Topic 9.4 Notes – Setting Up a Test for the Slope of a Regression Model

What a t-test for the slope is actually testing

In simple linear regression, we model the population relationship as

$$\mu_y = \alpha + \beta x$$

• $\beta$ = true population slope
• It represents the expected change in the mean of y for each 1-unit increase in x.

A t-test for the slope answers:

Is there convincing evidence that the true slope $\beta$ differs from a hypothesized value $\beta_0$?

Most of the time:

• $H_0: \beta = 0$
  (No linear relationship between x and y.)
• If we reject this, we have evidence of a statistically significant linear association.

Because we don’t know the population standard deviation of y, the test statistic follows a t-distribution with $\text{df} = n - 2$.

If you see:

• Two quantitative variables
• A regression output
• A question asking whether there is a linear relationship

→ This is a t-test for the slope.

Writing hypotheses correctly

The hypotheses are always about the population slope $\beta$, not the sample slope $b$.

General form

• $H_0: \beta = \beta_0$
• $H_a: \beta < \beta_0$, $\beta > \beta_0$, or $\beta \ne \beta_0$

Where:

• $\beta$ = true population slope
• $\beta_0$ = hypothesized value (often 0)

Choosing the alternative

• Two-sided $(\beta \ne \beta_0)$
  Used when checking for any linear relationship.
• Right-tailed $(\beta > \beta_0)$
  When context suggests a positive association.
• Left-tailed $(\beta < \beta_0)$
  When context suggests a negative association.

On a free-response question, define $\beta$ in context. For example:

$\beta$ is the true slope relating number of practice problems completed to exam score for all students at this school.

That definition earns points.

Conditions for inference

Before doing the test, you must verify four conditions. These are about the residuals.

A residual plot like the one below is what you use to check linearity and equal variability.

Residuals versus fitted values plot

1. Linearity

The true relationship between x and y is linear.
Check the residual plot. It should show no clear pattern.

2. Equal variability

The standard deviation of y is the same for all values of x.
On the residual plot, look for no fanning out or funneling in.

3. Independence

• Data come from a random sample or randomized experiment.
• If sampling without replacement, check the 10% condition
  $n \le 0.10N$

4. Normality

For each value of x, the distribution of y-values is approximately normal.

We usually check a histogram or normal probability plot of residuals.

• If $n \ge 30$, slight skewness is okay.
• If $n < 30$, no strong skewness or outliers.

Students lose points by forgetting to mention how they checked these. Be specific.

Mechanics of the test

Once conditions are met:

1. State hypotheses (with $\beta$ defined in context).
2. Calculate the test statistic

$$t = \frac{b - \beta_0}{SE_b}$$

• $b$ = sample slope
• $SE_b$ = standard error of the slope
• $\text{df} = n - 2$

3. Find the p-value using the t-distribution.
4. Conclusion in context

Example structure:

Because the p-value is 0.012, which is less than $\alpha = 0.05$, we reject $H_0$. There is convincing evidence of a linear relationship between practice time and exam score for students at this school.

Notice the conclusion:

• Mentions linear relationship
• Names the variables and population

Calculator note: use LinRegTTest. It gives you $t$, the p-value, $b$, and $SE_b$.

What the result means

If you reject $H_0$:

• There is statistically significant evidence that $\beta \ne \beta_0$.
• If $\beta_0 = 0$, there is evidence of a linear association.

If you fail to reject $H_0$:

• Not enough evidence of a linear relationship.
• This does not prove the slope is 0.

And unless the data came from a randomized experiment, you cannot claim causation.