Friday, May 27, 2016

New Big Ten versus Old Big Ten -- a Small Sample Hypothesis Test

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 25th 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 25th 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 F-statistic. 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 t-statistics 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+4-2))0.5

Which is equal to   37.4


Plugging the pooled standard error into the t-statisiic I get a value for the t-statistic of  0.5076.


The p-value for the two-tailed test consistent with this t-statistic is 0.6210.   (I used the T.INV function in Excel.   The two-tailed t pvalue is 2*(1-T.INV(0.5075,12,CUMUATIVE).

The p-value for this t-statistic can be found directly from the T.TEST function where one specifies data arrays for the two groups, two tails, and common variance.  

The t-value from the t-test 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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