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. \(H_{0}: p=0.8, n=40\) b. \(H_{0}: p=0.4, n=100\) c. \(H_{0}: p=0.1, n=50\) d. \(H_{0}: p=0.05, n=750\)

Short answer

The large-sample z test is only appropriate for scenarios b and d.

Step-by-step solution

Check for Scenario a

Given \(H_{0}: p=0.8, n=40\). So np = 0.8*40 = 32 and n(1-p) = 40*(1-0.8) = 8. Since n(1-p) < 10, the large sample z-test is not appropriate for scenario a.

Check for Scenario b

Given \(H_{0}: p=0.4, n=100\). So np = 0.4*100 = 40 and n(1-p) = 100*(1-0.4) = 60. Both np and n(1-p) are greater than 10, so the large sample z-test is appropriate for scenario b.

Check for Scenario c

Given \(H_{0}: p=0.1, n=50\). So np = 0.1*50 = 5 and n(1-p) = 50*(1-0.1) = 45. Since np < 10, the large sample z-test is not appropriate for scenario c.

Check for Scenario d

Given \(H_{0}: p=0.05, n=750\). So np = 0.05*750 = 37.5 and n(1-p) = 750*(1-0.05) = 712.5. Both np and n(1-p) are greater than 10, so the large sample z-test is appropriate for scenario d.