Chapter 11: Q.11.51 (page 460)

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.

Expert verified

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

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

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

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

$α = 0.05$

It find that $z_{α / 2} = z_{0.05 / 2}$

$= 1.96$

The range of confidence for $p$ is form

$p^{'} - z_{α / 2} \sqrt{\frac{p^{'} (1 - p^{'})}{n}} to p^{'} + z_{α / 2} \sqrt{\frac{p^{'} (1 - p^{'})}{n}}$

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

$0.076 \pm 1.960 (0.0119)$

$0.076 \pm 0.0233$

$(0.0527, 0.0993)$

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

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