Question

A. Suppose we test the proportions of people who like having a cup of coffee early in the morning for two populations: H0 : p1= p2 vs Ha : p1< p2. The sample sizes for these two population are n1= n2=400 and the numbers of people who like coffee are x1 =160 and x2=200 respectively. What is the value of the test statistics?

a. -2.8571        b. -2.8427

c. -2.8866        d. -2.8284

B. Suppose we take Type I error level to be α = 0.01 in the above population proportion testing. What is your conclusion?

a. Reject H0.   b. Accept H0 c. Do not know.

Answer 1

For sample 1, we have that the sample size is N_1= 400 , the number of favorable cases is X_1 = 160 , so then the sample proportion is

For sample 2, we have that the sample size is N_2 = 400 , the number of favorable cases is X_2 = 200 , so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.01.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p_1 = p_2

Ha: p_1 < p_2

This corresponds to a left-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=0.01, and the critical value for a left-tailed test is z_c = -2.33

(3) Test Statistics

The z-statistic is computed as follows:

b. -2.8427

(4) Decision about the null hypothesis

Since it is observed that z = -2.843 < z_c = -2.33, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0022, and since p = 0.0022 < 0.01, 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 less than p_2​, at the 0.01 significance level.

a. Reject H0