Dependent Samples t-Test Cheat Sheet

The dependent samples t-test is used to determine if there is a statistically significant difference between two related groups or measurements. It analyzes the mean difference between paired observations.

Core Principles

Compares means of two related groups (e.g., before and after treatment).
Assumes differences are normally distributed.
Uses paired observations to reduce variability.
Calculates a t-statistic to assess significance.
Null hypothesis (H0): The mean difference is zero.
Alternative hypothesis (H1): The mean difference is not zero.

Action Steps

1. State null and alternative hypotheses.
2. Define the decision criterion (alpha level, degrees of freedom).
3. Calculate the mean of the differences ($M_D$).
4. Calculate the standard deviation of the differences ($S_D$).
5. Calculate the standard error of the mean difference ($S_{M_D}$).
6. Compute the t-statistic.
7. Compare the calculated t-statistic to the critical t-value.
8. Make a decision: Reject or fail to reject the null hypothesis.
9. Report the results, including effect size and confidence intervals.

Formulas

$t = \frac{M_D - \mu_D}{S_{M_D}}$
$M_D = \frac{\sum D}{n}$
$S_D = \sqrt{\frac{\sum (D - M_D)^2}{n-1}}$
$S_{M_D} = \frac{S_D}{\sqrt{n}}$

Key Terms

Paired Samples: Observations that are related or matched, such as measurements from the same individual at two different times.
Difference Scores (D): The result of subtracting the second measurement from the first for each pair.
Mean Difference ($M_D$): The average of the difference scores.
Standard Error of the Mean Difference ($S_{M_D}$): An estimate of the standard deviation of the sampling distribution of the mean difference.
Degrees of Freedom (df): Calculated as n-1, where n is the number of pairs.

Timeline

Step 1: Formulate Hypotheses (H0: μD = 0, H1: μD ≠ 0).
Step 2: Determine Decision Criterion (df = n-1, alpha, critical t-value).
Step 3: Calculate the t-statistic using sample data.
Step 4: Compare t-statistic to critical value; make a decision.
Step 5: Calculate effect size (e.g., Cohen's d) and confidence intervals.

People

William Sealy Gosset (Student): Developed the t-distribution and t-test.