This
post creates and discusses confidence intervals on means of four different SAT
scores for 14 Big Ten schools.
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. One school in the league, Northwester, appears to be
an outlier. Will soon examine the impact
of this outlier on size of confidence intervals.
Nice
video explaining the construction of small sample confidence intervals around
means.
No comments:
Post a Comment