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In a sample of 131 women with cosmetic dermatitis from using eye shadow, 12 were diagnosed with a nickel allergy.

In a sample of 250 women with cosmetic dermatitis from using mascara, 25 were diagnosed with a nickel allergy.

a.

The sample proportion is the point estimator of the population proportion p.

The proportion of women with cosmetic dermatitis using eye shadow is,

$\begin{array}{rcl}\hat{\mathrm{p}}& =& \frac{x}{n}\\ & =& \frac{12}{131}\\ & =& 0.0916\end{array}$

Then the level of $100\left(1-\mathrm{伪}\right)\%$confidence interval for p (proportion) is,

$\hat{\mathrm{p}}卤{\mathrm{z}}_{\frac{\mathrm{伪}}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}$

For a 95% confidence interval, the value of$\frac{\mathrm{伪}}{2}$ is,

$\begin{array}{rcl}100(1-\mathrm{伪})\%& =& 95\%\\ (1-\mathrm{伪})& =& 0.95\end{array}$

For, $\mathrm{伪}=0.05\mathrm{and}\frac{\mathrm{伪}}{2}=0.025$

The 95% confidence interval is,

$\begin{array}{rcl}\hat{\mathrm{p}}卤{\mathrm{z}}_{\frac{\mathrm{伪}}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}& =& 0.0916卤1.960\sqrt{\frac{0.0916\left(1-0.0916\right)}{131}}\left[\mathrm{From}\mathrm{Standard}\mathrm{Normal}\mathrm{Table}\right]\\ & =& (0.0916卤1.960\sqrt{0.000635})\\ & =& \left(0.0916卤0.0494\right)\\ & =& \left(0.0422,0.141\right)\end{array}$

Therefore, a 95% confidence interval for the proportion of women with cosmetic dermatitis from using eye shadow is $\left(0.0422,0.141\right)$.

b.

The sample proportion is the point estimator of the population proportion p.

The proportion of women with cosmetic dermatitis using eye shadow is,

$\begin{array}{rcl}\hat{\mathrm{p}}& =& \frac{x}{n}\\ & =& \frac{25}{250}\\ & =& 0.1\end{array}$

Then the level of $100\left(1-\mathrm{伪}\right)\%$confidence interval for p (proportion) is,

$\hat{\mathrm{p}}卤{\mathrm{z}}_{\frac{\mathrm{伪}}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}$

For a 95% confidence interval, the value of$\frac{\mathrm{伪}}{2}$ is,

$\begin{array}{rcl}100(1-\mathrm{伪})\%& =& 95\%\\ (1-\mathrm{伪})& =& 0.95\end{array}$

For, $\mathrm{伪}=0.05\mathrm{and}\frac{\mathrm{伪}}{2}=0.025$

The 95% confidence interval is,

$\begin{array}{rcl}\hat{\mathrm{p}}卤{\mathrm{z}}_{\frac{\mathrm{伪}}{2}}\sqrt{\frac{\hat{\mathrm{p}}\left(1-\hat{\mathrm{p}}\right)}{\mathrm{n}}}& =& 0.1卤1.960\sqrt{\frac{0.1\left(1-0.1\right)}{250}}\left[\mathrm{From}\mathrm{Standard}\mathrm{Normal}\mathrm{Table}\right]\\ & =& (0.1卤1.960\sqrt{0.00036})\\ & =& \left(0.1卤0.0372\right)\\ & =& \left(0.0628,0.1372\right)\end{array}$

Therefore, a 95% confidence interval for the proportion of women with cosmetic dermatitis from using mascara is $\left(0.0628,0.1372\right)$.

c.

Suppose, the true proportion with a nickel allergy for two groups is 0.12.

Here, the 95% confidence interval for part (a) is large and the 95% confidence interval for part (b) is narrow.

So, part (b) is referenced.