366 Chapter 8 Confidence Intervals which has the form ̄x + z����∕ √ n, or a lower confidence bound, which has the form ̄x − z����∕ √ n. Note that the critical value is z�� rather than z��∕2. 67. Computers in the classroom: A simple random sample of 50 middle-school children participated in an experimental class designed to introduce them to computer programming. At the end of the class, the students took a final exam to assess their learning. The sample mean score was 78 points, and the population standard deviation is 8 points. Compute a lower 99% confidence bound for the mean score. 75.4 68. Charge it: A random sample of 75 charges on a credit card had a mean of $56.85, and the population standard deviation is $21.08. Compute an upper 95% confidence bound for the mean amount charged. $60.85 Answers to Check Your Understanding Exercises for Section 8.1 1. a. 1.645 b. 2.326 c. 2.81 d. 1.28 2. a. 95% b. 97% c. 80% d. 99.9% 3. a. 2.352 b. 1.030 c. 5.758 Tech: 5.757 d. 6.397 4. a. 105.2 b. 1.645 c. 1.1547 d. 1.90 e. 103.3 < �� < 107.1 f. Yes. We are 90% confident that �� is between 103.3 and 107.1, so it is likely that �� > 100. 5. a. 120.1 b. 1.96 c. 2.8284 d. 5.54 e. 114.6 < �� < 125.6 f. No. We are 95% confident that �� is between 114.6 and 125.6, so it is not likely that �� > 130. 6. a. True b. False 7. 102.9 < �� < 107.5 8. 113.5 < �� < 126.7 9. The confidence interval does not contain the value 100. Therefore, it is not likely that the claim that �� = 100 is true. 10. a. 2.576 b. 101.2143 < �� < 103.4911 11. a. 1.96 b. 1.568 c. 95% 12. a. 2.576 b. 2.061 c. 99% 13. 97 14. 136 15. Yes, the probability that a 95% confidence interval constructed by an appropriate method will cover the true value is 0.95. 16. No. Once a specific confidence interval is constructed, there is no probability attached to it. SECTION 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown Objectives 1. Describe the properties of the Student’s t distribution 2. Construct confidence intervals for a population mean when the population standard deviation is unknown Objective 1 Describe the properties of the Student’s t distribution The Student’s t Distribution In Section 8.1, we showed how to construct a confidence interval for the mean �� of a normal population when the population standard deviation �� is known. The confidence interval is ̄x ± z��∕2 �� √ n The critical value is z��∕2 because the quantity ̄x − �� ��∕ √ n has a normal distribution. In practice, it is more common that �� is unknown. When we don’t know the value of ��, we replace it with the sample standard deviation s. However, we cannot then use z��∕2 as the critical value, because the quantity ̄x − �� s∕ √ n does not have a normal distribution. One reason is that s is, on the average, a bit smaller than ��, so replacing �� with s tends to increase the magnitude. Another reason is that s is random whereas �� is constant, so replacing �� with s increases the spread. The distribution of this quantity is called the Student’s t distribution. It was discovered in 1908 by William Sealy Gosset, a statistician who worked for the Guinness Brewing Company in Dublin, Ireland. The management at Guinness considered the
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