Problem 58E

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.

Answer:

Step 1 of 4:

Given that, 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.

That is, and