## Tuesday, May 24, 2016

### Confidence intervals for Big Ten SAT Scores

Question:   Below are SAT scores -- 25th percentile math and verbal and 75th percentile math and verbal -- for the 14 Big Ten schools.   Calculate the mean and standard deviation of the four SAT scores for the 14 Big Ten Schools.   Use these values to construct the upper bound and the lower bound of 90% confidence intervals for the four mean SAT scores?

What SAT scores have the smallest/largest confidence intervals for Big Ten schools?

The data:

 SAT Scores for Big Ten Schools Big Ten School Verbal SAT 25 Math SAT 25 Verbal SAT 75 Math SAT 75 Ohio State 540 610 660 720 University of Michigan 620 660 720 760 Michigan State 420 550 580 690 University of Minnesota 550 620 690 740 University of Iowa 540 620 620 680 Purdue 520 560 630 690 Indiana University 520 540 630 660 Rutgers 520 570 640 690 University of Maryland 580 620 690 730 Northwestern 690 700 760 790 University of Illinois 560 700 670 780 Penn State 530 560 630 670 University of Wisconsin 530 630 650 750 University of Nebraska 490 520 660 670

The analysis:  Below are the mean, standard deviation, standard error, the t-statistics needed for a 90% confidence interval, and the lower and upper bounds for the 90% confidence interval for the four SAT scores on the 14 Big Ten Schools.

Observations:

Point estimates of verbal scores are a lot lower than math scores at both percentiles.

Dispersion of verbal scores among Big Ten schools is a lot higher than dispersion of math scores among Big Ten schools.   Verbal SAT confidence intervals are a low wider than Math SAT confidence intervals.

The confidence intervals can be used to determine what schools fall outside the norm on each attribute for the Big Ten.  For example, consider the confidence interval around the mean of the observed 25th percentiles for verbal SAT.    Two schools, Michigan State and University of Nebraska had a verbal SAT score below the lower bound for the verbal SAT at 25th percentile.   Three schools, University of Michigan, University of Maryland, and Northwester had verbal 25th percentiles over the upper bounds.

 Big Ten SAT Statistics and Confidence Intervals Verbal SAT 25 Math SAT 25 Verbal SAT 75 Math SAT 75 Mean 543.6 604.3 659.3 715.7 STD 61.5 56.9 45.3 43.1 n 14.0 14.0 14.0 14.0 STDERR 16.4 15.2 12.1 11.5 t13 1.771 1.771 1.771 1.771 LB 514.5 577.3 637.8 695.3 UB 572.7 631.2 680.7 736.1 UB-LB 58.2 53.9 42.9 40.8

Note:  The t-statistics used in the construction of these confidence intervals has 13 degrees of freedom.   The sample size is 14 Big Ten schools so subtract one from the sample size to get the degrees of freedom.

Concluding Thought:  The confidence intervals allow us to determine quickly what schools have SAT scores in the normal range for the league and what schools are atypical for this league.   Note the atypical schools are included in the construction of the confidence interval and this can cause the technique to understate the extent to which one school is a outlier.   In the context of this question, I am thinking of Northwestern the one private school in the list.    My next post will consider the impact of the exclusion of this one school from the league on the size and position of the confidence interval.

Nice video explaining the construction of small sample confidence intervals around means.