Overview
This lecture demonstrates how to perform a negative correlation hypothesis test using StatKey, focusing on the relationship between miles per gallon and horsepower in car data.
Hypothesis Test Setup
- We are testing whether there is a negative correlation (inverse relationship) between car weight and miles per gallon.
- The alternative hypothesis: population correlation (ρ) < 0 or slope (β₁) < 0.
- The test uses a 5% significance level (α = 0.05).
- Data assumptions and conditions are met.
Using StatKey for Hypothesis Testing
- Copy and paste the data (miles per gallon and horsepower) into StatKey under "Randomization Hypothesis Test" → "Test for Slope/Correlation."
- Select the header row if present.
- StatKey provides a scatter plot, sample size, correlation coefficient (R), slope, and y-intercept.
Analyzing Output and Interpreting Results
- A strong negative correlation is indicated when R is close to -1.
- StatKey simulates thousands of R values or slope values under the null hypothesis to create a reference distribution.
- For a left-tail test, use α = 0.05 to determine the critical value.
- Actual sample R or slope falling in the left tail indicates significant evidence against the null hypothesis.
- Calculating the p-value: Enter the sample statistic (R or slope) in StatKey to check the proportion of simulations as extreme or more extreme.
Calculating Test Statistics
- The t-test statistic for slope: (sample slope - 0) / standard error.
- StatKey provides the approximate standard error for this calculation.
Conclusion and Interpretation
- Both sample R and slope fell in the left tail; p-value ≈ 0.
- The sample data provides significant evidence to reject the null hypothesis.
- There is significant evidence supporting an inverse relationship between car weight and miles per gallon.
- The regression line is an effective model for predicting miles per gallon based on weight.
Key Terms & Definitions
- Correlation coefficient (R) — measures the strength and direction of linear relationship between two variables.
- Slope (β₁) — rate at which the dependent variable (e.g., miles per gallon) changes with the independent variable (e.g., weight).
- Null hypothesis (H₀) — assumption that there is no relationship (ρ = 0 or β₁ = 0).
- Alternative hypothesis (H₁) — claim of a negative relationship (ρ < 0 or β₁ < 0).
- Significance level (α) — probability threshold (commonly 0.05) for rejecting H ₀.
- P-value — probability of observing data as extreme as the sample result under H₀.
- Standard error — estimate of variability in the sampling distribution of a statistic.
Action Items / Next Steps
- Practice running a hypothesis test for negative correlation using StatKey with similar datasets.
- Review how to interpret StatKey outputs, scatterplots, and p-values.
- Prepare for questions on hypothesis testing for correlation and regression models.