Question

For each of the following values of $\alpha$find the values of z for which $H_0:\left(p_1-p_2\right)=0$would be rejected in favor of $H_a:\left(p_1-p_2\right)<0$.

$$\begin{gathered} \left(a\right)\alpha =.01 \\ \left(b\right)\alpha =.025 \\ \left(c\right)\alpha =.05 \\ \left(d\right)\alpha =.10 \end{gathered}$$

Short answer

a. For $\alpha =.01$,the value of Zis -2.326.

b. For $\alpha =.025$,the value of Zis -1.96.

c. For $\alpha =.05$ ,the value of Zis -1.645.

d. For $\alpha =.10$,the value of Z is -1.28.

Step-by-step solution

Given information

We have to find critical values for the hypotheses

$H_0:p_1-p_2=0$

And

$H_a:p_1-p_2<0$

Definition of Critical value

The critical value is the cut-off value for the test statistic, which decides whether to reject the null hypothesis or not.

Calculating Critical value

Here

$\alpha :$The level of significance

$\alpha =.01$

Since we have a left-tailed test (as an alternative hypothesis is left-tailed)

Using the standard normal table, the critical value at the 1% significance level for the left-tailed test is -2.326

That is

$z=-2.326$.

The null hypothesis will be rejected for any test statistic value less than -2.326.