In this unit, you'll learn how to conduct inference on the slope of a population regression line using data from a linear regression model. You’ll construct confidence intervals, perform hypothesis tests, and select the appropriate procedure for bivariate quantitative data.
We can estimate the true slope of the population regression line using a confidence interval.
Linear: The relationship between the variables is linear.
Independent: The observations are independent (check the design or residual plot).
Normal: The residuals are approximately normally distributed.
\[ b \pm t^* \cdot SE_b \] where:
\( b \) = sample slope
\( t^* \) = critical value from the t-distribution with \( df = n - 2 \)
\( SE_b \) = standard error of the slope (from computer or calculator output)
A linear regression model estimates how temperature affects sales of ice cream. The slope of the least-squares line is 2.3 with standard error 0.5 and \( df = 18 \). A 95% confidence interval is:
\[ 2.3 \pm t^* \cdot 0.5 = 2.3 \pm 2.101 \cdot 0.5 = (1.25, 3.35) \]
- Run LinRegTInt on your calculator.
- Enter the x- and y-lists, confidence level, and choose "Calculate".
- It returns the interval for the slope \( b \).
We can test whether there is a significant linear relationship between two quantitative variables by testing whether the slope of the population regression line is zero.
\( H_0: \beta = 0 \) (no linear relationship)
\( H_a: \beta \ne 0 \), \( \beta > 0 \), or \( \beta < 0 \) (depending on context)
\[ t = \frac{b - \beta_0}{SE_b} \] where:
\( b \) = sample slope
\( \beta_0 \) = hypothesized slope (usually 0)
\( SE_b \) = standard error of slope
\( df = n - 2 \)
A regression of hours studied vs. exam scores yields \( b = 4.1 \), \( SE_b = 1.3 \), and \( n = 16 \). Is there evidence of a positive association?
\[ t = \frac{4.1 - 0}{1.3} = 3.15, \quad df = 14 \] Using a t-distribution table or calculator, find the p-value and compare it to α.
If the p-value is less than the significance level (e.g., 0.05), reject \( H_0 \). Conclude that there is significant evidence of a linear relationship.
- Use LinRegTTest from the STAT → TESTS menu.
- Enter the x- and y-lists, hypothesis for \( \beta \), and press Calculate.
- You’ll get the t-statistic, p-value, and regression info.
Regression output usually includes:
Slope (\( b \)): coefficient for the explanatory variable
Standard error (SE): for the slope
t: test statistic for \( H_0: \beta = 0 \)
p-value: used to make decisions about the null hypothesis
\( R^2 \): percent of variation in response variable explained by the regression
s: standard deviation of the residuals
Tip: Always use the row for the explanatory variable (not the constant/intercept row) when conducting inference for slope.
Use regression inference procedures when:
You have two quantitative variables (explanatory and response)
You are using a linear regression model to describe the relationship
You want to estimate or test the slope of the population regression line
Use a confidence interval if the question asks to estimate the slope.
Use a significance test if the question asks whether there is convincing evidence of a relationship.
Note: Don’t use a chi-square test (used for categorical data) or a proportion test (used for categorical proportions).
| Procedure | Calculator | Formula | df |
|---|---|---|---|
| CI for slope | LinRegTInt |
\( b \pm t^* \cdot SE_b \) | \( n - 2 \) |
| Test for slope | LinRegTTest |
\( t = \frac{b - 0}{SE_b} \) | \( n - 2 \) |