Questions: The finishing time for cyclists in a race are normally distributed with a population standard deviation of 11 minutes and an unknown population mean. If a random sample of 22 cyclists is taken and results in a sample mean of 135 minutes, use Excel to find a 98 % confidence interval for the population mean. Round your final answer to two decimal places.

Solution Steps

For a confidence level of \( 98\% \), the Z-Score (Z) is determined to be:

\[ Z = 2.3263 \]

The margin of error (E) is calculated using the formula:

\[ E = \frac{Z \times \sigma}{\sqrt{n}} \]

Substituting the known values:

\[ E = \frac{2.3263 \times 11}{\sqrt{22}} \approx 5.4558 \]

The confidence interval for the population mean is calculated as follows:

\[ \text{Lower Bound} = \bar{x} - E = 135 - 5.4558 \approx 129.5442 \] \[ \text{Upper Bound} = \bar{x} + E = 135 + 5.4558 \approx 140.4558 \]

Rounding these values to two decimal places gives:

\[ \text{Lower Bound} \approx 129.54, \quad \text{Upper Bound} \approx 140.46 \]

Final Answer