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.


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