Question

You wish to test the following claim (Ha) at a significance level of α=0.001.

      Ho:p1=p2
      Ha:p1<p2

You obtain 421 successes in a sample of size n1=684 from the first population. You obtain 409 successes in a sample of size n2=589 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 critical value for this test? (Report answer accurate to three decimal places.)
critical value =

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

The test statistic is...

• in the critical region
• not in the critical region

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

H0: p1 = p2
Ha: p1 < p2

n1 = 684
n2 = 589
p1cap = 0.6155
p2cap = 0.6944

Here the significance level, 0.001. This is right tailed test; hence rejection region lies to the right. -3.09 i.e. P(z < -3.09) = 0.001

Reject H0 if test statistic, z < -3.09

pooled proportion, pcap = (421 + 409)/(684 + 589) = 0.6520

SE = sqrt(pcap * (1-pcap) * (1/n1 + 1/n2))
SE = sqrt(0.6520 * (1-0.6520) * (1/684 + 1/589))
SE = 0.0268

Test statistic,
z = (p1cap - p2cap)/SE
z = (0.6155 - 0.6944)/0.0268
z = -2.944

critical value = -3.090

The test statistic is not in the critical region

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