Questions: What does it mean when you calculate a 95% confidence interval? a) The process you used will capture the true mean 95% of the time, in the long run b) You can be "95% confident" that your interval will include the population mean c) You can be "5% confident" that your interval will not include the population mean d) All of the above statements are true

Solution Steps

To determine the Z-score for a 95% confidence level, we use the formula:

\[ Z = \text{PPF}\left(1 - \frac{1 - 0.95}{2}\right) = \text{PPF}(0.975) = 1.96 \]

Using the Z-score and the specified standard deviation (\(\sigma = 10.0\)) and margin of error (\(E = 2.0\)), we calculate the required sample size (\(n\)) using the formula:

\[ \text{Sample Size} = \left(\frac{Z \cdot \sigma}{E}\right)^2 = \left(\frac{1.96 \cdot 10.0}{2.0}\right)^2 = 96.0365 \approx 97 \]

Thus, the calculated sample size is:

\[ \text{Calculated Sample Size} = 97 \]

The margin of error (\(E\)) can be calculated using the formula:

\[ E = \frac{Z \cdot \sigma}{\sqrt{n}} = \frac{1.96 \cdot 10.0}{\sqrt{97}} \approx 1.99 \]

Thus, the calculated margin of error is:

\[ \text{Calculated Margin of Error} = 1.99 \]

To compute the confidence interval for the mean, we use the formula:

\[ \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}} = 50.0 \pm 1.96 \cdot \frac{10.0}{\sqrt{97}} \]

Calculating the interval gives:

\[ \text{Confidence Interval} = (50.0 - 1.96 \cdot \frac{10.0}{\sqrt{97}}, 50.0 + 1.96 \cdot \frac{10.0}{\sqrt{97}}) = (48.01, 51.99) \]

Final Answer