Question

find each of the following. enter your answers rounded to at least two decimal places.
part 1 of 5
$z_{\alpha / 2}$ for the $95 \\%$ confidence interval
$z_{\alpha / 2}=\square$
part 2 of 5
$z_{\alpha / 2}$ for the $96 \\%$ confidence interval
$z_{\alpha / 2}=\square$

Explanation:

Step1: Calculate $\alpha$ for 95% confidence interval

For a 95% confidence interval, $\alpha = 1 - 0.95=0.05$. Then $\frac{\alpha}{2}=\frac{0.05}{2} = 0.025$.

Step2: Find $z_{\alpha/2}$ using standard normal distribution

We look up the $z -$ value in the standard normal distribution table (or use a calculator with a normal - distribution function). The $z -$ value corresponding to a right - tail area of $0.025$ is $z_{0.025}$. Using a standard normal table or a calculator (e.g., in Excel: =NORM.S.INV(1 - 0.025)), we get $z_{0.025}=1.96$.

Step3: Calculate $\alpha$ for 96% confidence interval

For a 96% confidence interval, $\alpha = 1 - 0.96 = 0.04$. Then $\frac{\alpha}{2}=\frac{0.04}{2}=0.02$.

Step4: Find $z_{\alpha/2}$ using standard normal distribution

We look up the $z -$ value corresponding to a right - tail area of $0.02$. Using a calculator (e.g., in Excel: =NORM.S.INV(1 - 0.02)), we get $z_{0.02}\approx2.05$.

Answer:

For the 95% confidence interval: $z_{\alpha/2}=1.96$
For the 96% confidence interval: $z_{\alpha/2}\approx2.05$