Question

You wish to test the following claim (HaHa) at a significance level of α=0.05α=0.05.

      Ho:p1=p2Ho:p1=p2
      Ha:p1>p2Ha:p1>p2

You obtain 62.7% successes in a sample of size n1=201n1=201 from the first population. You obtain 54.5% successes in a sample of size n2=618n2=618 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 =

Answer 1

For sample 1, we have that the sample size is N_1= 201,

the number of favorable cases is X_1 = 126,

so then the sample proportion is

For sample 2, we have that the sample size is N_2 = 618,

the number of favorable cases is X_2 = 337,

so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​=p2​

Ha: :p1​>p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05,

and the critical value for a right-tailed test is z_c = 1.64

The rejection region for this right-tailed test is

R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

z=2.026>zc​=1.64,

it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is p = 0.0214,

and since p = 0.0214<0.05,

it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected.

Therefore, there is enough evidence to claim that population proportion p_1 is greater than p_2​, at the 0.05 significance level.

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