Chapter 9: Hypothesis Testing with Two Samples
Chapter 9 Review
9.1 Chapter Review
Two population means from independent samples where the population standard deviations are not known
- Random Variable: [latex]{\overline{X}}_{1}-{\overline{X}}_{2}[/latex] = the difference of the sampling means
- Distribution: Student’s t-distribution with degrees of freedom (variances not pooled)
Formula Review
Standard error: SE = [latex]\sqrt{\frac{{\left({s}_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({s}_{2}\right)}^{2}}{{n}_{2}}}[/latex]
Test statistic (t-score): t = [latex]\frac{\left({\overline{x}}_{1}-{\overline{x}}_{2}\right)-\left({\mu }_{1}-{\mu }_{2}\right)}{\sqrt{\frac{{\left({s}_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({s}_{2}\right)}^{2}}{{n}_{2}}}}[/latex]
Degrees of freedom:
[latex]df=\frac{{\left(\frac{{\left({s}_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({s}_{2}\right)}^{2}}{{n}_{2}}\right)}^{2}}{\left(\frac{1}{{n}_{1}-1}\right){\left(\frac{{\left({s}_{1}\right)}^{2}}{{n}_{1}}\right)}^{2}+\left(\frac{1}{{n}_{2}-1}\right){\left(\frac{{\left({s}_{2}\right)}^{2}}{{n}_{2}}\right)}^{2}}[/latex]
where:
s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
[latex]{\overline{x}}_{1}[/latex] and [latex]{\overline{x}}_{2}[/latex] are the sample means.
Cohen’s d is the measure of effect size:
[latex]d=\frac{{\overline{x}}_{1}-{\overline{x}}_{2}}{{s}_{pooled}}[/latex]
where [latex]{s}_{pooled}=\sqrt{\frac{\left({n}_{1}-1\right){s}_{1}^{2}+\left({n}_{2}-1\right){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}}[/latex]
9.2 Chapter Review
A hypothesis test of two population means from independent samples where the population standard deviations are known will have these characteristics:
- Random variable: [latex]\overline{{X}_{1}}–\overline{{X}_{2}}[/latex] = the difference of the means
- Distribution: normal distribution
Formula Review
Normal Distribution:
[latex]{\overline{X}}_{1}–{\overline{X}}_{2}\sim N\left[{\mu }_{1}–{\mu }_{2},\sqrt{\frac{{\left({\sigma }_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({\sigma }_{2}\right)}^{2}}{{n}_{2}}}\right][/latex].
Generally µ1 – µ2 = 0.
Test Statistic (z-score):
[latex]z=\frac{\left({\overline{x}}_{1}–{\overline{x}}_{2}\right)–\left({\mu }_{1}–{\mu }_{2}\right)}{\sqrt{\frac{{\left({\sigma }_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({\sigma }_{2}\right)}^{2}}{{n}_{2}}}}[/latex]
Generally µ1 – µ2 = 0.
where: σ1 and σ2 are the known population standard deviations. n1 and n2 are the sample sizes. x¯1 and x¯2 are the sample means. μ1 and μ2 are the population means.
9.3 Chapter Review
Test of two population proportions from independent samples.
Formula Review
Pooled Proportion: pc = [latex]\frac{{x}_{F}\text{ }+\text{ }{x}_{M}}{{n}_{F}\text{ }+\text{ }{n}_{M}}[/latex]
Distribution for the differences:
[latex]{{p}^{\prime }}_{A}-{{p}^{\prime }}_{B}\sim N\left[0,\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right][/latex]
where the null hypothesis is H0: pA = pB or H0: pA – pB = 0.
Test Statistic (z-score): [latex]z=\frac{\left({p}^{\prime }{}_{A}-{p}^{\prime }{}_{B}\right)}{\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}[/latex]
where the null hypothesis is H0: pA = pB or H0: pA − pB = 0.
where
p′A and p′B are the sample proportions, pA and pB are the population proportions,
Pc is the pooled proportion, and nA and nB are the sample sizes.
9.4 Chapter Review
A hypothesis test for matched or paired samples (t-test) has these characteristics:
- Test the differences by subtracting one measurement from the other measurement
- Random Variable: [latex]{\overline{x}}_{d}[/latex] = mean of the differences
- Distribution: Student’s-t distribution with n – 1 degrees of freedom
- If the number of differences is small (less than 30), the differences must follow a normal distribution.
- Two samples are drawn from the same set of objects.
- Samples are dependent.
Formula Review
Test Statistic (t-score): t = [latex]\frac{{\overline{x}}_{d}-{\mu }_{d}}{\left(\frac{{s}_{d}}{\sqrt{n}}\right)}[/latex]
where:
[latex]{\overline{x}}_{d}[/latex] is the mean of the sample differences. μd is the mean of the population differences. sd is the sample standard deviation of the differences. n is the sample size.
