## 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.