AP STATISTICS
LESSON 14 – 1
(DAY 2)
CONFIDENCE INTERVALS FOR
THE REGRESSION LINE
ESSENTIAL QUESTION:
How are confidence intervals and
significance tests made and used for
inference when working with regression
lines?
Objectives:
• To create significance tests for regression.
• To create confidence intervals for regression
lines.
• To make inferences dealing with regression
situations.
Technology Tips:
Here’s a quick way to calculate s:
• With x-values in L1 / list1 and the y-values in
L2 / list 2.
• Perform least-squares regression.
• The calculator creates or updates a list
named RESID. Store the results in L3 /list 3.
Then specify 1 – var Stat L3 and look at the
value ∑ x2.
• Dividing this number by (n – 2) to get s2 .
• Take the square root to obtain s.
Confidence Intervals for the
Regression Slope
The slope β of the true regression line is
usually the most important parameter in
a regression problem.
The slope is the rate of change of the
mean response as the as explanatory
variable increases.
Confidence Interval for
Regression Slope
A level C confidence interval for the slope the
slope β of the true regression line is
b ± t* SEb
In this recipe, the standard error of the leastsquares slope b is
SEb =
s
√∑ ( x – x )2
And t* is the upper ( 1 – C)/2 critical value with
n – 2 degrees of freedom.
Testing the Hypothesis of
No Linear Relationship
We can also test hypothesis about the slope
β. The most common hypothesis is
Ho : β = 0
A regression line with a slope of 0 is
horizontal. That is, there is no true linear
relationship between x and y.
You can use the test for zero slope to test the
hypothesis of zero correlation between any
two quantitative variables.
Significance Tests for Regression
Slopes
To test the hypothesis Ho : β = 0, compute the
t statistic
t= b
Seb
To test the hypothesis
Ho : β = 0, compute the t statistic t = b/ SEb
Significance tests for Regression
Slopes (continued…)
In terms of a random variable T
having the t(n – 2) distribution, the Pvalue for a test of Ho against
Ha : β > 0 is P( T ≥ t )
Ha : β < 0 is P( T ≤ t )
Ha : β ≠ 0 is 2P( T ≥ l t l)
This test is also a test of the
hypothesis that the correlation is 0 in
the population.
Testing the Hypothesis of No
Linear Relationship
H0 : β = 0
A regression line with slope 0 is horizontal.
That is, the mean of y does not change at all when
x changes.
Put another way Ho says that straight-line
dependence on x is of no value for predicting y.
Regression output from statistical software usually
gives t and its two-sided P-value.
For a one-sided test, divide the P-value in the
output by 2.
Example 14.4
Page 789
Regression Output: Crying and IQ
Minitab regression output fo the crying and IQ data
You can find a confidence interval for the intercept a of
the true regression line in the same way, using a and
SEa from the “Constant” line of the printout. We rarely
need to estimate a.
Example 14.5 Page 791
Testing Regression Slope
The hypothesis H0 : β = 0 says that crying has no
straight-line relationship with IQ. Figure 14.1 shows that
relationship, so it is not surprising that the computer
output in Figure 14.4 gives t = 3.07 with two-sided Pvalue 0.004. There is very strong evidence that IQ is
correlated with crying.
Figure 14.1
Figure 14.4
Example 14.6 Page 791
Beer and Blood Alcohol
How well does the number of beers a student drinks
predict his or her blood alcohol content?
• 16 volunteers drank a
randomly assigned number
of beers
• 30 minutes later, their
blood alcohol content (BAC)
was measured.
The scatterplot shows one unusual point: student number 3
may also be influential , though the point is not extreme in
the x direction (solid line). To verify that our results are not
too dependent o n this one observation, do the regression
again omitting student # 3 (dashed line).