Question

For which of the following sample sizes would the sampling distribution of \(\hat{p}\) be approximately normal when $$ \begin{array}{rl} p= & 0.2 ? \text { When } p=0.8 ? \text { When } p=0.6 ? \\ n=10 & n=25 \\ n=50 & n=100 \end{array} $$

Short answer

For \(p = 0.2\) and \(p = 0.8\), the sampling distribution of \(\hat{p}\) would be approximately normal for \(n > 25\). For \(p = 0.6\), the distribution would be approximately normal for \(n > 20\). Thus, sample sizes of 25, 50, and 100 could feasibly result in an approximately normal distribution.

Step-by-step solution

Evaluate for p = 0.2

First, substitute \(p = 0.2\) into conditions \(np > 5\) and \(n(1 - p) > 5\). This will result in the conditions \(n > 25\) and \(n > 6.25\). Therefore, you need a sample size bigger than 25 for the sampling distribution to be approximately normal.

Evaluate for p = 0.8

Substitute \(p = 0.8\) into the same conditions. The result will be \(n > 6.25\) and \(n > 25\). Again, choosing the larger, we find that the distribution will be approximately normal for samples of more than 25.

Evaluate for p = 0.6

Now take \(p = 0.6\) and substitute it into \(np > 5\) and \(n(1 - p) > 5\). The conditions now are \(n > 8.33\) and \(n > 20\). So, for this probability, the sample needs to be larger than 20 for it to be approximately normal.