Questions: Find the z* values based on a standard normal distribution for each of the following. (a) A 78% confidence interval for a proportion. Round your answer to two decimal places. ± z* = ± i (b) An 82% confidence interval for a slope. Round your answer to two decimal places. ± z* = ± i

Solution Steps

To find the critical value \( z^* \) for a 78% confidence interval, we first determine the significance level \( \alpha \):

\[ \alpha = 1 - 0.78 = 0.22 \]

Since this is a two-tailed test, we divide \( \alpha \) by 2:

\[ \frac{\alpha}{2} = \frac{0.22}{2} = 0.11 \]

Next, we find the critical z-value corresponding to \( 1 - 0.11 = 0.89 \) using the standard normal distribution:

\[ z^* = \text{invCDF}(0.89) \approx 1.23 \]

Thus, the critical value for the 78% confidence interval is:

\[ \pm z^* = \pm 1.23 \]

For the 82% confidence interval, we again calculate the significance level \( \alpha \):

\[ \alpha = 1 - 0.82 = 0.18 \]

Dividing \( \alpha \) by 2 gives:

\[ \frac{\alpha}{2} = \frac{0.18}{2} = 0.09 \]

We then find the critical z-value corresponding to \( 1 - 0.09 = 0.91 \):

\[ z^* = \text{invCDF}(0.91) \approx 1.34 \]

Thus, the critical value for the 82% confidence interval is:

\[ \pm z^* = \pm 1.34 \]

Final Answer