Questions: Find the sample size needed to estimate the population mean within 0.60, given sigma= 4 : (a) using a confidence level of 90% (b) using a confidence level of 96% Remember to round up to an integer. (a) at the 90% confidence level n= (b) at the 96% confidence level n=

Solution Steps

For a confidence level of \(90\%\), the Z-score is calculated as follows: \[ Z = \text{PPF}\left(1 - \frac{1 - 0.9}{2}\right) = \text{PPF}(0.95) = 1.6449 \]

Using the Z-score, the sample size \(n\) is determined by the formula: \[ n = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2 \] Substituting the values: \[ n = \left(\frac{1.6449 \cdot 4}{0.6}\right)^2 = 120.2464 \approx 121 \]

For a confidence level of \(96\%\), the Z-score is calculated as follows: \[ Z = \text{PPF}\left(1 - \frac{1 - 0.96}{2}\right) = \text{PPF}(0.98) = 2.0537 \]

Using the Z-score, the sample size \(n\) is determined by the formula: \[ n = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2 \] Substituting the values: \[ n = \left(\frac{2.0537 \cdot 4}{0.6}\right)^2 = 187.4615 \approx 188 \]

Final Answer