354 Chapter 8 Confidence Intervals Step 5: Interpret the result. We are 98% confident that the population mean score �� is between 63.81 and 70.79. Another way to say this is that we are 98% confident that �� is in the interval 67.30 ± 3.49. If we were to draw many different samples and use this method to construct the corresponding confidence intervals, then in the long run, 98% of the intervals would cover the true population mean ��. So unless we were quite unlucky in the sample we drew, the population mean �� will be between 63.81 and 70.79. Check Your Understanding 4. An IQ test was given to a simple random sample of 75 students at a certain college. The sample mean score was 105.2. Scores on this test are known to have a standard deviation of �� = 10. It is desired to construct a 90% confidence interval for the mean IQ score of students at this college. a. What is the point estimate? 105.2 b. Find the critical value. 1.645 c. Find the standard error. 1.1547 d. Find the margin of error. 1.90 e. Construct the 90% confidence interval. (103.3, 107.1) f. Is it likely that the population mean �� is greater than 100? Explain. Yes 5. The lifetime of a certain type of battery is known to be normally distributed with standard deviation �� = 20 hours. A sample of 50 batteries had a mean lifetime of 120.1 hours. It is desired to construct a 95% confidence interval for the mean lifetime for this type of battery. a. What is the point estimate? 120.1 b. Find the critical value. 1.96 c. Find the standard error. 2.8284 d. Find the margin of error. 5.54 e. Construct the 95% confidence interval. (114.6, 125.6) f. Is it likely that the population mean �� is greater than 130? Explain. No 6. In a survey of a simple random sample of students at a certain college, the sample mean time per week spent watching television was 18.3 hours and the margin of error for a 95% confidence interval was 1.2 hours. True or false: a. A 95% confidence interval for the mean number of hours per week spent watching television by students at this college is 17.1 < �� < 19.5. True b. Approximately 95% of the students at this university watch between 17.1 and 19.5 hours of television per week. False 7. Use the data in Exercise 4 to construct a 95% confidence interval for the mean IQ score. (102.9, 107.5) 8. Use the data in Exercise 5 to construct a 98% confidence interval for the mean lifetime for this type of battery. (113.5, 126.7) Answers are on page 366. Constructing confidence intervals with technology In Example 8.4, we found a 98% confidence interval for the mean test score, based on a sample size of n = 100, a sample mean of ̄x = 67.30, and a population standard deviation �� = 15. The following TI-84 Plus display presents the results.
navidi_monk_elementary_statistics_2e_ch7-9
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