Questions: The weights of all newborn babies are normally distributed with a mean of 8.1 pounds and a standard deviation of 3.1 pounds. Determine the sample size required to produce a standard deviation of x̄ equal to 0.41

Solution Steps

We are given the following parameters for the weights of newborn babies, which are normally distributed:

• Population mean (\( \mu \)) = 8.1 pounds
• Population standard deviation (\( \sigma \)) = 3.1 pounds
• Desired standard deviation of the sample mean (\( \sigma_{\bar{x}} \)) = 0.41 pounds

The standard deviation of the sample mean is related to the population standard deviation and the sample size by the formula:

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

To find the required sample size (\( n \)), we rearrange the formula:

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

Squaring both sides gives:

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

Substituting the known values into the equation:

\[ n = \left(\frac{3.1}{0.41}\right)^2 \]

Calculating the right-hand side:

\[ n = \left(7.561\right)^2 \approx 57.0 \]

Since the sample size must be a whole number, we round \( n \) to the nearest whole number:

\[ n \approx 57 \]

Final Answer