Question

Smallpox Vaccine. ABCNEWS.com published the results of a poll that asked U.S. adults whether they would get a smallpox shot if it were available. Sampling, data collection, and tabulation were done by TNS Intersearch of Horsham, Pennsylvania. When the risk of the vaccine was described in detail, $4$ in $10$ of those surveyed said they would take the smallpox shot. According to the article, "the results have a three-point margin of error" (for a $0.95$ confidence level). Use the information provided to obtain a $95\%$ confidence interval for the percentage of all U.S. adults who would take a smallpox shot, knowing the risk of the vaccine.

Short answer

The confidence interval for the percentage of all adults who would take a smallpox vaccination may be $95\%$confidence, indicating a risk of $0.37$to $0.43$for the vaccine.

Step-by-step solution

Given Information

$p=$percentage of all adults who would take a smallpox shot indicates the risk of the vaccine.

Here $x=4$

and $n=10$.

Explanation

The confidence interval for $p$:

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

Where

$z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\text{Margin of error}$

$\hat{p}=\frac{x}{n}$

$=\frac{4}{10}$

$=0.40$

The confidence interval for $p$:

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

$=0.40\pm 0.03\text{[margin of error is given as}3-\mathrm{point}]$$\text{i.e,}0.37\text{and}0.43$

$\text{i.}e,37\%\text{and}43\%$