Using a confidence interval to perform a test

A hypothesis test at significance level \(\alpha\) for

can be performed in the following way.

  1. Find a \((1 - \alpha)\) confidence interval for \(\theta\).
  2. Reject H0 if the confidence interval does not include \(\theta_0\).

T-shirt sizes

A retail clothing outlet has collected the following data from random sampling of invoices of T-shirts over the past month.

  Small Medium Large XL Total
North Island 2 15 24 9 50
South Island 4 17 23 6 50

We earlier found a 95% confidence interval for the proportion of Small T-shirts sold in the North Island using the pivot

\[ \frac{x - n\pi}{\sqrt{n \pi(1 - \pi)}} \;\;\underset{\text{approx}}{\sim} \;\; \NormalDistn(0,1) \]

where \(x = 2\) and \(n = 50\). This was rearranged to get

\[ -1.96 \;\;\lt\;\; \frac{2 - 50\pi}{\sqrt{50 \pi(1 - \pi)}} \;\;\lt\;\; 1.96 \]

with the resulting confidence interval

\[ 0.011 \;\;\lt\;\; \pi \;\;\lt\;\; 0.135 \]

In order to perform a hypothesis test for whether 15% of T-shirts sold in the North Island are of the Small size,

we note that 0.15 is not in the 95% confidence interval. This means that we would reject the null hypothesis in a test at the 5% significance level.