Cut-off share to reject a 25 percent preference claim

Question

A watchmaker asserts that at least a quarter of the public prefers its brand to a rival's. You ask $80$ randomly chosen people. Using a one-sided test at $\alpha=0.05$ with the normal approximation, below what sample proportion (to two decimals) would you reject the claim?

Short answer

About $0.17$ . Under $p=0.25$ the standard error is $\sqrt{0.25\cdot0.75/80}\approx0.0484$ , so rejection needs $\hat p<0.25-1.645\times0.0484\approx0.170$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Covariance and Correlation
Populations, Samples, and Sampling Distributions

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