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:
- [latex]\text{SE} = \sqrt{\frac{{\left({s}_{1}\right)}^{2}}{{n}_{1}}+\frac{{\left({s}_{2}\right)}^{2}}{{n}_{2}}}[/latex]
- Test statistic (t-score):
- [latex]t = \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]
- [latex]{\overline{x}}_{1}[/latex] and [latex]{\overline{x}}_{2}[/latex] are the sample means.
- [latex]\mu_1[/latex] and [latex]\mu_2[/latex] are the population means.
- 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]
- [latex]s_1[/latex] and [latex]s_2[/latex] are the sample standard deviations.
- [latex]n_1[/latex] and [latex]n_2[/latex] are the sample sizes.
- Cohen’s d is the measure of effect size:
- [latex]d=\frac{{\overline{x}}_{1}-{\overline{x}}_{2}}{{s}_{pooled}}[/latex]
- [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 [latex]\mu_1 - \mu_2 = 0[/latex].
- 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 [latex]\mu_1 - \mu_2 = 0[/latex].
- [latex]\sigma_1[/latex] and [latex]\sigma_2[/latex] are the known population standard deviations.
- [latex]n_1[/latex] and [latex]n_2[/latex] are the sample sizes.
- [latex]{\overline{x}}_{1}[/latex] and [latex]{\overline{x}}_{2}[/latex] are the sample means.
- [latex]\mu_1[/latex] and [latex]\mu_2[/latex] are the population means.
9.3 Chapter Review
Test of two population proportions from independent samples.
Random variable: [latex]{p}_{A}–{p}_{B}=[/latex] difference between the two estimated proportions
Distribution: normal distribution
Formula Review
- Pooled Proportion:
- [latex]p_c = \frac{{x}_{F}+{x}_{M}}{{n}_{F}+{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]
- The null hypothesis is [latex]H_0: p_A = p_B[/latex] or [latex]H_0: p_A - p_B = 0[/latex].
- 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]
- The null hypothesis is [latex]H_0: p_A = p_B[/latex] or [latex]H_0: p_A - p_B = 0[/latex].
- [latex]{{p}^{\prime }}_{A}[/latex] and [latex]{{p}^{\prime }}_{B}[/latex] are the sample proportions, [latex]{{p}^{\prime }}_{A}[/latex] and [latex]{{p}^{\prime }}_{B}[/latex] are the population proportions,
- [latex]{p}_{c}[/latex] is the pooled proportion, and [latex]n_A[/latex] and [latex]n_B[/latex] 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 [latex]n – 1[/latex] 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):
-
- [latex]t = \frac{{\overline{x}}_{d}-{\mu }_{d}}{\left(\frac{{s}_{d}}{\sqrt{n}}\right)}[/latex]
- [latex]{\overline{x}}_{d}[/latex] is the mean of the sample differences.
- [latex]\mu_d[/latex] is the mean of the population differences.
- [latex]s_d[/latex] is the sample standard deviation of the differences.
- [latex]n[/latex] is the sample size.
