EXAMPLE 8.5

Comparing two sunblock lotions. Your company produces a sunblock lotion designed to protect the skin from both UVA and UVB exposure to the sun. You hire a company to compare your product with the product sold by your major competitor. The testing company exposes skin on the backs of a sample of 20 people to UVA and UVB rays and measures the protection provided by each product. For 13 of the subjects, your product provided better protection, while for the other 7 subjects, your competitor’s product provided better protection. Do you have evidence to support a commercial claiming that your product provides superior UVA and UVB protection? For the data we have n = 20 subjects and X = 13 successes. The parameter p is the proportion of people who would receive superior UVA and UVB protection from your product. To answer the claim question, we test

H₀: p = 0.5

H_a: p ≠ 0.5

The expected numbers of successes (your product provides better protection) and failures (your competitor’s product provides better protection) are 20 × 0.5 = 10 and 20 × 0.5 = 10. Both are at least 10, so we can use the z test. The sample proportion is

The test statistic is

From Table A, we find P(Z < 1.34) = 0.9099, so the probability in the upper tail is 1 − 0.9099 = 0.0901. The P-value is the area in both tails, P = 2 × 0.0901 = 0.1802.

We conclude that the sunblock testing data are compatible with the hypothesis of no difference between your product and your competitor’s product (, z = 1.34, P = 0.18). The data do not support your proposed advertising claim.