Unit 9: Inference for Quantitative Data: Slopes

This unit extends linear regression from Unit 2 into inference. Instead of only describing the least-squares line for a sample, we ask whether there is evidence of a linear relationship in the population.

Population Regression Model

For a population, the linear regression model is

\[Y = \alpha + \beta x + \epsilon.\]

Here:

  • \(\alpha\) is the population intercept.
  • \(\beta\) is the population slope.
  • \(\epsilon\) is random error around the line.

The sample least-squares line is

\[\hat{y} = a + bx,\]

where \(b\) estimates the true slope \(\beta\).

The parameter of interest in AP Statistics is usually \(\beta\), the true slope of the population regression line.

Conditions For Regression Inference

Use the acronym LINER:

  1. Linear: scatterplot and residual plot show an approximately linear relationship.
  2. Independent: observations are independent; if sampling without replacement, check the 10% Condition.
  3. Normal: residuals are approximately normally distributed for each value of \(x\).
  4. Equal variance: vertical spread of residuals is roughly constant across \(x\).
  5. Random: data come from a random sample or randomized experiment.

Regression residual plots placeholder

Standard Error Of The Slope

The standard error of the slope is written \(SE_b\). Technology usually reports it directly.

Conceptually, \(SE_b\) measures how much the sample slope \(b\) would vary from sample to sample. Smaller residual scatter and more spread in the explanatory variable make the slope estimate more precise.

The residual standard deviation is often written \(s\):

\[s = \sqrt{\frac{\sum (y_i-\hat{y}_i)^2}{n-2}}.\]

The degrees of freedom for regression inference are

\[df = n-2.\]

Confidence Interval For Slope

A confidence interval for the true slope \(\beta\) is

\[b \pm t^*SE_b.\]

Use

\[df=n-2.\]

Interpretation: “We are __% confident that the true slope of the population regression line relating [x-context] to [y-context] is between __ and ___ [y-units per x-unit].”

If the interval contains 0, then 0 is a plausible true slope at that confidence level. If it does not contain 0, there is evidence of a linear relationship.

t-Test For Slope

The usual hypotheses are

\[H_0:\beta=0\]

and one of

\[H_a:\beta>0,\qquad H_a:\beta<0,\qquad H_a:\beta\ne0.\]

The test statistic is

\[t = \frac{b-0}{SE_b},\]

with

\[df=n-2.\]

A small p-value gives evidence that the true slope is not 0 in the direction of the alternative.

Interpreting Regression Output

Technology output often includes:

  • Coefficient estimate for intercept \(a\).
  • Coefficient estimate for slope \(b\).
  • Standard error of slope \(SE_b\).
  • t statistic for slope.
  • p-value for slope.
  • \(s\), the standard deviation of residuals.
  • \(R^2\), the percent of variability in \(y\) explained by the linear relationship with \(x\).

Remember: \(R^2\) is descriptive, while the p-value for slope is inferential. A small p-value does not prove causation, especially with observational data.

Residual Plots And Diagnostics

Before inference, check:

  • Scatterplot of \(y\) versus \(x\) for form and outliers.
  • Residual plot for randomness around 0.
  • Residual plot for equal spread.
  • Normal probability plot or histogram of residuals for approximate normality.

Curvature suggests the linear model is wrong. Fanning suggests unequal variance. Outliers or high-leverage points can strongly affect the slope and p-value.

Calculator Notes

Common calculator tool:

  • LinRegTTest: tests slope and gives regression output.

Many calculators also provide LinRegTInt or interval output depending on model/software. If not, compute

\[b \pm t^*SE_b\]

from the regression output.

Working Checklist

  1. Identify the explanatory variable \(x\) and response variable \(y\).
  2. State the parameter \(\beta\) in context.
  3. Check LINER conditions using plots and study design.
  4. Use \(df=n-2\).
  5. Compute the confidence interval or t-test for slope.
  6. Interpret slope in units of \(y\) per one unit of \(x\).
  7. Avoid causal language unless the data come from a randomized experiment.

Key Equations

Idea Equation
Population model \(Y=\alpha+\beta x+\epsilon\)
Sample regression line \(\hat{y}=a+bx\)
Residual standard deviation \(s=\sqrt{\sum(y_i-\hat{y}_i)^2/(n-2)}\)
Confidence interval for slope \(b\pm t^*SE_b\)
Test statistic for slope \(t=(b-0)/SE_b\)
Degrees of freedom \(df=n-2\)
Made by Joshua Bie