Inference for Regression

1
Chapter 10

• Inference for Regression

2
(No Transcript)
3
(No Transcript)
4
(No Transcript)
5
y
Note bi ? 0 could also be a negative slope...
b1 ? 0
b1 0
x
6
Hypothesis Tests for Slope Coefficients
Since we are interested in knowing whether the
independent variable has a significant effect on
the dependent variable, the Null hypothesis that
we set up is that there is NO effect Null
?10 ? .05 Alt ?1? 0 Test
Statistic
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
Apartment Rent Scenario

• Suppose you wish to study apartment rentals in a
  college town. You feel that the primary
  characteristics related to rent are
• Size (in square feet)
• Age (in years)
• Amount of amenities (1 if many 0 if minimal)
• You take a random sample of 100 apartment rentals
  and use EXCEL to estimate the multiple regression
  function.

14
Your (example) output from 100 apartment rentals
(y monthly rent).
Coefficients Standard Error t Stat P-value
Intercept 50.3 62.2 0.81
0.421 SqFeet 0.61 .049
12.6 0.000 Age -9.05
3.9 -2.32 0.043 Amenity 95.7
19.5 4.91 0.000
We can use this output to conduct tests for the
population slope coefficients (?i).
2-tailed
15
Your (example) output from 100 apartment rentals
(y monthly rent).
Coefficients Standard Error t Stat P-value
Intercept 50.3 62.2 0.81
0.421 SqFeet 0.61 .049
12.6 0.000 Age -9.05
3.9 -2.32 0.023 Amenity 95.7
19.5 4.91 0.000
Note that the t Stat gives
16
Does Square Footage Have a Significant Impact on
Rent?

• Ho ?1 0
• Ha ?1 ? 0 Let a .01

Since p-value lt .001 lt a (.01) we reject the
null, conclude Ha Square Footage is a
significant independent variable.
t .61/.049 12.6. With df 100 largest t-table
value 3.390 gives tail area .0005. Since 12.6 gt
3.390, 1-tailed p-value lt .0005, two tailed
p-value lt .001
-3.390
3.390
17
Your (example) output from 100 apartment rentals
(y monthly rent).
Coefficients Standard Error t Stat P-value
Intercept 50.3 62.2 0.81
0.421 SqFeet 0.61 .049
12.6 0.000 Age -9.05
3.9 -2.32 0.023 Amenity 95.7
19.5 4.91 0.000
The P-value gives the probability (if the null
were true) of getting a t-test statistic as far
(or farther) from 0 than our sample t. If b10
were true, the chance that b1 is 12.6 std. errors
away from 0 is close to 0 (000.0).
18
Does Rent Decrease with Apartment Age?

• Ho ?2 0
• Ha ?2 lt 0 Let a .05

t -9.05/3.9 -2.32. With df 100, 2.32 lies
between 2.081 2.364 so one-tailed p-value lies
between .02 .01
Since p-value lt .05 ( a) we reject the null,
conclude Ha. Rent significantly decreases with
apartment age.
- 2.364 - 2.05 - 2.081
19
Your (example) output from 100 apartment rentals
(y monthly rent).
Coefficients Standard Error t Stat P-value
Intercept 50.3 62.2 0.81
0.421 SqFeet 0.61 .049
12.6 0.000 Age -9.05
3.9 -2.32 0.023 Amenity 95.7
19.5 4.91 0.000

• The p-value for Age is .023. If b20 were true,
  the prob.
• of obtaining a b2 that is 2.32 or more std.
  errors away
• from 0 is .023/2 .011 Since .011 is less than
  a .05,
• this is consistent with our estimated p-value
  rejection

20
Does the Amount of Amenities Have a Positive
Impact on Rent?

• Ho ?3 0
• Ha ?3 gt 0 Let a .01

t 95.7/19.5 4.9. With df 100 largest
t-table value 3.390 gives tail area .0005. Since
4.9 gt 3.390, one-tailed p-value lt .0005
Since p-value lt .0005 lt a (.01) we reject the
null, conclude Ha. The amount of amenities has a
positive impact on rent.
3.390
21
Testing Significance of Slope Coefficients with
Confidence Intervals
If confidence interval contains zero the slope
coefficient is not significant and the
corresponding X-variable is not important in
predicting Y variable
22
Testing Significance of Slope Coefficients with
Confidence Intervals
Since none of the confidence intervals for the
slope coefficients in the model (sqft, age,
amenity) contains zero, all the variables are
important in predicting the monthly rental
23
Testing Significance of Slope Coefficients with
Confidence Intervals
Does rent decrease with apt. age? Yes, since we
are 95 confident that for each additional year
in age the mean apt. rental will decrease by an
amount between 16.80 and 1.30 (all other
factors being equal) Do amenities have a positive
impact on monthly rental? Yes, since we are 95
confident that apts. with amenities will rent for
an amount between 56.99 and 134.48 more than
apts. without (all other factors being equal)
24
Group Work Refer to the handout on condo prices.
Does condo price a) decrease with floor? b)
increase with distance from elevator? Does ocean
view affect condo price? Conduct formal
hypothesis tests. Use a .01. Construct a 99
confidence interval for the floor coefficient and
interpret the interval.
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
Output from PredInt Macro
29
(No Transcript)
30
Prediction and Estimation in Multiple Regression
Ex. 11.30, pg.657 Predicting College GPA for
C-Students Based on High School Math, Science and
English Scores