Question

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes \(n\) is the large-sample \(z\) test appropriate: a. \(H_{0}: \pi=.2, n=25\) b. \(H_{0}: \pi=.6, n=210\) c. \(H_{0}: \pi=.9, n=100\) d. \(H_{0}: \pi=.05, n=75\)

Short answer

The large-sample \(z\) test is appropriate for the hypotheses \(a: H_{0}: \pi=.2, n=25\), \(b: H_{0}: \pi=.6, n=210\) and \(c: H_{0}: \pi=.9, n=100\). Hypothesis \(d: H_{0}: \pi=.05, n=75\) does not meet the criteria, so the \(z\) test would not be suitable.

Step-by-step solution

Check Conditions for First Hypothesis

For the first hypothesis \(H_{0}: \pi=.2, n=25\), calculate \(np\) and \(n(1-p)\). In this case, \(np = 25*.2 = 5\) and \(n(1-p) = 25*(1-.2) = 20\). Both numbers are equal to or greater than 5, so the \(z\) test is applicable.

Check Conditions for Second Hypothesis

Examine the second hypothesis \(H_{0}: \pi=.6, n=210\). Calculate \(np\) which equals \(210*.6 = 126\) and \(n(1-p)\) which equals \(210*(1-.6) = 84\). Again, both numbers fulfill the criteria, hence the \(z\) test is valid.

Check Conditions for Third Hypothesis

Analyze the third hypothesis \(H_{0}: \pi=.9, n=100\). Here, \(np = 100*.9 = 90\) and \(n(1-p) = 100*(1-.9) = 10\). This hypothesis satisfies both conditions, meaning the \(z\) test can be applied.

Check Conditions for Fourth Hypothesis

Finally, look into the fourth hypothesis \(H_{0}: \pi=.05, n=75\). Calculate \(np\) which is \(75*.05 = 3.75\) and \(n(1-p)\) which is \(75*(1-.05) = 71.25\). For this hypothesis, \(np < 5\) does not meet the required criteria, thus the \(z\) test would not be appropriate.