Question

Independent random samples of 180 observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 104 successes, and sample 2 had 113 successes.

Suppose that, for practical reasons, you know that

p₁

cannot be larger than

p₂.

Test the appropriate hypothesis using α = 0.10.

Given: H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) < 0

Solve:

Find the test statistic. (Round your answer to two decimal places.)

z = ??

Find the rejection region. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

z>??

z<??

Answer 1

For sample 1, we have that the sample size is N; = 180, the number of favorable cases is Xi = 104, so then the sample proportion is = = 104 = 0.5778 For sample 2, we have that the sample size is Ng = 180, the number of favorable cases is Xo = 113, so then the sample proportion is pe = = = 0.6278 The value of the pooled proportion is computed as p= *** = 10:113 -0.6028 Also, the given significance level is a = 0.10 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: P = P2 Ha:P <P 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 a = 0.10, and the critical value for a left-tailed test is ze = -1.28 The rejection region for this left-tailed test is R={:::< -1.28} (3) Test Statistics The Z-statistic is computed as follows: P -P2 0.5778 -0.6278 VP(1-2)(1/n +1/na) 40.6028 - (1 – 0.6028)(1/180 +1/180)

So fom the above calculations

Answer

• test statistic = -0.97
• rejection region : reject H0 if z < -1.28