Question

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.

Short answer

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

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

c. Part (b) is referenced.

Step-by-step solution

Given information

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.

95% confidence interval for the proportion of women with cosmetic dermatitis from using eye shadow

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 $\left(0.0422,0.141\right)$.

95% confidence interval for the proportion of women with cosmetic dermatitis from using mascara

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 $\left(0.0628,0.1372\right)$.

checking which group is referenced

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.