370 Chapter 8 Confidence Intervals EXAMPLE 8.12 Constructing a confidence interval A food chemist analyzed the calorie content for a popular type of chocolate cookie. Following are the numbers of calories in a sample of eight cookies. 113 114 111 116 115 120 118 116 Find a 98% confidence interval for the mean number of calories in this type of cookie. Solution We check the assumptions. We have a simple random sample. Because the sample size is small, the population must be approximately normal. We check this with a dotplot of the data. 110 111 112 113 114 115 116 117 118 119 120 121 There is no evidence of strong skewness, and no outliers. Therefore, we may proceed. Step 1: Find the sample mean and sample standard deviation. We compute the mean and standard deviation of the sample values. We obtain ̄x = 115.375 s = 2.8253 Step 2: Find the number of degrees of freedom and the critical value t��∕2. The number of degrees of freedom is n − 1 = 8 − 1 = 7. Using Table A.3, we find that the critical value corresponding to a level of 98% is t��∕2 = 2.998. Step 3: Compute the margin of error. The margin of error is t��∕2 s√ n We substitute t��∕2 = 2.998, s = 2.8253, and n = 8 to obtain t��∕2 s√ n = 2.998 2.8253 √ 8 = 2.9947 Step 4: Construct the confidence interval. The 98% confidence interval is given by ̄x − t��∕2 s√ n < �� < ̄x + t��∕2 s√ n 115.375 − 2.9947 < �� < 115.375 + 2.9947 112.4 < �� < 118.4 Note that we round the final result to one decimal place, because the sample values were whole numbers. Step 5: Interpret the result. We are 98% confident that the mean number of calories per cookie is between 112.4 and 118.4. EXAMPLE 8.13 Constructing a confidence interval The General Social Survey is a survey of opinions and lifestyles of U.S. adults, conducted by the National Opinion Research Center at the University of Chicago. A sample of 123 people aged 18–22 reported the number of hours they spent on the Internet in an average week. The sample mean was 8.20 hours, with a sample standard deviation of 9.84 hours. Assume this is a simple random sample from the population of people aged 18–22 in the United States. Construct a 95% confidence interval for ��, the population mean number of hours per week spent on the Internet by people aged 18–22 in the United States. Solution We check the assumptions. We have a simple random sample. Now either the sample size must be greater than 30, or the population must be approximately normal. Since the sample size is n = 123, the assumptions are met.
navidi_monk_elementary_statistics_2e_ch7-9
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