9.5 Carrying Out a Test for the Slope of a Regression Model

To carry out a $t$ test for the slope of a regression model, calculate the test statistic $t = \frac{b - \beta_0}{SE_b}$, find the p-value from a $t$ distribution, and compare it to your significance level $\alpha$. If the p-value is at or below $\alpha$, reject the null hypothesis and use the context to state what the evidence says about the true population slope.

Why This Matters for the AP Statistics Exam

This topic is the "do" and "conclude" part of a regression slope hypothesis test. After you set up the test in 9.4, this is where you turn the regression output into a test statistic, a p-value, and a conclusion in context. On the AP Statistics exam, you may need to compute or read a t-statistic from computer output, interpret a p-value correctly, and write a conclusion that ties back to the research question. Free-response questions in this unit often start with a familiar slope task and build toward extended reasoning, so being fluent with the test steps helps you earn early points and set up later parts.

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Key Takeaways

• The appropriate procedure is a t-test for a slope, and under the null hypothesis the slope follows a t-distribution.
• The test statistic is t = (b - β₀)/SE_b, where b is the sample slope and β₀ is the value from the null hypothesis.
• Degrees of freedom are usually n - 2 for simple linear regression because the regression model estimates both slope and intercept; the CED also notes a one-parameter slope-test case that uses df = n - 1.
• The p-value is calculated assuming the null hypothesis is true (assuming the true population slope equals β₀).
• The decision rule is simple: if p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.
• Always write your conclusion in context, naming both variables and referencing the sample and population.

The t-Test Statistic

If the conditions for regression inference are met and the null hypothesis is true, the slope estimate follows a t-distribution. The test statistic measures how far your sample slope falls from the null slope, scaled by the standard error.

Take your observed slope, subtract the hypothesized slope, and divide by the standard error of the slope:

t = (b - β₀)/SE_b

Here b is your sample slope and β₀ is the value stated in the null hypothesis (often 0, meaning no linear relationship).

For degrees of freedom in simple linear regression, use df = n - 2, because two parameters (the slope and intercept) are estimated. The CED also notes a one-parameter slope-test case that uses df = n - 1, but most AP regression-output questions for simple linear regression use n - 2.

The p-Value

The p-value is the probability of getting a slope as extreme as, or more extreme than, your sample slope, assuming the null hypothesis is true. A small p-value means your observed slope would rarely happen by random chance if the true population slope really equaled β₀.

How you calculate the p-value depends on your alternative hypothesis:

• For Hₐ: β ≠ β₀, use a two-tailed p-value (both directions).
• For Hₐ: β > β₀ or Hₐ: β < β₀, use a one-tailed p-value (one direction).

Remember the interpretation: the p-value assumes the true population slope equals the value in the null hypothesis. It is not the probability that the null is true.

Making the Decision and Writing the Conclusion

Once you have your t-statistic and p-value, compare the p-value to your significance level α to make a formal decision.

• If p-value ≤ α, reject H₀.
• If p-value > α, fail to reject H₀.

Your conclusion should connect the decision back to the research question and name both variables in context. Use these as templates:

If the p-value is at or below α:

• "Since our p-value is less than our significance level, we reject H₀. We have convincing evidence that the true slope of the regression line between ________ and ________ is (the value in the alternative hypothesis, usually not 0)."

If the p-value is above α:

• "Since our p-value is greater than our significance level, we fail to reject H₀. We do not have convincing evidence that the true slope of the regression line between ________ and ________ is (the value in the alternative hypothesis, usually not 0)."

A test result that fails to reject H₀ does not prove there is no relationship. It only means you did not find enough evidence to support the alternative claim.

How to Use This on the AP Statistics Exam

Free Response

• Read computer output carefully. The slope estimate b, standard error SE_b, t-statistic, and p-value are usually right there in the regression table. Pull the correct values instead of recomputing when output is given.
• Show your test statistic formula with numbers plugged in, then state your p-value and df. Clear setup makes your work easy to follow.
• Watch whether the alternative is one-sided or two-sided. Some output reports a two-sided p-value, so you may need to halve it for a one-sided test.
• Always conclude in context and reference the population and sample. A correct decision with no context usually loses credit.

Problem Solving

• Confirm the conditions were addressed (linearity, equal variance, independence, normality) before trusting the test.
• Use the right degrees of freedom: n - 2 for simple linear regression.
• Tie the result to the research question. The test about the sample slope is your statistical reasoning to support a claim about the population slope.

Common Trap

• Do not interpret the p-value as the chance the null is true. It is computed assuming the null is true.
• Do not use deterministic language like "a 1-unit increase causes" when describing slope. Frame it as a predicted change.

Worked Example

A researcher studies the relationship between income and happiness using a sample of 50 individuals and runs a t-test for the slope.

• H₀: The slope of the regression line is 0 (no linear relationship between income and happiness).
• Hₐ: The slope is not 0 (there is a linear relationship).

The test statistic is t = 2.5 with df = 48 (since n = 50 and two parameters are estimated in simple linear regression). The resulting p-value is 0.01, and the significance level is α = 0.05.

Conclusion: Since the p-value of 0.01 is less than α = 0.05, reject H₀. There is convincing evidence of a linear relationship between income and happiness in the population. The reasoning: a p-value this small means a t-statistic as extreme as 2.5 would rarely occur by chance if the true slope were 0, so the null is unlikely to be correct.

For more practice and to work through choosing the right procedure, move on to Section 9.6.

Common Misconceptions

• The p-value is the probability the null hypothesis is true. It is not. The p-value is calculated by assuming the null is true, then measuring how unusual your sample slope is under that assumption.
• Failing to reject H₀ proves no relationship exists. It only means your sample did not provide enough evidence for the alternative. There may still be a real relationship you did not detect.
• The slope test always uses df = n - 1. For simple linear regression, the standard degrees of freedom are n - 2 because two parameters are estimated. The n - 1 form applies when the slope is the single parameter being tested.
• You always halve or always double the p-value. Match the p-value calculation to the alternative hypothesis: two-tailed for ≠, one-tailed for > or <.
• Slope describes a guaranteed change. Describe it as a predicted or expected change, not a deterministic one, and reference both variables.

Related AP Statistics Guides

• Unit 9 Overview: Slopes
• 9.2 Confidence Intervals for the Slope of a Regression Model
• 9.1 Introducing Statistics: Do Those Points Align?
• 9.4 Setting Up a Test for the Slope of a Regression Model
• 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval
• 9.6 Skills Focus: Selecting an Appropriate Inference Procedure