# Statistical Tests and Correlation

Psychology I · Statistics · https://tryals.app/learn/psychology-i/statistical-tests-and-correlation

## Choosing the Right Statistical Tool

The **Pearson correlation coefficient (r)** quantifies the strength and direction of a **linear relationship** between two continuous variables. It ranges from $-1$ (perfect negative) to $+1$ (perfect positive), with $0$ indicating no linear relationship.

| Test | Used for |
| --- | --- |
| **Pearson correlation** | Linear relationship between two continuous variables |
| **T-test** | Comparing means of exactly TWO groups |
| **ANOVA** | Comparing means of THREE OR MORE groups |
| **Chi-square test** | Relationships between CATEGORICAL variables |

A scatterplot is essential here because $r$ only captures **linear patterns** and can badly understate a strong, curved non-linear relationship.

## Avoiding Type I Errors with ANOVA

**ANOVA (Analysis of Variance)** tests all groups simultaneously in a single analysis. This controls the **Type I error** (false positive) rate, which would otherwise balloon if you ran multiple separate t-tests.

If ANOVA is significant, researchers use targeted **post-hoc tests** to pinpoint exactly which specific group pairs differ from one another.

**Common pitfall:** Running multiple separate t-tests across group pairs instead of ANOVA. Each extra test compounds your overall Type I error rate across the full set of comparisons, which is the exact problem ANOVA solves.

## Practice questions

3 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Sort each pair of variables by the direction of correlation you would expect.

**Answer:**

- Likely POSITIVE correlation: Outdoor temperature and ice-cream sales, Hours studied and exam score
- Likely NEGATIVE correlation: Hours of exercise per week and resting heart rate, Altitude up a mountain and air temperature, Car speed and travel time over a fixed distance

**Why:** Temperature/sales and study/score rise together (positive); exercise/heart-rate, altitude/temperature, and speed/travel-time move in opposite directions (negative). Reasoning about direction is the skill, not memorizing pairs.

Page: https://tryals.app/practice/psychology-i/statistical-tests-and-correlation/sort-each-pair-of-variables-by-the-direction-of-correlation-you-would

### 2. Athletes who appear on a magazine cover after an exceptional season usually do worse the following year. The 'cover jinx' has a duller explanation. What is it?

A. Regression to the mean, since luck does not persist
B. Selection bias in which athletes are chosen for covers
C. Complacency following recognition
D. The pressure of publicity, which measurably degrades performance

**Answer:** A. Regression to the mean, since luck does not persist

**Why:** Extreme observations are selected partly for their luck, and luck does not repeat, so the follow-up drifts back toward the true level with no causal story needed. The dangerous version is in evaluation: praise appears to hurt and punishment appears to work, purely because each follows an extreme.

Page: https://tryals.app/practice/psychology-i/statistical-tests-and-correlation/athletes-who-appear-on-a-magazine-cover-after-an-exceptional-season

### 3. Comparing four groups by running a t-test on every pair means six tests. Why does that inflate the false positive rate, and what does ANOVA do instead?

A. Because t-tests are less accurate than ANOVA, so running several compounds the inaccuracy.
B. Because each test carries its own 5% risk across six comparisons.
C. Because the six tests are not independent, which makes each individual p-value wrong.
D. Because ANOVA uses a stricter alpha level than 0.05, which is where the protection comes from.

**Answer:** B. Because each test carries its own 5% risk across six comparisons.

**Why:** Each test is fine on its own and the problem is arithmetic across tests: 0.95 to the sixth power is about 0.74, so the chance of at least one spurious result is roughly 26%. ANOVA asks one omnibus question at one alpha. It is the same multiple-comparisons problem that appears as undisclosed analytic flexibility, wearing a second costume.

Page: https://tryals.app/practice/psychology-i/statistical-tests-and-correlation/comparing-four-groups-by-running-a-t-test-on-every-pair-means-six