1. Inference for Regression Lines
Chapter 12

2. Conditions LINER Linear relationship between x and y
Independent (10% rule)
Normal (check residuals)
Equal variance (equal scatter above/below “residual=0” line)
Random

3. WE MAY ALWAYS ASSUME ALL CONDITIONS ARE MET (EVEN ON THE AP EXAM)
WE MAY ALWAYS ASSUME ALL CONDITIONS ARE MET (EVEN ON THE AP EXAM)!!!!! WOOT WOOT!

4. Example
Women made significant gains in the 1970’s in terms of their acceptance into professions that had been traditionally populated by men. To measure just how big these gains were, we will compare the percentage of professional degrees award to women in to the percentage awarded in for selected fields of student from two random samples. (Statistics and Data Analysis, Siegel, Morgan, p.549)

5. Example continued
b) For every 1% increase in 72-73, there is an approximate increase of 1.72% in
c) We know that 88.6% of the variation in the percent of degrees awarded in can be explained by percent awarded in in the regression model..
d) Since the residual plot is random scatter, the data are app. linear

6. Example continued Residual = yactual – ypredicted
= 7.2 – ( (1.1))
= 7.2 – 8.9 = – 1.7
f) Linear Regression t-test
b = true slope for predicting percent of degrees in using
degrees in 72-73
Assume all conditions are met

7. Example continued p-value = .0005 df = n – 2 = 6 0.253 Let a = .05
We reject Ho. Since the p-value is less than a there is enough
evidence to believe that there is a linear relationship between
the percent of degrees in and in

8. Computer output Regression equation Estimate of a Explanatory variable
MTB> Regress ‘F%78-79’ 1 ‘F%72-73’ ‘SRES2’ ‘FITS2’;
The regression equation is
F%78-79 = F%72-73
PREDICTOR COEF STDEV T-RATIO P
Constant F%
s = R-sq = 88.6% R-sq(adj) = 86.7%
Analysis of Variance
SOURCE DF SS MS F P
Regression
Error
Total
Regression equation
Estimate of a
Explanatory variable
Estimate of b
t-score
p-value
SEb
Standard error of the line
Coefficient of determination

9. Example continued
g) The standard error (S) is the standard deviation for the residuals.
“On average, the difference between the actual and
predicted % of degrees awarded in ‘78-’79 (y) is 2.97%”

10. Example continued
g) The standard error (S) is the standard deviation for the residuals.
“On average, the difference between the actual and
predicted % of degrees awarded in ‘78-’79 (y) is 2.97%”
h) The standard error of the slope (SEb) is
“Over repeated sampling, the slope of the sample regression line
would typically vary by about from the slope in the true
regression line for predicting % of degrees awarded in ‘78-’79
using % of degrees awarded in ‘72-’73.”

11. For Your Notes… Standard Error (S) write-up:
“On average, the difference between the actual and predicted [y-variable] is [#].”
Standard Error for the Slope (SEb)write-up:
“Over repeated sampling, the slope of the sample regression line would typically vary by about [#] from the slope of the true regression line for predicting [y-variable] using [x-variable].”

12. 2) The following is a MiniTab output for chocolate shakes using ounces to predict calories.
On average, the difference between the actual and predicted calories is
b) Over repeated sampling, the slope of the sample regression line would typically vary by about from the slope of the true regression line for predicting calories using ounces.

13. Example continued c) Linear Regression t-interval
b = true slope for predicting calories based on number of ounces
Given all conditions are met.
We are 95% confident that the true slope for the regression line
for predicting calories using ounces lies between and 138.3
calories per ounce.