Cogito
AP Statistics · Chapter 5 · Lesson 3
Inference for the Slope
Is the relationship real?
12 problems · about 22 minutes · VAR-7.J, DAT-3.K
What this lesson teaches
The student tests and estimates the slope of a regression line.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsSlope 3.0 with standard error 0.5. What is t?
Answer 6
Why 6.
Why are the degrees of freedom n − 2?
Answer Two parameters were estimated: the slope and the intercept.
Why The slope and intercept each use one.
Slope 1.5, standard error 0.5. What is t?
Answer 3
Why 1.5 ÷ 0.5.
A regression on 30 points. Degrees of freedom?
Answer 28
Why n − 2.
What value does the null hypothesis give the slope?
Answer 0
Why No association.
Build It Up
The same ideas with more to keep track of.
3 problemsSlope 0.8, standard error 0.4. What is t?
Answer 2
Why 0.8 ÷ 0.4.
A slope interval runs from 0.4 to 2.1. Is a non-zero slope established? 1 yes, 0 no.
Answer 1
Why Zero is outside.
Does a significant slope prove causation? 1 yes, 0 no.
Answer 0
Why Association only.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the slope inference procedure in order.
Answer 1. State the null that the population slope is zero. 2. Check the conditions, including the residual plot. 3. Compute t as the slope over its standard error. 4. Use n minus 2 degrees of freedom and conclude in context.
Why Conditions come before computation.
A regression on 50 points. Degrees of freedom?
Answer 48
Why n − 2.
The Statistic: Slope 2.4 with standard error 0.6. What is t?
Answer 4
Why 4.
The Degrees of Freedom: A regression on 22 points. What are the degrees of freedom?
Answer 20
Why 20.