7. Comparing Means

We have seen that a hypothesis test concerning the mean of a normal population can be carried out by standardizing the sample mean, i.e., by subtracting the population mean specified by the null hypothesis and dividing by the standard error of the mean, \( \sigma /\sqrt{n} \). The resulting statistic is a z-score, and may be referred to the standard normal distribution for tail area probabilities. In large samples we have also used a consistent estimate for σ when it is unknown. For example, in the two-sample binomial problem, the unknown standard error of the difference between proportions under the null hypothesis p ₁ = p ₂, \( {\left[p\left(1-p\right)\left({n}_1^{-1}+{n}_2^{-1}\right)\right]}^{1/2} \), is estimated bywhere \( \widehat{p} \) is the pooled proportion \( \widehat{p}=\left({n}_1{\widehat{p}}_1+{n}_2{\widehat{p}}_2\right)/\left({n}_1+{n}_2\right) \). Use of the standard normal distribution in this case is justified as an approximation in large samples by the central limit theorem.