SESSION:10 Hypothesis Testing with Two SamplesFormula Review

Session:10 Hypothesis Testing with Two Samples

Formula Review

Introductory Business Statistics | Leadership Development – Micro-Learning Session

Rice University 2020 | Michael Laverty, Colorado State University Global Chris Littel, North Carolina State University| https://openstax.org/details/books/introductory-business-statistics

10.1 Comparing Two Independent Population Means

Standard error: SE = $\sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}}$

$\sqrt{\frac{{(�_{1})}^{2}}{�_{1}} + \frac{{(�_{2})}^{2}}{�_{2}}}$

Test statistic (t-score): t_c = $\frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - δ_{0}}{\sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}}}$

$\frac{({\bar{�}}_{1} - {\bar{�}}_{2}) - �_{0}}{\sqrt{\frac{{(�_{1})}^{2}}{�_{1}} + \frac{{(�_{2})}^{2}}{�_{2}}}}$

Degrees of freedom:
$d f = \frac{{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}^{2}}{(\frac{1}{n_{1} - 1}) {(\frac{{(s_{1})}^{2}}{n_{1}})}^{2} + (\frac{1}{n_{2} - 1}) {(\frac{{(s_{2})}^{2}}{n_{2}})}^{2}}$

$� � = \frac{{(\frac{{(�_{1})}^{2}}{�_{1}} + \frac{{(�_{2})}^{2}}{�_{2}})}^{2}}{(\frac{1}{�_{1} - 1}) {(\frac{{(�_{1})}^{2}}{�_{1}})}^{2} + (\frac{1}{�_{2} - 1}) {(\frac{{(�_{2})}^{2}}{�_{2}})}^{2}}$

where:

$s_{1}$

$�_{1}$ and $s_{2}$

$�_{2}$ are the sample standard deviations, and $n_{1}$

$�_{1}$ and $n_{2}$

$�_{2}$ are the sample sizes.

${\bar{x}}_{1}$

${\bar{�}}_{1}$ and ${\bar{x}}_{2}$

${\bar{�}}_{2}$ are the sample means.

10.2 Cohen’s Standards for Small, Medium, and Large Effect Sizes

Cohen’s d is the measure of effect size:

$d = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{s_{p o o l e d}}$

$� = \frac{{\bar{�}}_{1} - {\bar{�}}_{2}}{�_{� � � � � �}}$
where $s_{p o o l e d} = \sqrt{\frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2}}$

$�_{� � � � � �} = \sqrt{\frac{(�_{1} - 1) �_{1}^{2} + (�_{2} - 1) �_{2}^{2}}{�_{1} + �_{2} - 2}}$

10.3 Test for Differences in Means: Assuming Equal Population Variances

�_{�} = \frac{({\bar{�}}_{1} - {\bar{�}}_{2}) - �_{0}}{\sqrt{� \overset{2}{�} (\frac{1}{�_{1}} + \frac{1}{�_{2}})}}

where $S \overset{2}{p}$

$� \overset{2}{�}$ is the pooled variance given by the formula:

� \overset{2}{�} = \frac{(�_{1} - 1) �_{1}^{2} - (�_{2} - 1) �_{2}^{2}}{�_{1} + �_{2} - 2}

10.4 Comparing Two Independent Population Proportions

Pooled Proportion: p_c = $\frac{x_{A} + x_{B}}{n_{A} + n_{B}}$

$\frac{�_{�} + �_{�}}{�_{�} + �_{�}}$

Test Statistic (z-score): $Z_{c} = \frac{(p^{'}_{A} - p^{'}_{B})}{\sqrt{p_{c} (1 - p_{c}) (\frac{1}{n_{A}} + \frac{1}{n_{B}})}}$

$�_{�} = \frac{(�^{'}_{�} - �^{'}_{�})}{\sqrt{�_{�} (1 - �_{�}) (\frac{1}{�_{�}} + \frac{1}{�_{�}})}}$

where

$p_{A}^{'}$

$�_{�}^{‘}$ and $p_{B}^{'}$

$�_{�}^{‘}$ are the sample proportions, $p_{A}$

$�_{�}$ and $p_{B}$

$�_{�}$ are the population proportions,

P_c is the pooled proportion, and n_A and n_B are the sample sizes.

10.5 Two Population Means with Known Standard Deviations

Test Statistic (z-score):

$Z_{c} = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - δ_{0}}{\sqrt{\frac{{(σ_{1})}^{2}}{n_{1}} + \frac{{(σ_{2})}^{2}}{n_{2}}}}$

$�_{�} = \frac{({\bar{�}}_{1} - {\bar{�}}_{2}) - �_{0}}{\sqrt{\frac{{(�_{1})}^{2}}{�_{1}} + \frac{{(�_{2})}^{2}}{�_{2}}}}$

where:
$σ_{1}$

$�_{1}$ and $σ_{2}$

$�_{2}$ are the known population standard deviations. n₁ and n₂ are the sample sizes. ${\bar{x}}_{1}$

${\bar{�}}_{1}$ and ${\bar{x}}_{2}$

${\bar{�}}_{2}$ are the sample means. μ₁ and μ₂ are the population means.

10.6 Matched or Paired Samples

Test Statistic (t-score): t_c = $\frac{{\bar{x}}_{d} - μ_{d}}{(\frac{s_{d}}{\sqrt{n}})}$

$\frac{{\bar{�}}_{�} - �_{�}}{(\frac{�_{�}}{\sqrt{�}})}$

where:

${\bar{x}}_{d}$

${\bar{�}}_{�}$ is the mean of the sample differences. μ_d is the mean of the population differences. s_d is the sample standard deviation of the differences. n is the sample size.