Questions: Suppose the standard deviation of the ages of all Florida panthers is 13.7 years. Let x̄ be the mean age for a sample of a certain number of Florida panthers. What sample size will give the standard deviation of x̄ equal to 0.8 years? Round the solution up to the nearest whole number, if necessary. n = □

Solution Steps

We are provided with the following information:

• The standard deviation of the ages of all Florida panthers is \( \sigma = 13.7 \) years.
• The desired standard deviation of the sample mean is \( \sigma_{\bar{x}} = 0.8 \) years.

To find the required sample size \( n \), we use the formula for the standard deviation of the sampling distribution of the sample mean:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

Rearranging this formula to solve for \( n \):

\[ n = \left(\frac{\sigma}{\sigma_{\bar{x}}}\right)^2 \]

Substituting the known values into the formula:

\[ n = \left(\frac{13.7}{0.8}\right)^2 \]

Calculating the value:

\[ n = \left(17.125\right)^2 = 293.281625 \]

Since the sample size must be a whole number, we round up to the nearest whole number:

\[ n = 294 \]

Final Answer