Inference for a regression slope helps you decide whether a linear relationship seen in sample data is likely to represent a real relationship in the population. A scatterplot may show an upward or downward trend, but random variation can create patterns even when the true slope is zero. The goal is to use a t-test or confidence interval to judge whether the slope of the population regression line differs from zero.
This is important in science, economics, engineering, and social research whenever data are used to support a claim about association.
The test focuses on the sample slope b, which estimates the unknown population slope beta. If the null hypothesis says beta = 0, then the explanatory variable has no linear predictive effect on the response in the population. The test statistic compares the observed slope to its standard error, using t = (b - 0) / SE_b with n - 2 degrees of freedom.
A confidence interval b ± t*SE_b gives a range of plausible values for the true slope and shows both the direction and likely size of the linear effect.
Key Facts
- Population regression model: y = alpha + beta x + epsilon
- Sample regression line: yhat = a + bx
- Null hypothesis for no linear relationship: H0: beta = 0
- Test statistic for slope: t = (b - beta0) / SE_b, usually beta0 = 0
- Degrees of freedom for slope inference: df = n - 2
- Confidence interval for the slope: b ± t* SE_b
Vocabulary
- Regression slope
- The slope describes the predicted change in the response variable y for a one-unit increase in the explanatory variable x.
- Population slope
- The population slope beta is the true slope of the linear relationship in the entire population.
- Standard error of the slope
- The standard error of the slope measures how much the sample slope would typically vary from sample to sample.
- t-test for slope
- A t-test for slope tests whether the population slope is equal to a hypothesized value, most often zero.
- Confidence interval
- A confidence interval gives a range of plausible values for the population slope based on sample data.
Common Mistakes to Avoid
- Saying the slope proves causation, which is wrong because regression inference shows evidence of linear association, not cause and effect unless the study design supports causality.
- Using the normal z distribution instead of the t distribution, which is wrong because the slope standard error is estimated from the data and the correct degrees of freedom are n - 2.
- Interpreting a confidence interval for the slope as a range of y-values, which is wrong because the interval estimates the change in predicted y for each one-unit increase in x.
- Ignoring the scatterplot and conditions, which is wrong because slope inference depends on a roughly linear pattern, independent observations, nearly normal residuals, and roughly constant residual spread.
Practice Questions
- 1 A sample of n = 18 gives a regression slope b = 2.40 with SE_b = 0.80. Test H0: beta = 0 using t = b / SE_b. What is the test statistic and the degrees of freedom?
- 2 A regression study has b = -1.25, SE_b = 0.30, n = 25, and t* = 2.069 for a 95% confidence interval. Compute the 95% confidence interval for the population slope.
- 3 A 95% confidence interval for a slope is (0.15, 0.62). Explain what this interval says about the direction of the relationship and whether a test of H0: beta = 0 would be significant at the 0.05 level.