
Model 
Sum of Squares 
degrees 
Mean 
Fstatistics 
Significance 
Regression: 
100.888 
2 
50.444 
.874 
.425 
a) 49. Degrees of freedom for
the total are n1.
b) There is no evidence that make of car matters. The
differences in the results we got were small enough to be
nothing more than random noise. If there is no difference at
all, we could have gotten a result that explains as much as
this one does 42.5% of the time by random
chance.
2. Every fall 20 teams from around the state of Indiana would meet in the crosscountry state championships. Some coaches maintain that the four semistate regions, each of which contributes 5 teams to the state finals, are not equal in talent. They argue that a 6th or 7th place team in a strong semistate final would easily go to the state finals if they could compete in a weak semistate final. (The data for this problem was collected before the rules changed and six teams were allowed to advance.)
a) In the graph below we have taken the final placing of the twenty women's teams that made it to the state finals and grouped them by the semistate that they came from. You can see that the first place team came from the group 2, and the 20th place team came from group 4. Based on this graph, which groups look weak and which look strong? Does the contention of the coaches mentioned above look like it may have substance? Explain.
Groups one and two look stronger than groups three and four; the coaches' contention looks like it might be valid.
b) Ultimately, we need to do a statistical analysis to see what we can determine. The null hypothesis will be that all the semistates are equally strong. We can do an Analysis of Variance test on the rank (place 1 to 20) or on the points scored (in cross country, like golf, fewer points are better). Based on the Analysis of Variance results, should we accept the hypothesis that all semistate regions are equally strong, or should we reject it and decide that the coaches in the introduction are right? Explain.
ANOVA
Sum of Squares 
df 
Mean Square 
F 
Sig. 

RANK 
Between Groups 
341.400 
3 
113.800 
5.627 
.008 
Within Groups 
323.600 
16 
20.225 

Total 
665.000 
19 

Points 
Between Groups 
153059.800 
3 
51019.933 
6.630 
.004 
Within Groups 
123126.400 
16 
7695.400 

Total 
276186.200 
19 
It appears that the coaches are correct. We would get results like what we are seeing by random chance if the semistates were all equal less than 1% of the time.
c) If we try to predict how well a team does, we can use regression with final rank as the dependent variable and as independent variables semistate rank plus a variable to indicate in which semistate the team ran. Valparaiso finished first in the NP semistate. What rank do we predict for it in the state meet?
8.450  9.2 + 2.250 = 1.5. We predict that they would finish first or second.
d) Penn High School finished sixth in the NP semistate, and did not go on to the state meet. If they had been allowed to go, where does this regression predict they would have finished?
8.450  9.2 + 2.250 *6= 12.75. We predict that they would have finished about thirteenth.
Model Summary
Model 
R 
R Square 
Adjusted R Square 
Std. Error of the Estimate 
1 
.904 
.818 
.769 
2.8414 
Coefficients
Unstandardized Coefficients 
Standardized Coefficients 
t 
Sig. 

Model 
B 
Std. Error 
Beta 

1 
(Constant) 
8.450 
1.852 
4.562 
.000 

NP 
9.200 
1.797 
.691 
5.120 
.000 

FC 
8.400 
1.797 
.631 
4.674 
.000 

MAN 
1.200 
1.797 
.090 
.668 
.514 

SEMIRANK 
2.250 
.449 
.552 
5.008 
.000 
3. After running a regression, I received the following Analysis of Variance results:
Source
Sum of Squares
Deg Freedom
Mean Square
F
Regression
8.70
4
2.18
7.18
Residuals
6.97
23
.30
Total
15.67
27
.58
a) Rsquare = 8.7/15.67 =
.555
b) We need the significance of the F statistics.