Question

a soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.16 inch. (a) determine the minimum sample size required to construct a 99% confidence interval for the population mean. assume the population standard deviation is 0.7 inch. (b) the sample mean is 27 inches. with a sample size of 137, a 99% level of confidence, and a population standard deviation of 0.7 inch, does it seem possible that the population mean could be less than 27.1 inches? explain. click here to view page 1 of the standard normal table. click here to view page 2 of the standard normal table. (a) the minimum sample size required to construct a 99% confidence interval is 128 soccer balls. (round up to the nearest whole number.) (b) the 99% confidence interval for a sample size of 137 is (, ). it seem possible that the population mean could be less than 27.1 inches because values below 27.1 inches fall the confidence interval. (round to two decimal places as needed.)

Explanation:

Step1: Find the z - value

For a 99% confidence interval, the significance level \(\alpha=1 - 0.99 = 0.01\). Then \(\frac{\alpha}{2}=0.005\). Looking up in the standard normal table, \(z_{\frac{\alpha}{2}}=z_{0.005} = 2.576\).

Step2: Calculate the margin of error formula for confidence interval

The formula for the confidence interval for the population mean when the population standard deviation \(\sigma\) is known is \(\bar{x}\pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}\). Here, \(\bar{x} = 27\), \(\sigma=0.7\), \(n = 137\).
The margin of error \(E=z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}=2.576\times\frac{0.7}{\sqrt{137}}\).
First, calculate \(\sqrt{137}\approx11.7047\). Then \(\frac{0.7}{\sqrt{137}}\approx\frac{0.7}{11.7047}\approx0.0598\). And \(E = 2.576\times0.0598\approx0.154\).

Step3: Calculate the confidence interval

The lower limit of the confidence interval is \(\bar{x}-E=27 - 0.154=26.85\).
The upper limit of the confidence interval is \(\bar{x}+E=27+0.154 = 27.15\).

Answer:

The 99% confidence interval for a sample size of 137 is \((26.85,27.15)\). It does seem possible that the population mean could be less than 27.1 inches because values below 27.1 inches fall within the confidence interval.