9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval

A confidence interval for the slope gives you a range of plausible values for the true slope of a population regression line. To justify a claim about the relationship between two variables, check whether 0 falls inside that interval: if 0 is not in the interval, you have evidence of a real linear relationship; if 0 is inside, you cannot rule out no linear relationship.

How Do You Use a Confidence Interval for Slope?

Use a confidence interval for slope by checking which slope values are plausible for the population regression line. If the interval is entirely positive or entirely negative, you have evidence of a linear relationship in that direction. If the interval contains 0, a slope of 0 is plausible, so the interval does not provide convincing evidence of a linear relationship.

Your AP Stats answer should always interpret the interval in context. Name the explanatory and response variables, reference the population represented by the sample, and connect the interval to the claim you are evaluating.

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Why This Matters for the AP Statistics Exam

Inference for slopes shows up in both multiple-choice and free-response questions, and this skill builds directly on the confidence intervals you constructed in earlier units. Slope questions often start with computer (regression) output, so you need to pull the right numbers and interpret them in context. Being able to interpret a slope interval and use it to support or reject a claim is exactly the kind of reasoning AP questions reward, and it sets you up for the significance test for slope that comes next in this unit.

Key Takeaways

• A confidence interval for the slope estimates the true population slope $\beta$, using the form $b \pm t^*(SE_b)$ with $n-2$ degrees of freedom.
• The confidence level (like 95%) describes how often intervals from repeated samples would capture the true slope, not the probability for one specific interval.
• To justify a claim about correlation, check whether 0 is inside the interval. If 0 is not included, there is evidence of a linear relationship.
• A correct interpretation references both the sample and the population it represents.
• The interval gets narrower as the sample size increases, and wider as the confidence level increases.
• Use the t critical value that matches your confidence level and degrees of freedom, not a t value printed in the output.

How the Confidence Interval Works

A confidence interval gives you plausible values for the true slope of the population regression line. The point estimate is the slope of the line of best fit, $b$, and the interval estimate is:

$b \pm t^*(SE_b)$

For example, if your confidence interval for the slope is $(1.35, 2.7)$, every value in that range is a plausible slope. Since both ends are positive, you can be fairly confident the relationship is positive and the true slope sits somewhere between 1.35 and 2.7.

Confidence Level

The confidence level sets how strict your interval is. It reflects the percentage of intervals that would contain the true slope if you took many random samples of the same size from the same population.

If you build a 95% confidence interval for the slope, that means if you took many random samples of the same size from the same population, about 95% of the resulting intervals would capture the true slope of the population regression line.

Interpreting the Interval

A good interpretation names the sample and describes the population. For an interval of $(1.35, 2.7)$, you could say:

• We are 95% confident that the true slope of the regression line relating variable A and variable B is between 1.35 and 2.7.
• In repeated random sampling with the same sample size, approximately 95% of the intervals created would capture the true slope of the population regression model.

This is similar to the interpretations for proportions and means from earlier units, just adjusted to estimate the true slope.

Two things change the width of the interval:

• Width decreases as the sample size increases, because a larger sample lowers the standard error.
• Width increases as the confidence level increases.

How to Use This on the AP Statistics Exam

Justifying a Claim

To use a slope interval to justify a claim about correlation, check whether 0 is inside the interval.

• If 0 is inside the interval, then 0 is a plausible value for the slope. A slope of 0 means no linear relationship, so you do not have convincing evidence of a linear relationship.
• If 0 is not inside the interval, you have evidence of a linear relationship. For the interval $(1.35, 2.7)$, 0 is not included, so you can be 95% confident the slope is positive and the two variables have a positive linear relationship.

Problem Solving

Many slope questions on the exam give you computer (regression) output and ask you to build and interpret the interval. Here is a worked example.

Assume a sample size of 40 and a 95% confidence level.

1. Find the degrees of freedom: $n - 2 = 40 - 2 = 38$.

2. Find the t critical value using invT for 95% confidence with 38 degrees of freedom. This gives $t^* \approx 2.02$.

3. Build the interval using the slope estimate and the standard error from the output. With a slope of $0.448$ and a standard error of $0.6565$:

$0.448 \pm 2.02(0.6565)$

This gives an interval of about $(-0.878, 1.774)$.

Since 0 is contained in this interval, you do not have evidence of a linear relationship. A low $R^2$ value and the matching low $r$ value (about 0.176) point to the same conclusion.

Common Trap

The output usually lists a t value for the slope already. That printed value is the test statistic for the sample, not the t critical value for your confidence interval. Always compute your own $t^*$ from the confidence level and degrees of freedom.

Common Misconceptions

• "95% confident" means a 95% probability the true slope is in this one interval. The confidence level describes the long-run success rate of the method across many samples, not the probability for a single interval.
• A nonzero slope estimate alone proves a relationship. You need the whole interval. If 0 is among the plausible values, you cannot rule out no linear relationship.
• Failing to reject no relationship proves there is no correlation. An interval that contains 0 means you lack convincing evidence of a linear relationship, not that the variables are definitely unrelated.
• Use the t value from the output for the interval. That printed value is the test statistic for the sample, not the critical value $t^*$ tied to your confidence level and degrees of freedom.
• Bigger confidence level is always better. Raising the confidence level widens the interval, which makes your estimate less precise. There is a tradeoff between confidence and precision.
• Forgetting context in the interpretation. A complete interpretation references the sample taken and the population it represents, not just the two numeric endpoints.

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
• 9.5 Carrying Out a Test for the Slope of a Regression Model
• 9.6 Skills Focus: Selecting an Appropriate Inference Procedure