Question

We survey a random sample of American River College students and ask if they drink coffee on a regular basis. The 90% confidence interval for the proportion of all American River College students who drink coffee on a regular basis is (0.262, 0.438). What will be true about the 95% confidence interval for these data? Group of answer choices The 95% confidence interval is narrower than the 90% confidence interval. The 95% confidence interval is wider than the 90% confidence interval. The two intervals will have the same width. It is impossible to say which interval will be wider.

Answer 1

Answer:

Correct option:

The 95% confidence interval is wider than the 90% confidence interval.

Step-by-step explanation:

The (1 - α)% confidence interval for the population proportion is:

$CI=\hat p\pm z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}}$

The width of this interval is:

$W=UL-LL\\=2\times z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}}$

The width of the interval is directly proportional to the critical value.

The critical value of a distribution is based on the confidence level.

Higher the confidence level, higher will be critical value.

The z-critical value for 95% and 90% confidence levels are:

$90\%: z_{\alpha /2}=z_{0.05}=1.645\\95\%: z_{\alpha /2}=z_{0.025}=1.96$

*Use a z-table.

The critical value of z for 95% confidence level is higher than that of 90% confidence level.

So the width of the 95% confidence interval will be more than the 90% confidence interval.

Thus, the correct option is:

"The 95% confidence interval is wider than the 90% confidence interval."