The Sampling Distribution of the Slope
Understand that the LSRL slope b is a statistic that varies from sample to sample. Describe its sampling distribution: approximately normal with mean β (true slope) and standard error SE_b. Interpret computer regression output.
A lesson of AP Statistics, in Inference for Regression, about 10 minutes.
Sampling Distribution of the Slope
By the end of this lesson, you will be able to describe the sampling distribution of the slope b, identify its center, shape, and spread.
Why a Sampling Distribution?
The slope $b$ of the LSRL is a statistic — it varies from sample to sample. If we took many random samples from the same population and fit a regression line to each, the slopes would form a distribution. Understanding this distribution lets us make probability statements about $b$ and perform inference about the true slope $\beta$.
Properties of the Sampling Distribution of b
When all regression conditions (LINER) are satisfied:
Center: $\mu_b = \beta$
The slope $b$ is an unbiased estimator of the true slope $\beta$. On average, the LSRL gives the right slope.
Spread: The standard deviation of the sampling distribution is:
$$\sigma_b = \frac{\sigma}{\sigma_x \sqrt{n}}$$
where $\sigma$ is the standard deviation of the responses around the true line and $\sigma_x$ is the standard deviation of the x values.
In practice: $\sigma$ is unknown and estimated by $s_e$ (the standard error of the residuals from computer output). The standard error of the slope is:
$$SE_b = \frac{s_e}{s_x\sqrt{n-1}}$$
This value appears in the "Std Error" column of computer regression output.
Shape: Approximately normal when sample size is large (or when the population is normal).
Reading Computer Output
Predictor Coef SE Coef T P
Constant 12.5 2.31 5.41 0.000
Hours 0.85 0.12 7.08 0.000
S = 4.23 R-Sq = 71.2% R-Sq(adj) = 70.8%
Most AP Statistics exams provide computer regression output in this form:
- Coef (b): the estimated slope = 0.85
- SE Coef: standard error of the slope = 0.12
- T: the test statistic = b / SE_b = 0.85/0.12 = 7.08
- P: the p-value for testing $H_0: \beta = 0$
- S: the standard error of the residuals ($s_e = 4.23$)