Question

Appropriate use of the interval $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. a. \(n=50\) and \(\hat{p}=0.30\) b. \(n=50\) and \(\hat{p}=0.05\) c. \(n=15\) and \(\hat{p}=0.45\) d. \(n=100\) and \(\hat{p}=0.01\)

Short answer

The sample size is large enough for the cases where \(n=50\) and \(\hat{p}=0.30\), and \(n=15\) and \(\hat{p}=0.45\). The sample sizes are not large enough for the cases where \(n=50\) and \(\hat{p}=0.05\), and \(n=100\) and \(\hat{p}=0.01\).

Step-by-step solution

For n=50 and \(\hat{p}\)=0.30

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(50*0.30 = 15\) and \(50*(1-0.30) = 35\). Because both of these values exceed 5, the sample size in this scenario is large enough.

For n=50 and \(\hat{p}\)=0.05

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(50*0.05 = 2.5\) and \(50*(1-0.05)=47.5\). Since the value of \(n\hat{p}\) is less than 5, the sample size in this scenario is not large enough.

For n=15 and \(\hat{p}\)=0.45

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(15*0.45=6.75\) and \(15*(1-0.45)=8.25\). Both of these are greater than 5, indicating that the sample size in this scenario is large enough.

For n=100 and \(\hat{p}\)=0.01

Calculate values of \(n\hat{p}\) and \(n*(1-\hat{p})\). Here, the calculations yield \(100*0.01=1\) and \(100*(1-0.01)=99\). The value of \(n\hat{p}\) is less than 5, which indicates that the sample is not large enough in this case.