Download Thus, with 95% confidence, you conclude that the mean

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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