Question: Once upon a time the Big Ten consisted of 10
schools. Four new schools Rutgers,
University of Maryland, Pennsylvania State, and University of Nebraska entered
the conference in recent years.
What did the entry of these four schools do to the mean of
the 25^{th} percentile of the Verbal SAT score in the Big Ten?
Conduct a hypothesis test for a difference in the mean for
Verbal SAT at the 25^{th} percentile between the two groups.
Discuss issues related to the implementation of this test?
How does the existence of Northwestern the outlier impact
the results presented here?
Big Ten Verbal and Math
SAT Averages


Original Big Ten Schools


School

Verbal SAT 25th
Percentile


1

Ohio State

540

2

University of Michigan

620

3

Michigan State

420

4

University of Minnesota

550

5

University of Iowa

540

6

Purdue

520

7

Indiana University

520

8

Northwestern

690

9

University of Illinois

560

10

University of Wisconsin

530

New Big Ten Schools


1

Rutgers

520

2

University of Maryland

580

3

Penn State

530

4

University of Nebraska

490

Analysis:
Let’s start with the calculation of the mean and standard deviation for the two groups.
25th Percentile of Verbal
SAT Scores Original Big Ten Schools
and New Entrants


Mean

STD


Original Big Ten Schools

549.0

69.8

Four New Entrants to
Conference

530.0

37.4

The test for difference in standard deviation is based on
the Fstatistic. Using the F.Test function and a two tailed test I get
0.3327. I am going to assume the same
variance for both populations.
The tstatistics is 549 530 / (sp x ( (1/4 + 1/10) ) ^{0.5}
The pooled average is
SP^{ } = ((9 x 69.8 + 3 x 37.4)/(10+42))^{0.5}
Which is equal to 37.4
Plugging the pooled standard error into the tstatisiic I
get a value for the tstatistic of 0.5076.
The pvalue for the twotailed test consistent with this
tstatistic is 0.6210. (I used the
T.INV function in Excel. The twotailed
t pvalue is 2*(1T.INV(0.5075,12,CUMUATIVE).
The pvalue for this tstatistic can be found directly from
the T.TEST function where one specifies data arrays for the two groups, two
tails, and common variance.
The tvalue from the ttest is within rounding error of
0.6201.
So what does this mean?
We fail to reject the hypothesis that the mean Verbal SAT scores
for the new schools in the Big Ten is identical to the mean for the original
ten schools.
The gap is even smaller when one removes Northwestern the
elite private school. The removal of
Northwestern from the sample reduces the average SAT score for the 9 original state schools in the
Big Ten to 533.
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