The sample slope estimates a population slope. Inference asks whether that population slope is really non-zero.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The null hypothesis
The null is that the population slope is zero, meaning no linear association between the variables.
The test statistic
t is the sample slope divided by its standard error, with n − 2 degrees of freedom.
Why n minus 2
Two parameters were estimated, the slope and the intercept, so two degrees of freedom are used up.
Reading computer output
Software prints the slope, its standard error, the t value and the p-value. Identifying the right row is most of the work.
Still not causation
A significant slope establishes association. Only a randomised experiment supports a causal claim.
Is the relationship real?
The sample slope is almost never exactly zero. Inference for the slope asks whether the true slope differs from zero, or whether the observed slope is within what sampling variation would produce.
The hypotheses
The null is that the population slope is zero — no linear relationship. The alternative is that it is not. As always, the hypotheses concern the population parameter, not the sample statistic.
The conditions
Linearity, independence, constant variance and roughly normal residuals — checked with a scatterplot and a residual plot. The exam expects both plots to be referenced, not merely the conditions listed.
A significant slope is not causation
Establishing that the slope is nonzero establishes association. Causation requires a randomised experiment. That distinction is examined repeatedly and is the most consequential idea in the course.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Ines Test a Slope
Output gives a slope of 2.4 with standard error 0.6, from a sample of 22.
- Step 1
The null is that the population slope is zero.
Step 4: Your Turn
Practice makes it stick.
The Statistic
Problem 1 of 2
Slope 2.4 with standard error 0.6. What is t?
The Degrees of Freedom
Problem 2 of 2
A regression on 22 points. What are the degrees of freedom?
Test the Relationship
1 of 8
Slope 1.5, standard error 0.5. What is t?
2 of 8
A regression on 30 points. Degrees of freedom?
3 of 8
What value does the null hypothesis give the slope?
4 of 8
Slope 0.8, standard error 0.4. What is t?
5 of 8
A slope interval runs from 0.4 to 2.1. Is a non-zero slope established? 1 yes, 0 no.
6 of 8
Does a significant slope prove causation? 1 yes, 0 no.
7 of 8
Put the slope inference procedure in order.
- 1Check the conditions, including the residual plot.
- 2Compute t as the slope over its standard error.
- 3Use n minus 2 degrees of freedom and conclude in context.
- 4State the null that the population slope is zero.
8 of 8
A regression on 50 points. Degrees of freedom?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Slope 3.0 with standard error 0.5. What is t?
Question 2 of 2
Why are the degrees of freedom n − 2?
What You Learned
- The null is that the population slope is zero, meaning no linear association.
- t is the slope divided by its standard error, on n − 2 degrees of freedom.
- A significant slope shows association, never causation.