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Chapter 9: Q.11.49 (page 460)
Find and interpret 95 % confidence interval for the proportion of all US adults who never clothes-shop online.
There is 95%confidence interval for the proportion of all US adults who never clothes-shop online between 0.291 and 0.349.
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Testing Claims About Proportions. In Exercises 7鈥�22, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Cell Phones and Handedness A study was conducted to investigate the association between cell phone use and hemispheric brain dominance. Among 216 subjects who prefer to use their left ear for cell phones, 166 were right-handed. Among 452 subjects who prefer to use their right ear for cell phones, 436 were right-handed (based on data from 鈥淗emi- spheric Dominance and Cell Phone Use,鈥� by Seidman et al., JAMA Otolaryngology鈥擧ead & Neck Surgery, Vol. 139, No. 5). We want to use a 0.01 significance level to test the claim that the rate of right-handedness for those who prefer to use their left ear for cell phones is less than the rate of right-handedness for those who prefer to use their right ear for cell phones. (Try not to get too confused here.)
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
Testing Claims About Proportions. In Exercises 7鈥�22, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Is Echinacea Effective for Colds? Rhinoviruses typically cause common colds. In a test of the effectiveness of Echinacea, 40 of the 45 subjects treated with Echinacea developed rhinovirus infections. In a placebo group, 88 of the 103 subjects developed rhinovirus infections (based on data from 鈥淎n Evaluation of Echinacea Angustifolia in Experimental Rhinovirus Infections,鈥� by Turner et al., New England Journal of Medicine, Vol. 353, No. 4). We want to use a 0.05 significance level to test the claim that Echinacea has an effect on rhinovirus infections.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, does Echinacea appear to have any effect on the infection rate?
Degrees of Freedom
For Example 1 on page 431, we used df\( = \)smaller of\({n_1} - 1\)and\({n_2} - 1\), we got\(df = 11\), and the corresponding critical values are\(t = \pm 2.201.\)If we calculate df using Formula 9-1, we get\(df = 19.063\), and the corresponding critical values are\( \pm 2.093\). How is using the critical values of\(t = \pm 2.201\)more 鈥渃onservative鈥� than using the critical values of\( \pm 2.093\).
In Exercises 5鈥�20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with 鈥淭able鈥� answers based on Table A-3 with df equal to the smaller of\({n_1} - 1\)and\({n_2} - 1\).)
Are Male Professors and Female Professors Rated Differently?
a. Use a 0.05 significance level to test the claim that two samples of course evaluation scores are from populations with the same mean. Use these summary statistics: Female professors:
n = 40, \(\bar x\)= 3.79, s = 0.51; male professors: n = 53, \(\bar x\) = 4.01, s = 0.53. (Using the raw data in Data Set 17 鈥淐ourse Evaluations鈥� will yield different results.)
b. Using the summary statistics given in part (a), construct a 95% confidence interval estimate of the difference between the mean course evaluations score for female professors and male professors.
c. Example 1 used similar sample data with samples of size 12 and 15, and Example 1 led to the conclusion that there is not sufficient evidence to warrant rejection of the null hypothesis.
Do the larger samples in this exercise affect the results much?
Variation of Heights Use the sample data given in Exercise 3 鈥淗eights鈥� and test the claim that women and men have heights with the same variation. Use a 0.05 significance level.
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