Comparing Means

  • Michael O. Finkelstein
  • Bruce Levin
Part of the Statistics for Social and Behavioral Sciences book series (SSBS)


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 by
$$ {\left[\widehat{p}\left(1-\widehat{p}\right)\left({n}_1^{-1}+{n}_2^{-1}\right)\right]}^{1/2}, $$
where \( \widehat{p} \) 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.


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Copyright information

© Springer Science+Business Media, LLC 2015

Authors and Affiliations

  • Michael O. Finkelstein
    • 1
  • Bruce Levin
    • 2
  1. 1.New YorkUSA
  2. 2.Department of Biostatistics, Mailman School of Public HealthColumbia UniversityNew YorkUSA

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