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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