Shoe Statistics

A statistics question asked in Quant Developer round at Tower Research Capital

A shoe manufacturer offers rock climbing shoes in black, blue, and pink. Of the first 1000 shoes sold, 400 were black. Calculate the value of the appropriate test statistic to determine if customers have a preference for black rock climbing shoes. Assume random sampling, variance homogeneity, and that preference is approximately normally distributed.

Try to solve this problem yourself before moving on to the solution below

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Solution

We are testing the null hypothesis

\( H_0 : p = \frac{1}{3} \)

against the alternative hypothesis

\( H_a : p > \frac{1}{3} \)

The sample proportion

\( \hat{p} = \frac{400}{1000} = 0.4\)

for n=1000

The Z statistic can be calculated as:

\( Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} = \frac{0.40 - \frac{1}{3}}{\sqrt{\frac{\frac{1}{3}(1-\frac{1}{3})}{1000}}} \approx 4.47 \)

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😃 Over to you: Were you able to solve this?

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