Question

question 14 of 25
a sample with a sample proportion of 0.4 and which of the following sizes
will produce the widest 95% confidence interval when estimating the
population parameter?

a. 90

b. 120

c. 60

d. 75

Explanation:

Step1: Recall the formula for the margin of error

The margin of error \(E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion and \(n\) is the sample size. For a 95% confidence interval, \(z_{\alpha/2}\) is a constant (\(z_{0.025}=1.96\)) and \(\hat{p} = 0.4\), \(1-\hat{p}=0.6\). The margin of error is inversely proportional to \(\sqrt{n}\) (since \(E\propto\frac{1}{\sqrt{n}}\) when other factors are fixed).

Step2: Analyze the relationship between sample size and margin of error

As \(n\) decreases, \(\sqrt{n}\) decreases, and \(\frac{1}{\sqrt{n}}\) increases. So, a smaller sample size will give a larger margin of error.

Step3: Compare the sample sizes

We have sample sizes \(n_A = 90\), \(n_B=120\), \(n_C = 60\), \(n_D=75\). Since \(60<75<90<120\)

Answer:

C. 60