Eye shadow, mascara, and nickel allergies. Pigmented makeup products like mascara and eye shadow may contain metal (e.g., nickel) allergens. Is a nickel allergy more likely to occur in women who report cosmetic dermatitis from using eye shadow or mascara? This was the question of interest in a paper published in the Journal of the European Academy of Dermatology and Venereology (June 2010). 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. Compute a 95% confidence interval for the proportion of women with cosmetic dermatitis from using eye shadow who have a nickel allergy. Interpret the result.

b. Compute a 95% confidence interval for the proportion of women with cosmetic dermatitis from using mascara who have a nickel allergy. Interpret the result.

c. Suppose you are informed that the true proportion with a nickel allergy for one of the two groups (eye shadow or mascara) is .12. Can you determine which group is referenced? Explain.

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 $100\left(1-\mathrm{\alpha }\right)\%$confidence interval for p (proportion) is,

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

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

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

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

The 95% confidence interval is,

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

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 $100\left(1-\mathrm{\alpha }\right)\%$confidence interval for p (proportion) is,

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

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

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

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

The 95% confidence interval is,

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

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.