390 Chapter 8 Confidence Intervals you believe that women are more effective at governing than men are?’’ A total of 2348 viewers answer the question, and 1247 of them answer ‘‘Yes.’’ Explain why these data should not be used to construct a confidence interval for the proportion of people who believe that women are more effective at governing than men are. Extending the Concepts Wilson’s interval: The small-sample method for constructing a confidence interval is a simple approximation of a more complicated interval known as Wilson’s interval. Let ̂p = x∕n. Wilson’s confidence interval for p is given by ̂p + 2 2n z��∕2 ± z��∕2 √ ̂p(1 − ̂p) n + z��∕2 2 4n2 1 + 2 n z��∕2 41. College-bound: In a certain high school, 9 out of 15 tenth-graders said they planned to go to college after graduating. Construct a 95% confidence interval for the proportion of tenth-graders who plan to attend college: a. Using Wilson’s method (0.357, 0.802) b. Using the small-sample method (0.357, 0.801) c. Using the traditional method (0.352, 0.848) 42. Comparing the methods: Refer to Exercise 41. a. Which of the three confidence intervals is the narrowest? Small-sample b. Does the small-sample method provide a good approximation to Wilson’s interval in this case? Yes c. Explain why the traditional interval is the widest of the three. 43. Approximation depends on the level: The small-sample method is a good approximation to Wilson’s method for all confidence levels commonly used in practice, but is best when z��∕2 is close to 2. Refer to Exercise 41. a. Use Wilson’s method to construct a 90% confidence interval, a 95% confidence interval, and a 99% confidence interval for the proportion of tenth-graders who plan to attend college. 90%: (0.392, 0.777); 95%: (0.357, 0.802); 99%: (0.296, 0.842) b. Use the small-sample method to construct a 90% confidence interval, a 95% confidence interval, and a 99% confidence interval for the proportion of tenth-graders who plan to attend college. 90%: (0.393, 0.765); 95%: (0.357, 0.801); 99%: (0.287, 0.871) c. For which level is the small-sample method the closest to Wilson’s method? Explain why this is the case. 95% Answers to Check Your Understanding Exercises for Section 8.3 1. a. 0.710 b. 0.647 < p < 0.773 c. Yes. We are 95% confident that the proportion who would improve their scores is between 0.647 and 0.773. Therefore, it is reasonable to conclude that the proportion is greater than 0.60. d. No. Because the sample contains only third-graders, it should not be used to construct a confidence interval for all elementary schoolchildren. 2. 151 3. 542 4. a. False b. True 5. a. False b. True c. False 6. 0.302 < p < 0.751 SECTION 8.4 Confidence Intervals for a Standard Deviation Objectives 1. Find critical values of the chi-square distribution 2. Construct confidence intervals for the variance and standard deviation of a normal distribution Objective 1 Find critical values of the chi-square distribution The Chi-Square Distribution Most confidence intervals constructed in practice are for means and proportions. However, when the population is normal, it is possible to construct confidence intervals for the standard deviation or variance. These confidence intervals are based on a distribution known as the chi-square distribution, denoted ��2. The symbol �� is the Greek letter chi CAUTION (pronounced ‘‘kigh’’; rhymes with sky). We will begin by describing this distribution. The methods of this section apply only for samples drawn from a normal distribution. If the distribution differs even slightly from normal, these methods should not be used. There are actually many different chi-square distributions, each with a different number of degrees of freedom. Figure 8.13 on page 391 presents chi-square distributions for several different degrees of freedom. There are two important points to notice: ∙ The chi-square distributions are not symmetric. They are skewed to the right. ∙ Values of the ��2 statistic are always greater than or equal to 0. They are never negative.
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