Abstract
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
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.
The original version of the book was revised. An erratum can be found at DOI 10.1007/978-1-4419-5985-0_15
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is because for fixed marginal rate, \( \widehat{p} \), z is a standardized hypergeometric variable with zero mean, \( E\left[z\Big|\widehat{p}\right]=0 \)], and unit variance, \( \mathrm{V}\mathrm{a}\mathrm{r}\left(z\Big|\widehat{p}\right)=1 \). The identity \( \mathrm{V}\mathrm{a}\mathrm{r}(z)=E\ \mathrm{V}\mathrm{a}\mathrm{r}\left(z\Big|\widehat{p}\right)+\mathrm{V}\mathrm{a}\mathrm{r}E\left[z\Big|\widehat{p}\right] \) then shows that Var(z) = 1 also unconditionally.
- 2.
An F variate with df 1 and df 2 degrees of freedom has mean df 2/(df 2 – 1) and variance \( 2\cdot d{f}_2^2\cdot \left(d{f}_1+d{f}_2-2\right)/\left[d{f}_1\cdot {\left(d{f}_2-2\right)}^2\cdot \left(d{f}_2\hbox{--} 4\right)\right] \).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Finkelstein, M.O., Levin, B. (2015). Comparing Means. In: Statistics for Lawyers. Statistics for Social and Behavioral Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5985-0_7
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5985-0_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-5984-3
Online ISBN: 978-1-4419-5985-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)