Section 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown 371 Step 1: Find the sample mean and sample standard deviation. These are given as ̄x = 8.20 and s = 9.84. Step 2: Find the number of degrees of freedom and the critical value t��∕2. The number of degrees of freedom is n − 1 = 123 − 1 = 122. Since this number of degrees of freedom does not appear in Table A.3, we use the next smaller value in the table, which is 100. The critical value corresponding to a level of 95% is t��∕2 = 1.984. Step 3: Compute the margin of error. The margin of error is t��∕2 s√ n We substitute t��∕2 = 1.984, s = 9.84, and n = 123 to obtain t��∕2 s√ n = 1.984 9.84 √ 123 = 1.7603 Step 4: Construct the confidence interval. The 95% confidence interval is given by ̄x − t��∕2 s√ n < �� < ̄x + t��∕2 s√ n 8.20 − 1.7603 < �� < 8.20 + 1.7603 6.44 < �� < 9.96 Note that we round the final result to two decimal places, because the value of ̄x was given to two decimal places. Step 5: Interpret the result. We are 95% confident that the mean number of hours per week spent on the Internet by people 18–22 years old is between 6.44 and 9.96. 0 10 20 30 40 50 20 15 10 5 0 Frequency Hours per week Figure 8.12 Note that in Example 8.13, the sample standard deviation of 9.84 is larger than the sample mean of 8.20. Since the minimum possible time to spend on the Internet is 0, the smallest sample value is less than one standard deviation below the mean. This indicates that the sample is fairly skewed. Figure 8.12 confirms this. Even though the sample is skewed, the t statistic is still appropriate, because the sample size of 123 is large. Constructing confidence intervals with technology CAUTION Confidence intervals constructed using technology may differ from those constructed by hand due to rounding. The differences are never large enough to matter. The following TI-84 Plus display presents the results of Example 8.13. The display is fairly straightforward. The quantity Sx is the sample standard deviation s. The TI-84 Plus uses the exact number of degrees of freedom, 122, rather than 100 as we did in the solution to Example 8.13. This does not make a difference when the answer is rounded to two decimal places. Note that the confidence level (95%) is not given in the display. The following MINITAB output presents the results of the same example. ���������������� ���������� �� �������� ������ �� ���������� ���������� �� ���������� ���� �������� �� ���������� ����~ ���� .�� ����������, �� ����������/ The output is fairly straightforward. Going from left to right, ‘‘N’’ represents the sample size, ‘‘Mean’’ is the sample mean ̄x, and ‘‘StDev’’ is the sample standard deviation s. The
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