Question

find the critical value for a two - tailed test with $\alpha=0.07$ and $n = 36$.
a. $\pm1.81$
b. $\pm1.96$
c. $\pm2.575$
d. $\pm2.33$

Explanation:

Step1: Determine the significance level for each tail

Since it's a two - tailed test, divide the significance level \(\alpha = 0.07\) by 2. So, \(\frac{\alpha}{2}=\frac{0.07}{2}=0.035\).

Step2: Use the standard normal distribution (z - distribution)

For a large sample size (\(n = 36\), \(n\geq30\) so we can use the z - distribution). We need to find \(z\) such that \(P(Z>z)=0.035\) and \(P(Z < - z)=0.035\). Looking up in the standard normal table (or using a calculator with a normal distribution function), the \(z\) - value corresponding to an area of \(1 - 0.035=0.965\) (since \(P(Z\leq z)=1 - P(Z > z)\)) is approximately \(z = 1.81\). But wait, let's check again.
Another way: Using the formula for the critical value of the z - test. The critical values \(z_{\alpha/2}\) for a two - tailed test. Using a standard normal table (z - table) or a statistical software.
If we use a calculator (e.g., TI - 84: invNorm\((1-\frac{\alpha}{2},0,1)\) and invNorm\((\frac{\alpha}{2},0,1)\)).
\(1-\frac{0.07}{2}=0.965\), invNorm\((0.965,0,1)\approx1.81\) and invNorm\((0.035,0,1)\approx - 1.81\)

Answer:

A. \(\pm1.81\)