Question

1. a study of 65 bolts of carpet showed that their average length was 78.2 yards. the standard deviation of the population is 4.6 yards. which of the following is the 99% confidence interval for the mean length per bolt of carpet?

Explanation:

Step1: Recall Confidence Interval Formula

For a population mean confidence interval when population standard deviation \(\sigma\) is known, the formula is \(\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the critical z - value, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Identify the Values

We know that \(\bar{x}=78.2\) yards, \(\sigma = 4.6\) yards, \(n = 65\), and for a 99% confidence interval, \(\alpha=1 - 0.99=0.01\), so \(\alpha/2=0.005\). The critical z - value \(z_{0.005}\) (from standard normal distribution table) is approximately 2.576.

Step3: Calculate the Margin of Error

First, calculate \(\frac{\sigma}{\sqrt{n}}\): \(\frac{4.6}{\sqrt{65}}\approx\frac{4.6}{8.0623}\approx0.5705\). Then, the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=2.576\times0.5705\approx1.47\).

Step4: Calculate the Confidence Interval

The lower limit is \(\bar{x}-E=78.2 - 1.47 = 76.73\) and the upper limit is \(\bar{x}+E=78.2+1.47 = 79.67\). So the 99% confidence interval is \((76.73,79.67)\) (approximate values, depending on the precision of \(z\) - value and calculations).

Answer:

The 99% confidence interval for the mean length per bolt of carpet is approximately \(\boldsymbol{(76.73, 79.67)}\) yards (the values may vary slightly depending on the precision of the critical z - value and intermediate calculations). If we use more precise calculation for \(\frac{4.6}{\sqrt{65}}\): \(\sqrt{65}\approx8.062257748\), \(\frac{4.6}{8.062257748}\approx0.57056\), \(2.576\times0.57056\approx1.47\), so the interval is \(78.2\pm1.47\), which is \((76.73, 79.67)\).