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

What a confidence interval for the slope means

When you build a regression model from sample data, you get a least squares slope $b$. That’s your point estimate of the true population slope $\beta$.

A confidence interval for $\beta$ gives a range of plausible values for the true slope of the population regression line.

The form is:

$$b \pm t^* \cdot SE_b$$

• $b$ = sample slope
• $SE_b$ = standard error of the slope
• $t^*$ = critical value with $df = n - 2$

What the slope represents

The slope tells you:

For each 1-unit increase in $x$, the predicted value of $y$ changes by about ___ units (on average).

So if a 95% CI for the slope is (2.1, 4.8), that means each additional 1 unit of $x$ increases predicted $y$ by somewhere between 2.1 and 4.8 units in the population.

Proper interpretation (in context)

You must reference:

• The sample
• The population
• The variables

Model language example:

We are 95% confident that the true slope of the regression line relating hours studied to exam score for all students at this school is between 2.1 and 4.8 points per hour.

Notice:

• “True slope”
• Specific variables
• Named population

What “95% confident” actually means

It does not mean there is a 95% probability the slope is in this interval.

It means:

If we took many random samples of the same size and built a 95% confidence interval each time, about 95% of those intervals would capture the true slope $\beta$.

The confidence level describes the method’s long-run success rate.

Using the interval to justify a claim

This is the skill your teacher and the AP exam care about most.

Everything revolves around whether 0 is in the interval.

Why 0? Because if $\beta = 0$, there is no linear relationship between $x$ and $y$.

If 0 is NOT in the interval

• $\beta = 0$ is not plausible.
• There is convincing evidence of a linear relationship.
• The sign tells you the direction.

Example: CI = (-3.2, -0.5)
Since 0 is not in the interval, there is evidence of a negative linear relationship in the population.

If 0 IS in the interval

• A slope of 0 is plausible.
• You do not have convincing evidence of a linear relationship.

Example: CI = (-1.4, 2.6)
Since 0 is in the interval, we do not have convincing evidence of a linear relationship between the variables in the population.

Be careful. Students often say “there is no relationship.” That’s too strong. You say:

We do not have convincing evidence of a linear relationship.

How this connects to hypothesis testing

A confidence interval for the slope is linked to a two-sided test of:

$$H_0: \beta = 0$$

At significance level $\alpha$:

• If 0 is outside a $100(1 - \alpha)\%$ CI → you would reject $H_0$.
• If 0 is inside → you would fail to reject $H_0$.

So the interval gives the same decision as a two-sided t-test about the slope.

How the interval is built from output

You’ll often get regression output with:

• Slope $b$
• Standard error $SE_b$
• Sample size $n$

To construct the interval:

1. Compute $df = n - 2$
2. Find $t^*$ for your confidence level
3. Calculate $b \pm t^* SE_b$

Important distinction:

• The reported t statistic in output is for testing $\beta = 0$.
• The critical value $t^*$ is used to build the confidence interval.

The interval is always centered at $b$, not 0.

What affects the width of the interval

The width is:

$$2 \cdot t^* \cdot SE_b$$

Two main drivers:

Sample size $n$

Larger $n$ → smaller $SE_b$ → narrower interval.

More data gives a more precise estimate of the slope.

Confidence level

Higher confidence → larger $t^*$ → wider interval.

You gain confidence but lose precision.

Common mistakes to avoid

• Saying “There is a 95% probability the slope is in this interval.”
• Forgetting to mention the population.
• Concluding causation from observational data.
• Ignoring 0 when justifying a claim.
• Switching to correlation language when the question asks about the slope.