z statistic for a favourite among three shoe colours

Question

Climbing shoes come in three colours: black, blue and pink. Of the first $1000$ pairs sold, $400$ were black. Compute the test statistic for checking whether buyers prefer black (i.e. its share is above one third), using the normal approximation.

Short answer

About $4.47$ . Under $p=\tfrac13$ , $z=\dfrac{0.4-\tfrac13}{\sqrt{\tfrac13\cdot\tfrac23/1000}}\approx\dfrac{0.0667}{0.0149}\approx4.47$ .

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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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