Confidence Intervals and Sample Size

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lu49076_ch07.qxd 5/20/2003 3:16 PM Page 362 362 Chapter 7 Confidence Intervals and Sample Size measurements was 2.6. Find the 95% confidence interval of the true variance. 15. A random sample of 15 snowmobiles was selected, and the lifetime (in months) of the batteries was measured. The variance of the sample was 8.6. Find the 90% confidence interval of the true variance. 7–38 Statistics Today Data Analysis Would You Change the Channel?—Revisited 16. The heights of 28 police officers from a large-city police force were measured. The standard deviation of the sample was 1.83 inches. Find the 95% confidence interval of the standard deviation of the heights of the officers. The estimates given in the survey are point estimates. However, since the margin of error is stated to be 3 percentage points, an interval estimate can easily be obtained. For example, if 45% of the people changed the channel, then the confidence interval of the true percentages of people who changed channels would be 42% � p � 48%. The article fails to state whether a 90%, 95%, or some other percentage was used for the confidence interval. Using the formula given in Section 7–4, a minimum sample size of 1068 would be needed to obtain a 95% confidence interval for p, as shown. Use pˆ and qˆ as 0.5, since no value is known for pˆ . n � pˆqˆ � z ��2 E � 2 � (0.5)(0.5) � � 1067.1 � 1068 1.96 0.03� 2 The Data Bank is found in Appendix D, or on the World Wide Web by following links from www.mhhe.com/math/stat/bluman/. 1. From the Data Bank choose a variable, find the mean, and construct the 95% and 99% confidence intervals of the population mean. Use a sample of at least 30 subjects. Find the mean of the population, and determine whether it falls within the confidence interval. 2. Repeat Exercise 1, using a different variable and a sample of 15. 3. Repeat Exercise 1, using a proportion. For example, construct a confidence interval for the proportion of individuals who did not complete high school. 4. From Data Set III in Appendix D, select a sample of 30 values and construct the 95% and 99% confidence intervals of the mean length in miles of major North American rivers. Find the mean of all the values, and determine if the confidence intervals contain the mean. 5. From Data Set VI in Appendix D, select a sample of 20 values and find the 90% confidence interval of the mean of the number of acres. Find the mean of all the values, and determine if the confidence interval contains the mean. 6. From Data Set XIV in Appendix D, select a sample of 20 weights for the Pittsburgh Steelers, and find the proportion of players who weigh over 250 pounds. Construct a 95% confidence interval for this proportion. Find the proportion of all Steelers who weigh more than 250 pounds. Does the confidence interval contain this value?
lu49076_ch07.qxd 5/20/2003 3:16 PM Page 363 Chapter Quiz Determine whether each statement is true or false. If the statement is false, explain why. 1. Interval estimates are preferred over point estimates since a confidence level can be specified. 2. For a specific confidence interval, the larger the sample size, the smaller the maximum error of estimate will be. 3. An estimator is consistent if, as the sample size decreases, the value of the estimator approaches the value of the parameter estimated. 4. To determine the sample size needed to estimate a parameter, one must know the maximum error of estimate. Select the best answer. 5. When a 99% confidence interval is calculated instead of a 95% confidence interval with n being the same, the maximum error of estimate will be a. Smaller b. Larger c. The same d. It cannot be determined 6. The best point estimate of the population mean is a. The sample mean b. The sample median c. The sample mode d. The sample midrange 7. When the population standard deviation is unknown and sample size is less than 30, what table value should be used in computing a confidence interval for a mean? a. z b. t c. Chi-square d. None of the above Complete the following statements with the best answer. 8. A good estimator should be _______, _______, and _______. 9. The maximum difference between the point estimate of a parameter and the actual value of the parameter is called _______. 10. The statement “The average height of an adult male is 5 feet 10 inches” is an example of a(n) _______ estimate. 11. The three confidence intervals used most often are the _______%, _______%, and _______%. 12. A random sample of 49 shoppers showed that they spend an average of $23.45 per visit at the Saturday Mornings Bookstore. The standard deviation of the sample Section 7–6 Summary 363 was $2.80. Find the 90% confidence interval of the true mean. 13. An irate patient complained that the cost of a doctor’s visit was too high. She randomly surveyed 20 other patients and found that the mean amount of money they spent on each doctor’s visit was $44.80. The standard deviation of the sample was $3.53. Find the 95% confidence interval of the population mean. Assume the variable is normally distributed. 14. The average weight of 40 randomly selected school buses was 4150 pounds. The standard deviation was 480 pounds. Find the 99% confidence interval of the true mean weight of the buses. 15. In a study of 10 insurance sales representatives from a certain large city, the average age of the group was 48.6 years and the standard deviation was 4.1 years. Find the 95% confidence interval of the population mean age of all insurance sales representatives in that city. 16. In a hospital, a sample of 8 weeks was selected, and it was found that an average of 438 patients were treated in the emergency room each week. The standard deviation was 16. Find the 99% confidence interval of the true mean. Assume the variable is normally distributed. 17. For a certain urban area, it was found that in a sample of 4 months, an average of 31 burglaries occurred each month. The standard deviation was 4. Find the 90% confidence interval of the true mean number of burglaries each month. 18. A university dean wishes to estimate the average number of hours that freshmen study each week. The standard deviation from a previous study is 2.6 hours. How large a sample must be selected if he wants to be 99% confident of finding whether the true mean differs from the sample mean by 0.5 hour? 19. A researcher wishes to estimate within $300 the true average amount of money a county spends on road repairs each year. If she wants to be 90% confident, how large a sample is necessary? The standard deviation is known to be $900. 20. A recent study of 75 workers found that 53 people rode the bus to work each day. Find the 95% confidence interval of the proportion of all workers who rode the bus to work. 21. In a study of 150 accidents that required treatment in an emergency room, 36% involved children under 6 years of age. Find the 90% confidence interval of the true proportion of accidents that involve children under the age of 6. 7–39
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Confidence Intervals and Sample Size

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