398 Chapter 8 Confidence Intervals 16. Credit card debt: In a survey of 1118 U.S. adults conducted in 2012 by the Financial Industry Regulatory Authority, 626 said they always pay their credit cards in full each month. Construct a 95% confidence interval for the proportion of U.S. adults who pay their credit cards in full each month. (0.531, 0.589) 17. Windy place: Mt. Washington, New Hampshire, is one of the windiest places in the United States. Wind speed measurements on a simple random sample of 50 days had a sample mean of 45.01 mph. Assume the population standard deviation is �� = 25.6 mph. Construct a 95% confidence interval for the mean wind speed on Mt. Washington. (37.91, 52.11) 18. An apple a day: Following are the numbers of grams of sugar per 100 grams of apple in a random sample of 6 Red Delicious apples. Assume the population is normally distributed. 12.0 12.6 13.1 13.5 12.1 10.5 Construct a 95% confidence interval for the standard deviation of the number of grams of sugar. (0.66, 2.58) 19. Pneumonia: In a simple random sample of 1500 patients admitted to the hospital with pneumonia, 145 were under the age of 18. Construct a 99% confidence interval for the proportion of pneumonia patients who are under the age of 18. (0.077, 0.116) 20. College tuition: A simple random sample of 35 colleges and universities in the United States had a mean tuition of $18,702 with a standard deviation of $10,653. Construct a 95% confidence interval for the mean tuition for all colleges and universities in the United States. (15,043, 22,361) Answers to Check Your Understanding Exercises for Section 8.5 1. The parameter is the population standard deviation. The 95% confidence interval is 2.55 < �� < 5.49. 2. The parameter is the population mean. The 99% confidence interval is 5.51 < �� < 9.11. 3. The parameter is the population proportion. The 99% confidence interval is 0.113 < p < 0.327. 4. The parameter is the population mean. The 95% confidence interval is 15.2 < �� < 19.6. Chapter 8 Summary Section 8.1: In this section, we presented the basic ideas behind confidence intervals. We learned that a point estimate is a single number that is used to estimate the value of an unknown parameter. For example, the sample mean ̄x is a point estimate of the population mean ��. The standard error of a point estimate tells us roughly how far from the true value the point estimate is likely to be. We multiply the standard error by a critical value to obtain the margin of error. By adding and subtracting the margin of error from the point estimate, we obtain a confidence interval. The level of a confidence interval is the proportion of samples for which the confidence interval will contain the true value. If the sample size √ is large (n > 30) or if the population is approximately normal, then the confidence interval for the population mean is ̄x ± z∕2��∕ n if the population mean �� is known. ��Section 8.2: When the population is approximately normal, or the sample size is large (n > 30), we can use the Student’s t distribution to construct a confidence interval for a population mean √ �� when the population standard deviation is unknown. Let s be the sample standard deviation. The confidence interval is ̄x ± t∕2s∕ n if �� is unknown. ��Section 8.3: In this section, we learned to construct confidence intervals for population proportions. The assumptions are that the individuals in the population can be divided into two categories, that the sample contains at least 10 individuals in each category, and that the population is at least 20 times as large as the sample. We denote the sample size by n and the number of individuals in the sample who fall into √ the specified category by x. When the assumptions are met, the confidence interval for the population proportion p is ̂p ± z��∕2 ̂p(1 − ̂p)∕n, where ̂p = x∕n. A small-sample method can also be used. In the small-sample method, we define ̃p = x + 2 n + 4 . The confidence interval is ̃p ± z��∕2 √ ̃p(1 − ̃p) n + 4 . The small-sample confidence interval is actually valid for any sample size. Section 8.4: When the population is almost exactly normal, it is possible to find a confidence interval for the variance or standard deviation of the population using the chi-square distribution. This method is very sensitive to the assumption of normality, and should not be used unless it is certain that the population is almost exactly normal. The lower bound of the confidence interval is (n − 1)s2∕��2 ��∕2 and the upper bound is (n − 1)s2∕��2 1−��∕2. Section 8.5: We have learned to construct confidence intervals for a population mean, a population proportion, and a population standard deviation or variance. There are two methods for constructing a confidence interval for a population mean, the z method and the t method. The method to use depends on whether the population standard deviation �� is known. Vocabulary and Notation chi-square distribution 390 critical value 349 point estimate 348 confidence interval 350 degrees of freedom 367 standard error 349 confidence level 350 margin of error 349 Student’s t distribution 366
navidi_monk_elementary_statistics_2e_ch7-9
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