Question

Independent random samples of sizes n₁ = 307 and n₂ = 309 were taken from two populations. In the first sample, 92 of the individuals met a certain criteria whereas in the second sample, 108 of the individuals met the same criteria.

Test the null hypothesis H0:p1=p2versus the alternative hypothesis HA:p1<p2.

a)  Calculate the z test statistic, testing the null hypothesis that the population proportions are equal.

Round your response to at least 3 decimal places.

    

b) What is the approximate value of the p-value?

Round your response to at least 3 decimal places.

   

c)  What is the appropriate conclusion that can be made?

		There is sufficient evidence to reject the null hypothesis at the 5% level of significance, but insufficienct evidence to reject the null hypothesis at the 10% level of significance.
		There is insufficient evidence to reject the null hypothesis at both the 5% and 10% levels of significance.
		There is sufficient evidence to reject the null hypothesis at the 10% level of significance, but insufficient evidence to reject the null hypothesis at the 5% level of significance.
		There is sufficient evidence to reject the null hypothesis at both the 10% and 5% levels of significance.

Answer 1

n₁ = 307 ,  n₂ = 309

x1= 92, x2= 108

Ho:p1​=p2​

Ha:p1​<p2​

a)  Calculate the z test statistic

Z = -1.321

test statistic = -1.321

b) Calculate the p-value

P-Value = P(Z < -1.321)

find P(Z < -1.321) using normal z table we get

P(Z < -1.321) = 0.0933

P-Value = 0.0933

c)

now if = 0.05

(P-Value = 0.0933) > ( = 0.05)

then null hypothesis is not rejected.

now if = 0.10

(P-Value = 0.0933) <  ( = 0.10)

then null hypothesis is rejected.

There is sufficient evidence to reject the null hypothesis at the 10% level of significance, but insufficient evidence to reject the null hypothesis at the 5% level of significance.