Question

You wish to test the following claim ($H_a$) at a significance level of $\alpha =0.02$.

      $H_o:p_1=p_2$
      $H_a:p_1<p_2$

You obtain 380 successes in a sample of size $n_1=641$ from the first population. You obtain 179 successes in a sample of size $n_2=259$ from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = 

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value = 

The p-value is...

• less than (or equal to) $\alpha$

• greater than $\alpha$

This test statistic leads to a decision to...

• reject the null

• accept the null

• fail to reject the null

As such, the final conclusion is that...

• There is sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.

• There is not sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.

• The sample data support the claim that the first population proportion is less than the second population proportion.

• There is not sufficient sample evidence to support the claim that the first population proportion is less than the second population proportion.

Answer 1

Solution :

Given that,

This a left (One) tailed test.

Critical value of  the significance level is α = 0.02, and the critical value for a two-tailed test is

$Z_{c}$ = -2.054

$\hat p$ = x / n = 423 / 558 = 0.76

Test statistics

z = ($\hat p$ -  ) / $\sqrt{}$ *(1-) / n

= ( 0.76 - 0.77) / $\sqrt{}$ (0.77*0.23) / 558

= -0.561