Questions: You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ=34.8. You would like to be 99% confident that your estimate is within 4 of the true population mean. How large of a sample size is required?

You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ=34.8. You would like to be 99% confident that your estimate is within 4 of the true population mean. How large of a sample size is required?

Transcript text: You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $\sigma=34.8$. You would like to be $99 \%$ confident that your estimate is within 4 of the true population mean. How large of a sample size is required?

Solution

Solution Steps

Step 1: Determine the Z-Score

To find the required sample size, we first need to determine the Z-score corresponding to a $99\%$ confidence level. The Z-score is calculated using the formula:

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

Step 2: Calculate the Sample Size

Next, we use the Z-score to calculate the sample size using the formula:

\[ \text{Sample Size} = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2 \]

Substituting the values:

\[ \text{Sample Size} = \left(\frac{2.5758 \cdot 34.8}{4}\right)^2 \]

Calculating this gives:

\[ \text{Sample Size} = (2.5758 \cdot 8.7)^2 = (22.42626)^2 \approx 502.1953 \]

Rounding this value, we find:

\[ \text{Sample Size} \approx 503.0 \]