Questions: (a) The minimum sample size required to construct a 99% confidence interval is 107 soccer balls. (Round up to the nearest whole number.) (b) The 99% confidence interval for a sample size of 117 is ( . the confidence interval. it seem possible that the population mean could be less than 27.7 inches because values below 27.7 inches fall (Round to two decimal places as needed.)

Solution Steps

To determine the minimum sample size required to construct a \(99\%\) confidence interval, we use the formula:

\[ Z = \text{PPF}\left(1 - \frac{1 - 0.99}{2}\right) = \text{PPF}(0.995) = 2.5758 \]

The sample size \(n\) is calculated as follows:

\[ n = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2 = \left(\frac{2.5758 \cdot 1.5}{0.5}\right)^2 = 59.7141 \approx 60 \]

Thus, the minimum sample size required is:

\[ \boxed{n = 60} \]

For a sample size of \(n = 117\) and a sample mean \(\bar{x} = 28.0\), the confidence interval is calculated using the formula:

\[ \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}} \]

Substituting the values:

\[ 28.0 \pm 2.5758 \cdot \frac{1.5}{\sqrt{117}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 2.5758 \cdot \frac{1.5}{\sqrt{117}} \approx 0.36 \]

Thus, the confidence interval is:

\[ (28.0 - 0.36, 28.0 + 0.36) = (27.64, 28.36) \]

The \(99\%\) confidence interval for a sample size of \(117\) is:

\[ \boxed{(27.64, 28.36)} \]

To determine if it seems possible that the population mean could be less than \(27.7\) inches, we check the lower bound of the confidence interval:

\[ 27.64 < 27.7 \]

Since the lower bound of the confidence interval is less than \(27.7\), it does seem possible that the population mean could be less than \(27.7\) inches.

Final Answer