Welcome to Part 4 — Applications (Cases and Examples) of this free online high school statistics textbook. This hands-on section brings statistical concepts to life through detailed, worked-out case studies and real-world examples. High school students explore complete applications of hypothesis testing—including t-tests, ANOVA designs, chi-square tests, and non-parametric methods—covering everything from formulating research questions and selecting the appropriate test to performing calculations, interpreting results, and drawing meaningful conclusions.
Ideal for AP Statistics practice and pre-college preparation, Part 4 features 10 comprehensive cases with step-by-step explanations, formulas, data examples, and practical scenarios (e.g., comparing teaching methods, stress reduction programs, and categorical associations). These worked examples reinforce descriptive statistics, inferential statistics, and critical statistical thinking in an engaging, example-driven format.
Case Studies in Part 4: Applications
- Case 1: Independent t-Test – Comparing two independent groups (e.g., different teaching methods).
- Case 2: Paired t-Test – Analyzing before-and-after data in the same subjects.
- Case 3: One-Way ANOVA – Testing differences across three or more groups.
- Case 4: Factorial ANOVA (2×2 Design) – Examining main effects and interactions.
- Case 5: Repeated-Measures ANOVA – Handling multiple measurements on the same subjects.
- Case 6: Mixed ANOVA – Combining between-subjects and within-subjects factors.
- Case 7: Chi-Square Goodness-of-Fit – Assessing observed vs. expected frequencies.
- Case 8: Chi-Square Test of Independence – Exploring relationships in categorical data.
- Case 9: Mann-Whitney U Test – Non-parametric alternative for two independent samples.
- Case 10: Wilcoxon Signed-Rank Test – Non-parametric option for paired data.
A practice self-test quiz is also available to reinforce learning (optional signup for full interactive access). Dive into these free high school statistics applications for real-world insight into hypothesis testing, statistical analysis examples, and building confidence with data interpretation!
Case 1 — Independent t-test (Two Groups)
Scenario: A teacher compares math scores of students taught by lecture vs. interactive software.
Question: Are the two teaching methods different in average score?
Design/Test: Independent-samples t-test.
Worked Example:
- Group A (Lecture): mean = 78, SD = 10, n = 20
- Group B (Software): mean = 85, SD = 12, n = 20
Formula:
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$
In words:
$$t = \frac{\text{mean}_1 - \text{mean}_2}{\sqrt{\tfrac{\text{variance}_1}{n_1} + \tfrac{\text{variance}_2}{n_2}}}$$
Plugging in values:
$$t = \frac{78 - 85}{\sqrt{\tfrac{100}{20} + \tfrac{144}{20}}} = \frac{-7}{\sqrt{12.2}} = \frac{-7}{3.49} = -2.01$$
Degrees of freedom = 38.
Case 2 — Paired t-test (Before and After)
Scenario: Students take a memory test before and after a week of practice.
Question: Did scores improve after training?
Design/Test: Paired-samples t-test.
Worked Example:
Differences (After – Before): 2, 4, 3, 5, 6
- Mean difference:
$$\bar{D} = \frac{2+4+3+5+6}{5} = 4$$ - Standard deviation of differences: $$s_D = 1.58$$
Formula:
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$
Plugging in values:
$$t = \frac{4}{1.58/\sqrt{5}} = \frac{4}{0.71} = 5.63$$
Degrees of freedom = 4.
Case 3 — One-way ANOVA (Three Groups)
Scenario: A psychologist tests meditation, exercise, and music as stress-reduction methods.
Question: Do the methods differ in mean stress score?
Design/Test: One-way ANOVA.
Worked Example:
- Group means: Meditation = 65, Exercise = 70, Music = 80
- $$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
- $$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$
Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$
$$F = \frac{150}{16.7} = 9.0, \quad df = (2,12)$$
Case 4 — Factorial ANOVA (2 × 2 Design)
Scenario: A researcher studies teaching method (Lecture vs. Online) × Time of Day (Morning vs. Afternoon).
Question: Do method, time, or their interaction affect performance?
Design/Test: Two-way (factorial) ANOVA.
Worked Example (summary):
- Lecture: Morning = 70, Afternoon = 90
- Online: Morning = 80, Afternoon = 80
Interaction: Lecture scores rise with time, Online stays flat.
Formulas:
- $$df_A = a - 1, , df_B = b - 1, , df_{A \times B} = (a-1)(b-1), , df_{\text{within}} = N - ab$$
Case 5 — Repeated-Measures ANOVA
Scenario: Five students are tested across three conditions.
Question: Do scores differ across conditions?
Design/Test: Repeated-measures ANOVA.
Worked Example (summary):
- Means increase steadily: 70 → 75 → 80
- df:
$$df_{\text{rows}} = n - 1, \quad df_{\text{columns}} = k - 1, \quad df_{\text{error}} = (n-1)(k-1)$$
Formula:
$$F = \frac{MS_{\text{columns}}}{MS_{\text{error}}}$$
Case 6 — Mixed ANOVA
Scenario: Two groups (Drug, Placebo) tested across three weeks.
Question: Is there an effect of group, time, or interaction?
Design/Test: Mixed (split-plot) ANOVA.
Worked Example (summary):
- Drug: 70 → 80 → 90
- Placebo: 70 → 72 → 74
- Drug improves over time, Placebo stays flat.
Formula:
$$F = \frac{MS_{\text{effect}}}{MS_{\text{error}}}$$
Case 7 — Chi-square Goodness-of-Fit
Scenario: A survey asks students to choose a favorite subject: Math, Science, or English.
Question: Is the distribution of responses different from equal chance?
Design/Test: Chi-square goodness-of-fit test.
Formula:
$$\chi^2 = \sum \frac{(O - E)^2}{E}$$
In words:
$$\chi^2 = \frac{\text{(Observed - Expected)}^2}{\text{Expected}}, , \text{summed across categories}$$
Case 8 — Chi-square Test of Independence
Scenario: A researcher tests whether gender (Male, Female) is related to sport preference (Soccer, Basketball, Tennis).
Question: Is there an association between gender and sport?
Design/Test: Chi-square test of independence.
Formula:
$$\chi^2 = \sum \frac{(O - E)^2}{E}$$
Case 9 — Mann–Whitney U Test
Scenario: Students in two different schools are ranked by teacher ratings.
Question: Do the two groups differ in median rank?
Design/Test: Mann–Whitney U test (non-parametric).
Formula:
$$U = n_1 n_2 + \frac{n_1 (n_1 + 1)}{2} - R_1$$
Where $$R_1$$ = sum of ranks for group 1.
Case 10 — Wilcoxon Signed-Rank Test
Scenario: The same students are ranked before and after training.
Question: Did the ranks change?
Design/Test: Wilcoxon signed-rank test (non-parametric).
Formula (summary):
- Compute differences (After – Before).
- Rank the absolute differences.
- Assign signs and sum.
- Test statistic = smaller of the two signed sums.
Practice self-test quiz
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