ASSIGNMENT 24
Assumptions, R-square adjusted, and Testing Hypotheses.
1. In class 22 we saw four clouds of points (A, B, C, D) that produced identical regression lines and
regression statistics. Match the data set with the regression assumption most clearly violated . (This is a
matching question, draw lines connecting each of the data sets with one (and only one) entry from the
second column.) One of the data sets did not clearly violate any of the three listed assumptions. [20
points]
Data Set
A
B
C
D
Assumption Violated
Linearity
Homoskedasticity
Normality
none
A definitely violates Linearity. B will have two large positive residuals and 9 small
negative residuals. Thus the distribution of residuals will not be symmetric. B
violates the normality assumption. C looks fine. D will have scattered residuals
for the high X value and a single residual of zero at the low X value. This might
indicate heteroskedasticiy. Thus D violates homoscedasticity.
Using n (X,Y) pairs, Al regressed Y on X. Bo appended a copy of the data set to the original data set and
also regressed Y on X. This meant that each X,Y pair in Al’s data set appeared twice in Bo’s. Bo’s sample
size was 2n. Assume X and Y are positively correlated.
2. This question asks how Bo’s regression line will compare to Al’s. (Circle one answer.) [10 points]
A. Bo’s line will be steeper than Al’s.
B. Bo’s line will be the same slope as Al’s.
C. Bo’s line will be less steep than Al’s.
D. How Bo’s line compares to Al’s will depend on the data.
The correct answer is B. Al and Bo’s lines will be identical.
3. Both Al and Bo test H0: b=0 versus Ha: b≠0. How will their p-values compare? (Circle one answer.)
[10 points]
A. Bo’s p-value will be lower than Al’s.
B. Bo’s p-value will be equal to Al’s
C. Bo’s p-value will be greater than Al’s.
D. It depends on the data.
The correct answer is A. The coefficients will be identical and the standard error
of the model will be about equal (Bo has the same ten residuals as Al…twice….the
scatter of Bo’s 20 residuals will be the same as Al’s.) But with n=20, the standard
error of Bo’s coefficient will be lower. Given the coefficient is positive (as
mentioned in the question), Bo’s t will be higher and p-value lower. Once again,
Bo has CHEATED by doubling his data….his cheating is rewarded with a lower pvalue.
4. (EMBS problem 22) PC World provided ratings for the top five small-office laser printers and five
corporate laser printers (PC World, Feb 2003). The following data show the speed for plain text printing
in pages per minute (ppm) and the price of the printer.
Name
Minolta-QMS PagePro 1250W
Brother HL-1850
Lexmark E320
Minolta-QMS PagePro 1250E
HP Laserjet 1200
Xerox Phaser 4400/N
Brother HL-2460N
IBM Infoprint 1120n
Lexmark W812
Oki Data B8300n
Type
Small Office
Small Office
Small Office
Small Office
Small Office
Corporate
Corporate
Corporate
Corporate
Corporate
Speed
12
10
12.2
10.3
11.7
17.8
16.1
11.8
19.8
28.2
Price
199
499
299
299
399
1850
1000
1387
2089
2200
a. Regress Price on Speed and report the resulting regression equation. (One can simply cut and paste
the data above into excel….or key in the data.) [15 points]
Intercept
Speed
Coefficients
-745.4806
117.91732
The predicted price is -745.5 + 117.9 * Speed (in ppm). As expected, the predicted
price increases with speed. The best guess for the rate of increase is $117.9 per
ppm.
b. What is the adjusted R-square? [5 points]
Regression Statistics
Multiple R
0.840892
R Square
0.7070994
Adjusted R Square
0.6704869
Standard Error
458.02486
Observations
10
The adjusted R square is 67%. 67% of the variation of price (in our ten data
points) is explained using regression and ppm.
c. One might expect that faster printers are higher priced. Do the data support that notion? (As always,
state relevant hypotheses, present a test statistic and p-value, and state your conclusion.) [20 points]
Let the null hypothesis be that b (the coefficient of the true regression line of price
on speed) is zero. The alternative is Ha: b>0. This is a one-tailed test.
Intercept
Speed
Coefficients
-745.480629
117.9173201
Standard
Error
427.4955998
26.83196475
t Stat
-1.743832286
4.394658432
P-value
0.119347079
0.002303147
The p-value reported in the output if for the 2-tailed alternative, so our p-value is
half that. Our p-value is 0.00115. We reject H0 in favor of our Ha. (We guessed
the correct tail. B-hat was positive….and consistent with our Ha.) The
relationship between price and speed is statistically significant.
d. One might also expect that printers positioned for corporate use are higher priced (on average) than
those marketed for small office use. Use two different methods (t-test two sample and regression with
a dummy variable) to test the relevant hypothesis. Be sure to state the hypotheses, give the test
statistics, p-values and conclusions. The two methods should produce identical results. [20 points]
First we do a t-test Two-Sample. The two samples are the four price for the small
office printers and the four prices for the corporate printers. I did not even bother
to put the data into two columns. In the “t-test two-sample” window I simply
highlighted the first four, then the second four prices.
t-Test: Two-Sample Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Variable 1
339
13000
5
132956.85
Variable
2
1705.2
252913.7
5
0
8
-5.92418929
0.000176015
1.859548038
0.00035203
2.306004135
The t-stat is -5.9 and the one-tailed p-value is 0.00017. We reject H0 in favor of
Ha. The difference in sample mean prices for the two kinds of printers is
statistically significant.
Next we use regression with a dummy variable. The conclusion will be identical.
This makes this an academic exercise. I will comply…..I need the points.
Name
Minolta-QMS PagePro
1250W
Brother HL-1850
Lexmark E320
Minolta-QMS PagePro
1250E
HP Laserjet 1200
Xerox Phaser 4400/N
Brother HL-2460N
IBM Infoprint 1120n
Lexmark W812
Oki Data B8300n
Type
Speed
Price
Dsmalloffice
Small Office
Small Office
Small Office
12
10
12.2
199
499
299
1
1
1
Small Office
Small Office
Corporate
Corporate
Corporate
Corporate
Corporate
10.3
11.7
17.8
16.1
11.8
19.8
28.2
299
399
1850
1000
1387
2089
2200
1
1
0
0
0
0
0
ANOVA
df
Regression
Residual
Total
Intercept
Dsmalloffice
1
8
9
Coefficients
1705.2
-1366.2
SS
4666256.1
1063654.8
5729910.9
MS
F
4666256.1 35.09601875
132956.85
Standard
Error
t Stat
P-value
163.0686052 10.45694846 6.07462E-06
230.6138331 -5.92418929 0.00035203
Significance
F
0.00035203
Lower 95%
1329.163122
-1897.996453
The p-value for our one-tail alternative is ½ of the p-value of 0.00035reported (in
two places) in the regression output.