1. Inference for Regression Slope
a.k.a., “the end”
AP Statistics
Chapter 27

2. What would a scatterplot look like if there is NO ASSOCIATION???
If there is no association, we say that the slope is “zero”

3. An Example: Body Fat and Waist Size

4. (test scores vs time?)

5. c2 goodness-of-fit test
Regression slope
t-test
and
t-interval
c2 goodness-of-fit test
c2 test of homogeneity
c2 test of independence
2-way table
(1 sample)
Association between CATEGORICAL variables?
1 row or column
(1 sample)
2-way table
(2+ samples)
Association between NUMERICAL variables?

6. Assumptions/Conditions for regression inference
Do we have a random sample?
Does the data appear to be linear? There should be NO pattern in the plot of residuals vs x (or resids vs predicted y)
Equal Variance in “y” The scatterplot of “y” vs “x” should have roughly equal thickness in “y”.
Nearly Normal Condition Histogram of residuals should be unimodal & roughly symmetric, without outliers
Degrees of freedom:
df = n – 2

7. (quick review about those conditions…)

8. Residuals Residual: Observed y – Predicted y e=𝑦− 𝑦
(difference between observed value and predicted value)
Believe it or not, our “best fit line” will actually MISS most of the points.
Residual:
Observed y – Predicted y e=𝑦− 𝑦

9. Every point has a residual...
and if we plot them all, we have a residual plot.
We do NOT want a pattern in the residual plot!
This residual plot has no distinct pattern… so it looks like a linear model is appropriate.

10. OOPS!!! Does a linear model seem appropriate?
Although the scatterplot is fairly linear… the residual plot has a clear curved pattern. A linear model is NOT appropriate here.

11. Note about residual plots
residuals vs. 𝒙
and
residuals vs. 𝒚
will look the same
but don’t plot
residuals vs. 𝒚
(that will look different)

12. If all assumptions are true, the IDEALIZED regression model would look like this:
We can depict our distribution like this:

13. (okay back to the foldable…)

14. Regression Inference Ho: 1 = 0 t-test for regression slope
obs – exp
st. dev
t-test for regression slope
Define 1 (“true mean change in y for each unit increase in x”)
Ho: 1 = 0
Ha: 1 ≠ 0
There is NO linear association between “x” and “y” OR
There IS a linear association between “x” and “y”
statistic ±crit. value × st. err
t-interval for regression slope
THERE IS NO CALCULATOR SHORTCUT FOR THIS INTERVAL!
MAKE SURE YOU KNOW THIS FORMULA!!!

15. (examples on computer printout worksheet)
Notes 10.1 – Slope t-test and t-interval