Question: Below are SAT scores  25^{th}
percentile math and verbal and 75^{th} 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 tstatistics 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 25^{th} 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
25^{th} percentile. Three
schools, University of Michigan, University of Maryland, and Northwester had
verbal 25t^{h} 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

UBLB

58.2

53.9

42.9

40.8

Note: The tstatistics 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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