Hypothesis Testing: 1-Sample & 2-Sample Cases Perform the required hypothesis test for two population means. You may presume that the assumptions for using the pooled t-test are satisfied. 1) 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. 1) Sample 1 (women) Sample 2 (men) x 1 = 11.3 x 2 = 17.0 s 1 = 4.3 s 2 = 4.8 n n 1 = 14 2 = 17 Do the data provide sufficient evidence to conclude that the mean time for women is less than the mean time for men? Perform a pooled t-test at the 5% significance level. Explanation: Comparing independent populations, standard deviations assumed equal. Perform a nonpooled t-test to compare the two population means. You may assume that the samples are independent and that the variable under consideration is normally distributed on both populations. The two population standard deviations cannot be assumed to be equal. 2) A researcher was interested in comparing the resting pulse rate of people who exercise regularly and people who do not exercise regularly. Independent random samples of 16 people aged 30-40 who do not exercise regularly (sample 1) and 12 people aged 30-40 who do exercise regularly (sample 2) were selected and the resting pulse rate of each person was measured. The summary statistics are as follows. Sample 1 Sample 2 x 1 = 72.7 x 2 = 69.7 s = 10.9 s = 8.2 1 2 n = 16 n = 12 1 2 Do the data provide sufficient evidence to conclude that the mean resting pulse rate of people who do not exercise regularly is greater than the mean resting pulse rate of people who exercise regularly? Perform a nonpooled t-test at the 2.5% significance level. Explanation: Comparing independent populations, standard deviations not assumed equal. 1 2) Use a paired t-test to perform the required hypothesis test for two population means. Assume that dependent samples have been selected from the two populations and that the differences are normally distributed. 3) The table below shows the weights of seven subjects before and after following a particular diet for two months. 3) Subject A B C D E F G Before 179 157 163 195 188 170 158 After 172 148 161 200 174 172 146 Do the data suggest that the diet is effective in reducing weight? Perform a paired t-test at the 1% significance level. Explanation: Hypothesis test of dependent or matched samples. Perform the required chi-square hypothesis test. Preliminary data analyses and other information indicate that it is reasonable to assume that the variable under consideration is normally distributed. 4) Heights of men aged 25 to 34 have a standard deviation of 2.9 in. Use a 0.05 significance level to test the claim that the heights of women aged 25 to 34 have a different standard deviation. The heights (in inches) of 16 randomly selected women aged 25 to 34 are listed below. (Note: s = 2.279 in.) 62.13 65.09 64.18 66.72 63.09 61.15 67.50 64.65 63.80 64.21 60.17 68.28 66.49 62.10 65.73 64.72 Explanation: Chi-Square Test for Differences in Population Standard Deviations 2 4) Perform the required hypothesis test for two population standard deviations. Assume that independent samples have been randomly selected from the two populations and that the variable under consideration is normally distributed on both populations. 5) A researcher is interested in comparing the amount of variation in womenʹs scores on a certain test with the amount of variation in menʹs scores on the same test. Random samples of 11 men and 13 women yielded the following scores. 5) Men: 72, 60, 52, 87, 66, 74, 95, 50, 81, 70, 72 Women: 70, 78, 62, 96, 75, 68, 41, 74, 80, 47, 73, 94, 65 Do the data provide sufficient evidence to conclude that there is less variation in menʹs scores than in womenʹs scores? Perform an F-test at the 5% significance level. (Note: s = 13.754 and s = 15.588) 1 2 Explanation: Perform an F-Test for two population standard deviations Perform a hypothesis test for a population proportion using the critical value approach. 6) An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 237 fathers from Littleton yielded 104 who did not help with child care. Do the data provide sufficient evidence to conclude that in Littleton the proportion is higher than 0.34? Use a 0.05 significance level. Explanation: Hypothesis Test for Population Proportion 3 6)