Math215 - Statistical Concepts Names: _______________________________ Confidence Intervals Worksheet Use your TI/84 calculator to practice finding confidence intervals for a population mean and a population proportion! First review the following worked examples: 1. Example of a Confidence Interval for the mean, , of a population with a known: Suppose a random sample of 240 credit card bills showed an average balance of $1250. Also assume that the population standard deviation is $350. Find a 95% confidence interval for the population mean credit card balance. To access the confidence interval choices, press STAT and select TESTS. The confidence interval choices are found in items 7 through B. Select 7:ZInterval because the population standard deviation, , is known. In this example, summary statistics are given, so select the STATS option for input. Enter the value of , the value of x and the sample size n. Use 0.95 for the C-Level. Highlight Calculate and press ENTER. 2. Example of a Confidence Interval for the mean, , of a population with an unknown: In a test of an exercise program that is designed for weight loss, 40 individuals participated in a randomized trial to see how effective the program is. After one month in this exercise program, the mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb. Find a 95% confidence interval for the mean weight loss of all adults who follow the exercise program for weight loss. To access the confidence interval choices, press STAT and select TESTS. The confidence interval choices are found in items 7 through B. Select 8:TInterval because the population standard deviation, , is unknown. In this example, summary statistics are given, so we will select the STATS option for input. We enter the value of x , s and n .Use 0.95 for the C-Level. Highlight Calculate and press ENTER. 1 130314 Math215 - Statistical Concepts Names: _______________________________ 3. Example of a Confidence Interval for a population proportion, ^p : The local NPR station of a major city in the mid-west wants to find the percent of its listening population that gives donations to the station. 360 randomly selected listeners were surveyed, and it was found that 145 listeners made contributions to the station. Find a 95% confidence interval for the proportion of all listeners in that city who also made donations to the station. To find a confidence interval for a proportion, press the STAT key and use option A:1-PropZlnt under TESTS. Notice that the normal z-distribution will be used. The sample size needed to construct a confidence interval for n= ^p (1 ^p )[ n=0.25[ z 2 m z 2 m if ^p is available. if ^p is unavailable. ^p with margin of error m: 2 130314 Math215 - Statistical Concepts Names: _______________________________ The sample size needed to constrcut a confidence interval for n=[ with margin of error m: z 2 . m Now, it is your turn to practice these skills. Each team should complete one section of the following worksheet. Each section includes two parts. Team ONE Part I Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 95 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60. Answer the following questions: 1. 2. 3. 4. Construct a 95% confidence interval using a sample size of 50, then of 100, then of 1,000. How did changing the sample size affect the size of the interval? What is the error of the estimate for each of these sample sizes? How large of a sample is required for the error of the estimate to be within +.04 of the population proportion? 5. How large would the sample have to be if the coach of the team were not available? Show your work. Be prepared to present your analysis to the class. Part II In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2. 1. Construct a 95% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture. 2. Interpret this interval in the context of this problem. 3 130314 Math215 - Statistical Concepts Names: _______________________________ 3. Construct a 95% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment. 4. Interpret this interval in the context of this problem. 5. Compare the two confidence intervals. Do the results suggest the effectiveness of one method over the other? 4 130314 Math215 - Statistical Concepts Names: _______________________________ Confidence Intervals Worksheet Team TWO Part I. Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 90 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60. Answer the following questions: 1. 2. 3. 4. Construct a 90% confidence interval using a sample size of 50, then of 100, then of 1,000. How did changing the sample size affect the size of the interval? What is the error of the estimate for each of these sample sizes? How large of a sample is required for the error of the estimate to be within +.04 of the population proportion? 5. How large would the sample have to be if the coach of the team were not available? Show your work. Be prepared to present your analysis to the class. Part II. In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2. 1. Construct a 97% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture. 2. Interpret this interval in the context of this problem. 3. Construct a 97% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment. 4. Interpret this interval in the context of this problem. 5. Compare the two confidence intervals. Do the results suggest the effectiveness of one method over the other? 5 130314 Math215 - Statistical Concepts Names: _______________________________ Confidence Intervals Worksheet Team THREE Part I. Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 75 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60. Answer the following questions: 1. 2. 3. 4. Construct a 75% confidence interval using a sample size of 50, then of 100, then of 1,000. How did changing the sample size affect the size of the interval? What is the error of the estimate for each of these sample sizes? How large of a sample is required for the error of the estimate to be within +.04 of the population proportion? 5. How large would the sample have to be if the coach of the team were not available? Show your work. Be prepared to present your analysis to the class. Part II. In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2. 1. Construct a 90% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture. 2. Interpret this interval in the context of this problem. 3. Construct a 90% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment. 4. Interpret this interval in the context of this problem. 5. Compare the two confidence intervals. Do the results suggest the effectiveness of one method over the other? 6 130314 Math215 - Statistical Concepts Names: _______________________________ Confidence Intervals Worksheet Team FOUR Part I. Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 93 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60. Answer the following questions: 1. 2. 3. 4. Construct a 93% confidence interval using a sample size of 50, then of 100, then of 1,000. How did changing the sample size affect the size of the interval? What is the error of the estimate for each of these sample sizes? How large of a sample is required for the error of the estimate to be within +.04 of the population proportion? 5. How large would the sample have to be if the coach of the team were not available? Show your work. Be prepared to present your analysis to the class. Part II. In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2. 1. Construct a 99% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture. 2. Interpret this interval in the context of this problem. 3. Construct a 99% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment. 4. Interpret this interval in the context of this problem. 5. Compare the two confidence intervals. Do the results suggest the effectiveness of one method over the other? 7 130314 Math215 - Statistical Concepts Names: _______________________________ Confidence Intervals Worksheet Team FIVE Part I. Suppose the coach of the football team wants to estimate the proportion of the population of fans who support his current starter lineup. The coach wants the estimate to be .04 of the true proportion. Assume a 99 percent level of confidence. The coach estimated the proportion supporting the current starter lineup to be .60. Answer the following questions: 1. 2. 3. 4. Construct a 99% confidence interval using a sample size of 50, then of 100, then of 1,000. How did changing the sample size affect the size of the interval? What is the error of the estimate for each of these sample sizes? How large of a sample is required for the error of the estimate to be within +.04 of the population proportion? 5. How large would the sample have to be if the coach of the team were not available? Show your work. Be prepared to present your analysis to the class. Part II. In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture and 80 subjects were given a sham treatment. The numbers of migraine attacks for acupuncture treatment group had a mean of 1.8 and a standard deviation of 1.4. The numbers of migraine attacks for the sham treatment group had a mean of 1.6 and a standard deviation of 1.2. 1. Construct an 85% confidence interval estimate of the mean number of migraine attacks for those treated with acupuncture. 2. Interpret this interval in the context of this problem. 3. Construct an 85% confidence interval estimate of the mean number of migraine attacks for those given a sham treatment. 4. Interpret this interval in the context of this problem. 5. Compare the two confidence intervals. Do the results suggest the effectiveness of one method over the other? 8 130314