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BEREMC08_0131536869.QXD 1/24/05 8:15 PM Page 269 Thus, with 95% confidence, you conclude that the mean amount of all the sales invoices is between $104.53 and $116.01. The 95% confidence level indicates that if you selected all possible samples of 100 (something that is never done in practice), 95% of the intervals developed would include the population mean somewhere within the interval. The validity of this confidence interval estimate depends on the assumption of normality for the distribution of the amount of the sales invoices. With a sample of 100, the normality assumption is not overly restrictive and the use of the t distribution is likely appropriate. Example 8.3 further illustrates how you construct the confidence interval for a mean when the population standard deviation is unknown. EXAM P L E 8 .3 ESTIMATING THE MEAN FORCE REQUIRED TO BREAK ELECTRIC INSULATORS A manufacturing company produces electric insulators. If the insulators break when in use, you are likely to have a short circuit. To test the strength of the insulators, you carry out destructive testing to determine how much force is required to break the insulators. You measure force by observing how many pounds are applied to the insulator before it breaks. Table 8.2 lists thirty values from this experiment. FORCE Construct a 95% confidence interval estimate for the population mean force required to break the insulator. TABLE 8 .2 Force (in Pounds) Required to Break the Insulator 1,870 1,866 1,820 1,728 1,764 1,744 1,656 1,734 1,788 1,610 1,662 1,688 1,634 1,734 1,810 1,784 1,774 1,752 1,522 1,550 1,680 1,696 1,756 1,810 1,592 1,762 1,652 1,662 1,866 1,736 SOLUTION Figure 8.7 shows that the sample mean is X = 1,723.4 pounds and the sample standard deviation is S = 89.55 pounds. Using Equation (8.2) on page 268 to construct the confidence interval, you need to determine the critical value from the t table for an area of 0.025 in each tail with 29 degrees of freedom. From Table E.3, you see that t29 = 2.0452. Thus, using X = 1,723.4 , S = 89.55, n = 30, and t29 = 2.0452, X ± t n −1 S n = 1,723.4 ± ( 2.0452 ) 89.55 30 = 1,723.4 ± 33.44 1,689.96 ≤ µ ≤ 1,756.84 FIGUR E 8 .7 Minitab Confidence Interval Estimate for the Mean Amount of Force Required to Break Electric Insulators You conclude with 95% confidence that the mean force required for the population of insulators is between 1,689.96 and 1,756.84 pounds. The validity of this confidence interval estimate depends on the assumption that the force required is normally distributed. Remember, however, that you can slightly relax this assumption for large sample sizes. Thus, with a sample