's degree only. Solve the problem. 6) A researcher is interested in comparing the resting pulse rate of women who exercise regularly and women who do not exercise regularly. She wants to perform a hypothesis test to determine whether resting pulse rate of women who exercise at least 6 hours per week is less than resting pulse rate of women who exercise less than 6 hours per week. Identify the two populations for the proposed hypothesis test. 7) A researcher is interested in comparing the resting pulse rate of women who exercise regularly and women who do not exercise regularly. She wants to perform a hypothesis test to determine whether resting pulse rate of women who exercise at least 6 hours per week is less than resting pulse rate of women who exercise less than 6 hours per week. Identify the variable for the proposed hypothesis test. 8) The forced vital capacity (FVC) is often used by physicians to assess a person's ability to move air in and out of their lungs. It is the maximum amount of air that can be exhaled after a deep breath. A researcher wants to perform a hypothesis test to determine whether forced vital capacity for women is less than forced vital capacity for men. Identify the two populations for the proposed hypothesis test. 1
Provide an appropriate response. 9) Suppose that you want to perform a hypothesis test based on independent simple random samples to compare s of two populations. Further suppose that either the variable under consideration is normally distributed on each of the two populations or the sample sizes are large. True or false? If the population standard deviations are quite different, then using the pooled t-test can result in a significantly smaller Type I error probability than the one specified. 10) Suppose that you want to perform a hypothesis test based on independent simple random samples to compare s of two populations. Further suppose that either the variable under consideration is normally distributed on each of the two populations or the sample sizes are large. True or false? If the population standard deviations are equal, then the pooled t-test gives a larger probability of a Type II error than the nonpooled t-test. 11) Suppose that you want to perform a hypothesis test based on independent simple random samples to compare s of two populations. Further suppose that the variable under consideration is normally distributed on each of the two populations and that the population standard deviations are unknown. If the sample standard deviations are 3.8 and 6.1, respectively, and the sample sizes are and 60, respectively, would you use the pooled or the nonpooled t-test? Explain your answer. 12) Suppose that you want to perform a hypothesis test based on independent simple random samples to compare s of two populations. Further suppose that either the variable under consideration is normally distributed on each of the two populations or the sample sizes are large. True or false? If the population standard deviations are equal, then, on average, the pooled t-test is slightly more powerful than the nonpooled t-test. Preliminary data analyses indicate that you can reasonably use nonpooled t-procedures on the given data. Apply a nonpooled t-test to perform the required hypothesis test, using either the critical-value approach or the P-value approach as indicated. 13) A researcher was interested in comparing the amount of time spent watching television by women and by men. Independent 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: Sample 1 (women) Sample 2 (men) x 1 = 12.3 x 2 = 13.8 s 1 = 3.9 s 2 = 5.2 n 1 = 14 n 2 = 17 At the 5% level, do the data provide amount of time spent watching television by women is less than amount of time spent watching television by men? Use the P-value approach. 14) A researcher wishes to determine whether the systolic blood pressure of people who follow a vegetarian diet is, on average, lower than the systolic blood pressure of those who do not follow a vegetarian diet. Independent simple random samples of 85 vegetarians and 75 nonvegetarians yielded the following sample statistics: Vegetarians Nonvegetarians n1 = 85 n2 = 75 x1 = 124.1 x2 = 138.7 s1 = 38.7 s2 = 39.2 Use a level of 0.01 to test the claim that systolic blood pressure for vegetarians is lower than systolic blood pressure for nonvegetarians. Use the P-value approach. 2
15) A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs. College A College B 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 At the 10% level, do the data provide GPA of students at college A differs from GPA of students at college B? Use the critical-value approach. (Note: x 1 = 3.11, x 2 = 3.4385, s1 = 0.4357, s2 = 0.5485.) Summary statistics are given for independent simple random samples from two populations. Use the nonpooled t-test to conduct the required hypothesis test. 16) x 1 = 12.2, s1 = 3.9, n1 = 14, x 2 = 13.5, s2 = 5.2, n2 = 17 Perform a left-tailed hypothesis test using a level of α = 0.05. 17) x 1 = 74.8, s1 = 4.5, n1 = 11, x 2 = 64.2, s2 = 5.1, n2 = 9 Perform a two-tailed hypothesis test using a level of α = 0.01. Apply the nonpooled t-interval procedure to obtain the required confidence interval. You may assume that the assumptions for using the procedure are satisfied. 18) 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 of 16 people ages 30-40 who do not exercise regularly and 12 people ages 30-40 who do exercise regularly were selected, and the resting pulse rate (in beats per minute) of each person was measured. The summary statistics are as follows. Do Not Exercise Do Exercise x 1 = 72.0 x 2 = 69.4 s1 = 10.9 s2 = 8.2 n1 = 16 n2 = 12 Determine a 95% confidence interval for the, μ1 - μ2, mean pulse rate of people who do not exercise and the mean pulse rate of people who do exercise. 19) A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs. College A College B 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 Determine a 95% confidence interval for the, μ1 - μ2, mean GPA of college A students and GPA of college B students. (Note: x 1 = 3.11, x 2 = 3.4385, s1 = 0.4357, s2 = 0.5485.) 3
20) A paint manufacturer wished 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 = 75.7 x 2 = 63.2 s1 = 4.5 s2 = 5.1 n1 = 11 n2 = 9 Determine a 98% confidence interval for the, μ1 - μ2, mean drying time for type A and drying time for type B. Provide an appropriate response. ) A researcher would like to conduct a hypothesis test to compare SAT scores of students who have received extra coaching and SAT score of students who have not received extra coaching. In this case, a paired t-test would be appropriate since a natural pairing exists (a pair consists of a student before coaching and the same student after coaching). If the researcher instead uses independent samples and a pooled or nonpooled t-test, how is this likely to affect the probability of a Type II error? ) Manufacturer X has designed athletic footwear which it hopes will improve the performance of athletes running the 100-meter sprint. It wishes to perform a hypothesis test to compare times of athletes in the 100-meter sprint running with Manufacturer X's footwear and with other shoes. Do you think that a paired t-test or a pooled t-test would be more appropriate in this situation? Why? If you think that a paired test is preferable, explain what would constitute a pair. 23) A teacher is interested in performing a hypothesis test to compare math score of the girls and math score of the boys. She randomly selects 10 girls from the class and then randomly selects 10 boys. She arranges the girls' names alphabetically and uses this list to assign each girl a number between 1 and 10. She does the same thing for the boys. She pairs each girl with the boy having the same number. She then performs a paired t-test. Do you think that in this situation a paired t-test is appropriate? Explain your answer. Preliminary data analyses indicates that use of a paired t-test is reasonable. Perform the hypothesis test by using either the critical-value approach or the P-value approach as indicated. Assume that the null hypothesis is H0 : μ1 = μ2. 24) A coach uses a new technique in training middle distance runners. The times, in seconds, for 8 different athletes to run 800 meters before and after this training are shown below. Athlete A B C D E F Before 113.5 118 118.6 108 111.8 108.7 After 114.1 116.7 116.2 108.8 110 108.8 At the 5% level, do the data provide evidence that the training helps to improve times for the 800 meters? Use the P-value approach. ) Five students took a math test before and after tutoring. Their scores were as follows. Subject A B C D E Before 65 66 79 74 72 After 69 75 77 77 84 At the 1% level, do the data provide score before tutoring differs from the mean score after tutoring? Use the P-value approach. 4
26) Ten different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. The results are shown below. Before 33 33 38 33 35 35 40 40 40 31 31 35 33 After 34 35 33 At the 5% level, do the data provide amount of water use after the viewing differs from amount of water use before the viewing? Use the critical-value approach. 27) The table below shows the weights, in pounds, of seven subjects before and after following a particular diet for two months. Subject A B C D E F G Before 200 189 186 160 176 151 168 After 193 180 184 165 162 153 156 At the 1% level, do the data provide the diet is effective in reducing weight? Use the critical-value approach. Use the paired t-interval procedure to obtain the required confidence interval. You may assume that the conditions for using the procedure are satisfied. ) A test of abstract reasoning is given to a random sample of students before and after completing a formal logic course. The results are shown below. Before After 74 68 81 82 85 78 67 73 92 95 71 66 61 67 78 81 64 71 80 88 Determine a 95% confidence interval for the mean score before completing the course and score after completing the course. 29) A coach uses a new technique in training middle distance runners. The times, in seconds, for 9 different athletes to run 800 meters before and after this training are shown below. Athlete A B C D E F Before 115.2 120.9 108.0 112.4 107.5 119.1 1 After 116.0 119.1 105.1 111.9 109.1 115.2 Determine a 99% confidence interval for the mean time before and after training. 5
30) The table below shows the weight, in pounds, of 9 subjects before and after following a particular diet for two months. Subject Before After A 168 165 B 180 172 C 157 164 D 132 1 E 202 183 F 124 126 G 190 180 H 0 195 I 171 163 Determine a 99% confidence interval for the weight loss that would be obtained, on average, by following the diet for two months. Provide an appropriate response. 31) A nutritionist wants to investigate whether her new diet will be effective in helping women aged 30-40 to lose weight. She will use a paired sample to determine whether weight of women before going on this diet is greater than weight of women after being on this diet for two months. Identify the paired- variable for the proposed hypothesis test. 32) A company which designs sports shoes has made an improvement to their popular running shoe. They hope that athletes wearing the new running shoe will be able to run faster over short distances. They will use a paired sample to determine whether time to run 100 meters for sprinters wearing the new running shoe is less than time to run the 100 meters for sprinters wearing the old running shoe. Identify the pairs for the proposed hypothesis test. 33) A nutritionist wants to investigate whether her new diet will be effective in helping women aged 30-40 to lose weight. She will use a paired sample to determine whether weight of women before going on this diet is greater than weight of women after being on this diet for two months. Identify the pairs for the proposed hypothesis test. 34) A researcher wants to use a paired sample to determine whether number of hours spent exercising per week for married men differs from number of hours spent exercising per week for married women. Identify the paired- variable for the proposed hypothesis test. 35) When performing a one-way ANOVA, two of the assumptions required are that the populations be normally distributed and that s have equal standard deviations. What rule of thumb can be used to assess the equal-standard deviations assumption? What other method can be used to assess the normality and equal-standard deviations assumptions? 36) True or false: In a one-way ANOVA, the null hypothesis will be rejected if the variation among the sample means is large relative to the variation within the samples. 37) In the context of a one-way ANOVA, explain what is meant by variation between samples and variation within samples. 38) A one-way ANOVA is to be performed. The following sample data are obtained. x 1 = 20, x 2 = 30, x 3 = 40 The common population standard deviation for the three populations is 2.5. Do you think that the sample means could be due to variation within the populations or does it seem clear that the sample means is due to a between population means? Do you think that the null hypothesis would be rejected? Explain your thinking. 6
39) A one-way ANOVA is to be performed. Independent random samples are taken from two populations. The sample data are depicted in the dotplot below. Is it reasonable to the sample means is due to a between means and not to variation within s? Do you think the null hypothesis would be rejected? Explain your thinking. Preliminary data analyses indicate that it is reasonable to consider the assumptions for one-way ANOVA satisfied. Use Minitab to perform the required hypothesis test using the p-value approach. 45) Random samples of four different models of cars were selected and the gas mileage of each car was measured. The results are shown below. Model A 23 24 26 Model B 26 29 30 Model C 30 32 27 Model D 26 Test the claim that the four different models have the same population mean. Use a level of 0.05. 40) True or false: In a one-way ANOVA, if the null hypothesis is rejected, we the population means are all different (i.e., no two of means are equal). 41) A one-way ANOVA is being performed. Suppose that SST = 135 and SSE = 74.5. Find the value of the third sum of squares, give its notation, state its name and the source of variation it represents. 42) A one-way ANOVA is being performed. True or false: The null hypothesis will be rejected if SSE is large relative to SSTR. 43) A one-way ANOVA is performed to compare s of four populations. The sample sizes are 16,, 17, and. Determine the degrees of freedom for the F-statistic. 44) True or false: When performing a one-way ANOVA, the error sum of squares can be obtained by subtracting the treatment sum of squares from the total sum of squares. 46) At the 0.01 level, do the data provide a exists population means of the three different brands? The sample data are given below. Brand A 44 47 44 40 39 Brand B 30 32 34 36 38 40 42 Brand C 27 31 32 36 47) At the 0.0 level, do the data provide a exists population means of the four different brands? The sample data are given below. Brand A 17 20 Brand B 18 18 23 26 Brand C 24 26 29 29 Brand D 27 29 35 36 37 7
48) At the 0.0 level, do the data provide a exists population means of the three different brands? The sample data are given below. Brand A 32 34 37 33 36 39 Brand B 27 24 33 30 Brand C 32 49) At the 0.0 level, do the data provide a exists population means of the four different brands? The sample data are given below. Brand A 15 23 20 Brand B 20 17 23 Brand C 20 19 18 Brand D 15 15 14 23 50) A consumer magazine wants to compare the lifetimes of ballpoint pens of three different types. The magazine takes a random sample of pens of each type in the following table. Brand 1 260 8 184 9 Brand 2 181 240 162 8 Brand 3 238 7 241 3 Do the data indicate that there is a in mean lifetime for the three brands of ballpoint pens? Use α = 0.01. 8
Answer Key Testname: 1342-MAY-PT-2MN-ANOVA 1) False 2) False 3) The two samples are not independent since they consist of 30 married couples. The sample of men should selected independently of the sample of women. 4) Let μ 1 denote length of marriages in California and let μ 2 denote length of marriages in Texas. The null and alternative hypotheses are H0: μ 1 = μ 2 and Ha: μ 1 μ 2. 5) Let μ 1 denote salary of men with a postgraduate degree and let μ 2 denote the mean salary of men with a Bachelor's degree only. The null and alternative hypotheses are H0: μ 1 = μ 2 and Ha: μ 1 > μ 2. 6) Women who exercise at least 6 hours per week and women who exercise less than 6 hours per week 7) Resting pulse rate 8) Women and men 9) False 10) False 11) The nonpooled t-test. In theory, the pooled t-test requires that standard deviations be If the population standard deviations are unequal, but not too unequal, and the sample sizes are nearly the same, using the pooled t-test will not cause serious difficulties. However, in this case, the significant sample standard deviations suggests that standard deviations are significantly un In addition, the sample sizes are also significantly different. Therefore, using the pooled t-test could result in a significantly larger Type I error probability than the one specified. (Explanations will vary.) 12) True 13) H0: μ1 = μ2 Ha: μ1 < μ2 α = 0.05 Test statistic: t = -0.917 P-value > 0.05 Do not reject H 0. At the 5% do not provide time for women is less than the mean time for men. 14) H0: μ1 = μ2 Ha: μ1 < μ2 α = 0.01 t = -2.365 0.005 < P < 0.01 Reject H0. At the 1% provide systolic blood pressure for vegetarians is lower than the mean systolic blood pressure for nonvegetarian s. 15) H0: μ1 = μ2 Ha: μ1 μ2 α = 0.10 t = -1.506 Critical values = ±1.740 Do not reject H0. At the 10% do not provide GPA of students at college A differs from GPA of students at college B. 16) Test statistic: t = -0.795 Critical value = -1.701 Do not reject H0 17) Test statistic: t = 4.873 Critical values = ±2.9 Reject H 0 18) -4.84 to 10.04 beats per minute 19) -0.78 to 0.13 20) 6.88 to 18.12 hours ) Using independent samples, the probability of a Type II error is likely to be larger than if a paired sample is used. 9
Answer Key Testname: 1342-MAY-PT-2MN-ANOVA ) A paired t-test is more appropriate since a natural pairing exists. By using a paired sample, extraneous sources of variation are removed, in this case, the variation in 100-meter sprint times among athletes. Thus, the sampling error made in estimating the population means should be smaller. A pair would consist of an athlete's time running the 100-meter sprint in Manufacturer X's footwear and that same athlete's time running the 100-meter sprint in his (or her) usual shoes. (Explanations will vary.) 23) No, a paired test is not appropriate since a natural pairing does not exist members of the two populations. 24) H0: μ1 = μ2 Ha: μ1 > μ2 α = 0.05 t = 2.7 0.0 < P < 0.05 Reject H0. At the 5% provide the training helps to improve times for the 800 meters. ) H0: μ1 = μ2 Ha: μ1 μ2 α = 0.01 t = -2.134 0.05 < P < 0.1 Do not reject H0. At the 1% do not provide score before tutoring differs from score after tutoring. 26) H0: μ1 = μ2 Ha: μ1 μ2 α = 0.05 t = 2.894 Critical values = ±2.262 Reject H0. At the 5% provide amount of water use after the viewing differs from amount of water use before the viewing. 27) H0: μ1 = μ2 Ha: μ1 > μ2 α = 0.01 t = 1.954 Critical value = 3.143 Do not reject H0. At the 1% do not provide the diet is effective in reducing weight. ) -2.4 to 5.6 29) -0.82 to 3.26 seconds 30) -1.9 to 16.3 lb 31) Difference between weight of a woman aged 30-40 before going on this diet and weight of same woman after being on this diet for two months. 32) Each pair consists of a sprinter wearing the old running shoe and the same sprinter wearing the new running shoe. 33) Each pair consists of a woman aged 30-40 before going on the diet and the same woman after being on the diet for two months. 34) Difference between hours of weekly exercise for a married man and hours of weekly exercise of his wife 35) As a rule of thumb, consider the equal-standar d deviations assumption satisfied if the ratio of the largest to the smallest sample standard deviation is less than 2. Also, the normality and equal-standar d deviations assumptions can be assessed by performing a residual analysis. This involves finding each residual (the observation and of the sample containing it) and making a normal probability plot of the residuals. If the assumptions are met, the residuals should be roughly linear and a plot of the residuals against the sample means should fall roughly in a horizontal band centered and symmetric about the horizontal axis. 36) True 10
Answer Key Testname: 1342-MAY-PT-2MN-ANOVA 37) The variation between samples gives a measure of the variation among the sample means and is a weighted average of their squared deviations about the mean of all the sample data. The variation within samples gives a measure of the variation among individual observations within each population. 38) It seems clear that the sample means is due to a between population means, not to the variation within the populations. The null hypothesis would be rejected. 39) Since the variation sample means is large relative to the variation within the samples, it is reasonable to the sample means is due to a population means and not to variation within the populations. The null hypothesis would be rejected. 40) False 41) SSTR = 60.5. SSTR = treatment sum of squares and represents the variation among the sample means. 42) False 43) (3, 71) 44) True 45) H0: μ1 = μ2 = μ3 = μ4. Ha: Not all the means are α = 0.05 p = 0.008 Reject the claim of equal means. The different models do not appear to have the same mean. 46) H0: μ1 = μ2 = μ3. Ha: Not all s are α = 0.01 p = 0.001 Reject the null hypothesis. There is a exists between means of the three different brands. 47) H0: μ1 = μ2 = μ3 = μ4. Ha: Not all the means are α = 0.0 p = 0.003 Reject the null hypothesis. There is a exists between means of the four different brands. 48) H0: μ1 = μ2 = μ3. Ha: Not all s are α = 0.0 p = 0.001 Reject the null hypothesis. There is a exists between means of the three different brands. 49) H0: μ1 = μ2 = μ3 = μ4. Ha: Not all the means are α = 0.0 p = 0.982 Fail to reject the null hypothesis. There is not a exists between means of the four different brands. 50) H0: μ1 = μ2 = μ3. Ha: Not all s are α = 0.01 p = 0.1 Fail to reject the claim of equal means. The data do not provide there is a in lifetimes of the three brands of ballpoint pen. 11