## Related questions with answers

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

Solution

VerifiedGiven:

$\begin{aligned} n&=\text{Sample size}=250 \\ x&=\text{Number of successes}=25 \\ c&=\text{Confidence coefficient}=95\%=0.95 \end{aligned}$

We need to determine a confidence interval about the population proportion $p$.

The confidence interval about the population proportion is $\hat{p}\pm E$ with $E=z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$.

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