SAT Scores vs. Acceptance Rates

The experiment must fulfill two goals: (1) to produce a professional
report of your experiment, and (2) to show your understanding of the topics
related to least squares regression as described in Moore & McCabe, Chapter 2.
In this experiment, I will determine whether or not there is a relationship
between average SAT scores of incoming freshmen versus the acceptance rate of
applicants at top universities in the country. The cases being used are 12 of
the very best universities in the country according to US News & World Report.
The average SAT scores of incoming freshmen are the explanatory variables. The
response variable is the acceptance rate of the universities.

I used September 16, 1996 issue of US News & World Report as my source.
I started out by choosing the top fourteen "Best National Universities". Next,
I graphed the fourteen schools using a scatterplot and decided to cut it down to
12 universities by throwing out odd data.

A scatterplot of the 12 universities data is on the following page (page 2)

The linear regression equation is:
ACCEPTANCE = 212.5 + -.134 * SAT_SCORE
R= -.632 R^2=.399

I plugged in the data into my calculator, and did the various regressions. I
saw that the power regression had the best correlation of the non-linear
transformations.

A scatterplot of the transformation can be seen on page 4.

The Power Regression Equation is
ACCEPTANCE RATE=(2.475x10^23)(SAT SCORE)^-7.002
R= -.683 R^2=.466

The power regression seems to be the better model for the experiment that I have
chosen. There is a higher correlation in the power transformation than there is
in the linear regression model. The R for the linear model is -.632 and the R in
the power transformation is -.683. Based on R^2 which measures the fraction of
the variation in the values of y that is explained by the least-squares
regression of y on x, the power transformation model has a higher R^2 which is .
466 compared to .399. The residual plot for the linear regression is on page 5
and the residual plot for the power regression is on page 6. The two residuals
plots seem very similar to one another and no helpful observations can be seen
from them. The outliers in both models was not a factor in choosing the best
model. In both models, there was one distinct outlier which appeared in the
graphs.

The one outlier in both models was University of Chicago. It had an
unusually high acceptance rate among the universities in this experiment. This
school is a very good school academically which means the average SAT scores of
incoming freshmen is fairly high. The school does not receive as many applicants
to the school as the others, this due in part because of the many other factors
besides academic where applicants would choose other schools than University of
Chicago. Although the number applicants is relatively low, most of these
applicants are very qualified which results in it having a high acceptance rate.

Rate = A*(SAT)^(B)
A=2.475x10^23
B=-7.002

From the model I have chosen, I predicted what the acceptance rate for a
school would be if the average SAT score was a perfect 1600.
SAT = 1600
Rate = A*(SAT)^B = (2.475x10^23) *(1600)^(-7.002) = 9.1%

From the equation found, we have determined this "university" would have a
acceptance rate of only 9.1%. This seems as a good prediction because such a
school would have a very low acceptance rate compared to the other top
universities. I believe causation does occur in this experiment. With there
being a higher average SAT scores of applicants admitted, it would be harder to
be admitted into that school. Although, I think the equation found is not very
accurate when predicting far away from the median.

I do not believe there would be any sources in collecting the data. All the
data was taken from the magazine, US News & World Reports. I strictly took
twelve of the top 14 universities based on this magazine. I believe some lurking
variable may be type of school, majors offered, and number of applicants. The
number of applicants a school has would have somewhat an effect on its
acceptance rate. If a school had a enormous amount of applicants, then this
school would have a relatively low acceptance rate. One reason I think this
experiment had a somewhat poor association is because of the schools selected.
Two of these schools were technical schools which meant only certain applicants
would want to apply to these schools while the other schools were more general
overall.

In conclusion, the data