Glossary
Alternative Hypothesis (Hₐ)
The statement that contradicts the null hypothesis, representing what the researcher is trying to find evidence for (e.g., the population slope is not zero, or is greater/less than a specific value).
Example:
If a nutritionist believes a new diet plan reduces weight, the alternative hypothesis would be Hₐ: β < 0, indicating a negative linear relationship between diet duration and weight.
Equal Variance (condition)
A condition for valid regression inference requiring that the variability of the residuals is roughly constant across all values of the explanatory variable, often checked by looking for 'fanning' or a megaphone shape in the residual plot.
Example:
When analyzing the relationship between car speed and fuel efficiency, a mechanic verifies the equal variance condition by observing that the spread of residuals is consistent across all speeds, avoiding a wider spread at higher speeds.
Hypothesis Testing for Regression Slopes
A statistical procedure used to determine if there is a statistically significant linear relationship between two variables in a population, based on sample data.
Example:
An environmental scientist uses hypothesis testing for regression slopes to assess if there's a significant linear relationship between the amount of pollution and the average temperature in a city.
Linearity (condition)
A condition for valid regression inference requiring that the relationship between the explanatory and response variables is approximately linear, typically assessed by examining the residual plot for any obvious patterns.
Example:
Before concluding that study time linearly predicts test scores, a student checks the linearity condition by ensuring the residual plot shows no curved pattern, suggesting a straight line is an appropriate model.
Null Hypothesis (H₀)
The statement of no effect, no difference, or no relationship that is assumed to be true until evidence suggests otherwise. For regression slopes, it typically states the population slope (β) is equal to a specific value, often zero (H₀: β = 0).
Example:
In a study examining the effect of fertilizer on crop yield, the null hypothesis would be H₀: β = 0, meaning the fertilizer has no linear impact on yield.
T-Statistic
A standardized test statistic used in a t-test that measures how many standard errors the sample slope is away from the hypothesized population slope.
Example:
A large t-statistic (e.g., 5.2) for the relationship between exercise and heart rate suggests a very strong and significant linear association.
T-Test for Slopes
A specific inferential statistical test used to determine if the slope of a population regression line is significantly different from a hypothesized value, most commonly zero.
Example:
A student performs a t-test for slopes to see if the number of hours spent on social media significantly predicts a student's GPA.