Test of Regression Coefficient, ß1, with Two-tail P-values
1. This is another way to evaluate the simple regression model—along with interpretation (sign & magnitude)
as well as the R2.
2. Recall symbols: Y = ß0 + ß1 X
Yˆ  ˆ0  ˆ1 X
true line of means
estimated line of means
3. Hypotheses about ß1, the regression coefficient:
a. Can test for any value, but usually test for 0. This is what most software packages assume.
b. H0: ß1 = 0 If we substitute the 0, we have Y = ß0 + 0 X or Y = ß0 (a horizontal line)
What does this imply about the effect on Y when X changes?
Does this indicate a good or poor model? What does it say about the way we formulated the
model?
c. H1: ß1 _ 0 The sign in H1 depends on how you think the world works.
a. If you think X and Y change in the same direction, then ß1 > 0 (positive coefficient)
Example: When advertising increases, we expect sales to increase.
b. If you think X and Y change in opposite directions, then ß1 < 0 (negative coefficient)
Example: When absences increase, when expect grades to decrease.
c. If you think a change in X will affect Y but you’re not sure of the direction Y will change,
then ß1  0 (positive or negative coefficient)
Example: When income increases, we expect the demand for a good to shift, but we’re
not always sure what the direction of change will be.
4. Test statistic is ˆ1 , the estimate of ß1. The sampling distribution of ˆ1 is normal, if e is normally distributed
(recall the assumptions about e).
two-tail p-value areas
5. Excel assumes the test is two-tail and computes a two-tail
p-value. This is the appropriate p-value if you want to do
a two-tail test for ß1, where we have H1: ß1  0. For
example, suppose you estimate S=f(Ad) and obtain
Sˆ  50  5Ad . ˆ1 = 5. The p-value includes the areas
above ˆ values that are more than 5 units from the mean,
1
0. Suppose Excel says the p-value is 0.09 and α = 0.05.
Then we cannot reject H0, because 0.09 > 0.05.
-10
-5
0
5
10
6. If we want to do a one-tail test, we can convert the Excel
two-tail p-value to a one-tail p-value. However, we do not need to do this in all cases. When the estimate,
ˆ1 , has the wrong sign (sign doesn’t agree with H1 for the one-tail test you’re doing), then it is obvious that
we cannot reject H0. A ˆ with the wrong sign could never be in the rejection region (for typical α values)
1
because it falls on the wrong side of the diagram. It could never be strong evidence for an H1 that says ß1
should be the opposite sign.
If ˆ1 ’s sign agrees with H1, then we only need ½ of the Excel two-tail p-value (see above diagram). For
instance, if H1 is ß1 > 0 for the sales-advertising model, and we obtain Sˆ  50  5Ad , then the one-tail p-value
is P( ˆ1 > 5 when H0 is true) or the area for ˆ1 values that are 5 or more. This is ½ the 0.09 we used earlier or
0.045. If α = 0.05, then we reject H0, because 0.045 < 0.05.
Rule for decision for one-tail test for ß1 = 0 when using Excel two-tail p-value:
Step 1: Check the sign of ˆ1 . If it is incorrect, you cannot reject H0. If it is correct, go to step 2.
Step 2: The appropriate one-tail p-value is ½ of the Excel two-tail p-value. If one-tail p-value  α, then
reject H0.
7. Interpretation.
Template: We [have or do not have] strong evidence of H1 (remember to put H1 in terms of the problem).
Examples: from #5 where we could not reject H0 for the two-tail test
We do not have strong evidence, that advertising affects sales.
We do not have strong evidence, that an increase in advertising will change sales.
from #6 where we rejected H0 for the upper-tail test
We have strong evidence, that the coefficient of advertising is positive (technical rendition of
H1 for us in class)
We have strong evidence, that advertising positively affects sales.
We have strong evidence that an increase in advertising will lead to an increase in sales.