8.23 A manufacturing company produces electric insulators. You define the variable of interest as the strength of the insulators. If the insulators break when in use, a short circuit is likely. 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. You collect the force data for 30 insulators selected for the experiment and organize and store these data in Force: 1,870 1,728 1,656 1,610 1,634 1,784 1,522 1,696 1,592 1,662 1,866 1,764 1,734 1,662 1,734 1,774 1,550 1,756 1,762 1,866 1,820 1,744 1,788 1,688 1,810 1,752 1,680 1,810 1,652 1,736 Construct a 95% confidence interval estimate for the population mean force. What assumption must you make about the population distribution in order to construct the confidence interval estimate in (a)

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally.

The sample selected here is n = 30.

Thus, the sampling distribution of the sample mean will be normal.

Compute the sample mean and standard deviation as follows:

[tex]\bar x=\frac{1}{n}\sum x=\frac{1}{30}\times 51702=1723.4\\\\s=\sqrt{\frac{1}{n-1}\sum (x-\bar x)^{2}}=\sqrt{\frac{1}{30-1}\times 232561.2}=89.55[/tex]

Construct a 95% confidence interval estimate for the population mean force as follows:

[tex]CI=\bar x\pm z_{\alpha /2}\times\frac{s}{\sqrt{n}}[/tex]

[tex]=1723.4\pm 1.96\times\frac{89.55}{\sqrt{30}}\\\\=1723.4\pm 32.045\\\\=(1691.355, 1755.445)\\\\\approx (1691, 1755)[/tex]

Thus, the 95% confidence interval estimate for the population mean force is (1691, 1755).