Comprehensive Study Notes: Hypothesis Testing, t-Tests, ANOVA, Correlation, Regression, and Chi-Square

Hypothesis Testing

Overview of Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0.

• Steps of Hypothesis Testing:

  1. State the hypotheses (null and alternative).

  2. Set the significance level (α).

  3. Choose the appropriate test statistic.

  4. Calculate the test statistic from sample data.

  5. Determine the p-value or critical value.

  6. Draw a conclusion (reject or fail to reject H0).

• Types of Errors:

  • Type I Error (α): Rejecting a true null hypothesis.

  • Type II Error (β): Failing to reject a false null hypothesis.

• Statistical Power: Probability of correctly rejecting a false null hypothesis (1 - β).

t-Tests

One-Sample t-Test

Used to compare the mean of a single sample to a known value or population mean.

• Test Statistic: $t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$

• Interpretation: Compare calculated t to critical t-value from t-distribution table.

• Effect Size: Cohen's d can be used to measure effect size.

Independent Samples t-Test

Compares the means of two independent groups to determine if they are significantly different.

• Equal Sample Sizes: $t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{2}{n}}}$

• Pooled Standard Deviation: $s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

• Different Sample Sizes: $t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Related (Paired) Samples t-Test

Used when the same subjects are measured twice (e.g., before and after treatment).

• Test Statistic: $t = \frac{\bar{D}}{s_D / \sqrt{n}}$

• Where: $\bar{D}$ is the mean of the difference scores, $s_D$ is the standard deviation of the difference scores.

Analysis of Variance (ANOVA)

One-Way ANOVA

Used to compare means across three or more independent groups.

• F Statistic: $F = \frac{MS_{between}}{MS_{within}}$

• Mean Square Between: $MS_{between} = \frac{SS_{between}}{df_{between}}$

• Mean Square Within: $MS_{within} = \frac{SS_{within}}{df_{within}}$

• Interpretation: Large F indicates significant differences among group means.

Two-Way ANOVA

Used to examine the effect of two independent variables (factors) on a dependent variable, including possible interaction effects.

• Main Effects: Effect of each factor independently.

• Interaction Effect: Combined effect of both factors.

• F Statistic for Each Effect: $F = \frac{MS_{effect}}{MS_{error}}$

Repeated Measures ANOVA

Used when the same subjects are measured under different conditions or at different times.

• Accounts for within-subject variability.

Correlation

Pearson Correlation Coefficient (r)

Measures the strength and direction of the linear relationship between two continuous variables.

• Formula: $r = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sqrt{\sum (X - \bar{X})^2 \sum (Y - \bar{Y})^2}}$

• df for correlation: $n - 2$

• Interpretation: r ranges from -1 (perfect negative) to +1 (perfect positive).

Linear Regression

Simple Linear Regression

Models the relationship between a dependent variable (Y) and an independent variable (X) using a straight line.

• Regression Equation: $Y = b_0 + b_1 X$

• Slope (b1): $b_1 = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (X - \bar{X})^2}$

• Intercept (b0): $b_0 = \bar{Y} - b_1 \bar{X}$

• Interpretation: Slope indicates the change in Y for a one-unit increase in X.

Chi-Square Tests

Chi-Square Goodness of Fit Test

Tests whether observed frequencies differ from expected frequencies in one categorical variable.

• Test Statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• O: Observed frequency

• E: Expected frequency

Chi-Square Test for Independence

Tests whether two categorical variables are independent.

• Same formula as above, applied to contingency tables.

Summary Table: Hypothesis Tests and Formulas

Test	Purpose	Formula
One-Sample t-Test	Compare sample mean to population mean	$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$
Independent Samples t-Test	Compare means of two independent groups	$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
Related Samples t-Test	Compare means of paired samples	$t = \frac{\bar{D}}{s_D / \sqrt{n}}$
One-Way ANOVA	Compare means of 3+ groups	$F = \frac{MS_{between}}{MS_{within}}$
Pearson Correlation	Measure linear relationship	$r = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sqrt{\sum (X - \bar{X})^2 \sum (Y - \bar{Y})^2}}$
Linear Regression	Predict Y from X	$Y = b_0 + b_1 X$
Chi-Square	Test categorical association	$\chi^2 = \sum \frac{(O - E)^2}{E}$

Additional info:

• Effect sizes (e.g., Cohen's d, eta squared) are important for interpreting the magnitude of results.

• SPSS is a common statistical software used for performing these analyses.

• Post-hoc tests are used after ANOVA to determine which groups differ.