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