Question

Asthmatics and Sulfites. In the article "Explaining au Unusual Allergy, " appearing on the Everyday Health Network, Dr. A. Feldweg explained that allergy to sulfites is usually seen in patients with asthma. The typical reaction is a sudden increase in asthma symptoms after eating a food containing sulfites. Studies are performed to estimate the percentage of the nation's $10$million asthmatics who are allergic to sulfites. In one survey, $38$of $500$randomly selected U.S. asthmatics were found to be allergic to sulfites. Find and interpret a $95\%$confidence interval for the proportion, $p$, of all U.S. asthmatics who are allergic to sulfites.

Short answer

The confidence interval for the proportion of all asthmatics in the United States who are allergic to sulfites falls between $0.0527$ and $0.0993$.

Step-by-step solution

Given Information

The sample proportion $p^{\text{'}}=\frac{x}{n}$

$\frac{38}{500}=0.076$

Explanation

When $x$and $n-x$are both $5$the one-proportion z-interval technique is appropriate since the difference is bigger.

It has a $95\%$confidence interval,

that is

$\alpha =0.05$

It find that $z_{\alpha /2}=z_{0.05/2}$

$=1.96$

The range of confidence for $p$is form

$p^{\text{'}}-z_{\alpha /2}\sqrt{\frac{p^{\text{'}}\left(1-p^{\text{'}}\right)}{n}}\text{to}{p}^{\text{'}}+z_{\alpha /2}\sqrt{\frac{p^{\text{'}}\left(1-p^{\text{'}}\right)}{n}}$

$\text{i.e.}0.076\pm 1.960\sqrt{\frac{0.076(1-0.076)}{500}}$

$0.076\pm 1.960(0.0119)$

$0.076\pm 0.0233$

$(0.0527,0.0993)$

Thus, the $95\%$the confidence range for the fraction of all asthmatics in the United States who are allergic to sulfites is $(0.0527,0.0993)$.