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Lesson 14 - 1
Testing the Significance of the
Least Squares Regression Model
Objectives
• Understand the requirements of the leastsquares regression model
• Compute the standard error of the estimate
• Verify that the residuals are normally
distributed
• Conduct inference on the slope and intercept
• Construct a confidence interval about the
slope of the least-squares regression model
Vocabulary
• Bivariate normal distribution – one variable
is normally distributed given any value of the
other variable and the second variable is
normally distributed given any value of the
first variable
• Jointly normally distributed – same as
bivariate normal distribution
Least-Squares Regression Model
yi = β0 + β1xi + εi
where
yi is the value of the response variable for the ith individual
β0 and β1 are the parameters to be estimated based on the
sample data
xi is the value of the explanitory variable for the ith individual
εi is a random error term with mean 0 and variance σ²εi = σ²
The error terms are independent and normally distributed
I = 1, … , n where n is the sample size (number of ordered
pairs in the data set)
Requirements for Inferences
• The mean of the responses depends linearly on the
explanatory variable
– Verify linearity with a scatter plot (as in Chapter 4)
• The response variables are normally distributed with
the same standard deviation
– We plot the residuals against the values of the explanatory
variable
– If the residuals are spread evenly about a horizontal line
drawn at 0, then the requirement of constant variance is
satisfied
– If the residuals increasingly spread outward (or
decreasingly contract inward) about that line at 0, then the
requirement of constant variance may not be satisfied
Hypothesis Tests
• Only after requirements are checked can we
proceed with inferences on the slope, β1, and
the intercept, β0
• Tests:
Two-Tailed
H0: β1 = 0
H1: β1 ≠ 0
Left Tailed
H0: β1 = 0
H1: β1 < 0
Right Tailed
H0: β1 = 0
H1: β1 > 0
Note: these procedures are considered robust (in
fact for large samples (n > 30), inferential procedures
regarding b1 can be used with significant departures
for normality)
Test Statistic
t0 =
(b1 – βi)
--------------- =
se
-------------Σ (xi – x)2
b1
-----s b1
Note: degrees of freedom = n – 2
H0: β1 = 0
tα/2 for two-tailed (≠0)
tα for one-tailed (>0 or <0)
Standard Error of the Estimate
2
(y
–
y
)
i
i
Σ
se = --------------- =
n–2
2
residuals
Σ
---------------n–2
Note: by divide by n – 2 because we have estimated two
parameters, β0 and β1
Conclusions from Test
Rejecting the null hypothesis means that for
• The two-tailed alternative hypothesis, H1: β1 ≠ 0
– The slope is significantly different from 0
– There is a significant linear relationship between the variables
x and y
• The left-tailed alternative hypothesis, H1: β1 < 0
– The slope is significantly less than 0
– There is a significant negative linear relationship between the
variables x and y
• The right-tailed alternative hypothesis, H1: β1 > 0
– The slope is significantly greater than 0
– There is a significant positive linear relationship between the
variables x and y
Hypothesis Testing on β1
Steps for Testing a Claim Regarding the Population Mean
with σ Known
0. Test Feasible (requirements)
1.Determine null and alternative hypothesis
(and type of test: two tailed, or left or right tailed)
2.Select a level of significance α based on
seriousness of making a Type I error
3.Calculate the test statistic
4.Determine the p-value or critical value using level of
significance (hence the critical or reject regions)
5.Compare the critical value with the test statistic
(also known as the decision rule)
6.State the conclusion
Example
Confidence Intervals for β1
Confidence intervals are of the form
Point estimate ± margin of error
se
Lower bound = b1 – tα/2 --------------- = b1 - tα/2 · sb1
2
(x
–
x)
Σ i
se
Upper bound = b1 + tα/2 --------------- = b1 + tα/2 · sb1
Σ (xi – x)2
note: tα/2 degrees of freedom = n – 2
pre-conditions:
1) data randomly obtained
2) residuals normally distributed
3) constant error variance
Using TI
• Enter explanatory variable in L1 and the
response variable in L2
• Press STAT, highlight TESTS and select
E:LinRegTTest
• Be sure Xlist is L1 and Ylist is L2. Make sure
that Freq is set to 1. Set the direction of the
alternative hypothesis. Highlight calculate
and ENTER.
Summary and Homework
• Summary
– Confidence intervals and prediction intervals
quantify the accuracy of predicted values from leastsquares regression lines
– Confidence intervals for a mean response measure
the accuracy of the mean response of all the
individuals in a population
– Prediction intervals for an individual response
measure the accuracy of a single individual’s
predicted value
• Homework
– pg 748 - 752; 1, 2, 3, 4, 7, 12, 13, 18