Question

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.2, n=25\) b. \(H_{0}: p=0.6, n=200\) c. \(H_{0}: p=0.9, n=100\) d. \(H_{0}: p=0.05, n=75\)

Short answer

The large-sample z test is appropriate for hypothesis A, B and C but not for D.

Step-by-step solution

Test for Hypothesis A

For Hypothesis A, we have \(p=0.2\) and \(n=25\). Check if both \(np = 25*0.2 = 5\) and \(n(1-p) = 25*0.8 = 20\) are greater than or equal to 5. Here, both values satisfy the condition, hence the large-sample z test is appropriate.

Test for Hypothesis B

For Hypothesis B, we have \(p=0.6\) and \(n=200\). Check if both \(np = 200*0.6 = 120\) and \(n(1-p) = 200*0.4 = 80\) are greater than or equal to 5. Here, both values satisfy the condition, hence the large-sample z test is appropriate.

Test for Hypothesis C

For Hypothesis C, we have \(p=0.9\) and \(n=100\). Check if both \(np = 100*0.9 = 90\) and \(n(1-p) = 100*0.1 = 10\) are greater than or equal to 5. Here, both values satisfy the condition, hence the large-sample z test is appropriate.

Test for Hypothesis D

For Hypothesis D, we have \(p=0.05\) and \(n=75\). Check if both \(np = 75*0.05 = 3.75\) and \(n(1-p) = 75*0.95 = 71.25\) are greater than or equal to 5. Here, \(np < 5\), hence the large-sample z test is not appropriate.