## Wednesday, August 10, 2016

### Difference between private and public college SAT scores

Difference between private and public college SAT scores

The problems in this post all involve SAT scores (Verbal and Math, 25th and 75th percentile) for 31 public universities and 20 private universities in the state of California.   The data used to address the issue in this post can be found at the following site.

The basic issues I want to address in this post involve the extent of the differences in SAT scores between private and public colleges in the state of California.

Questions:

For each SAT score provide the mean, standard deviation, skewness and kurtosis for the sample of 31 public schools and 20 private schools.

Descriptive Statistics for Public Schools in California

 Public Universities in California verbal25 verbal75 math25 math75 mean 452.26 563.55 471.94 591.61 STD 55.96 62.37 71.25 78.02 Skewness 0.91 0.77 0.87 0.83 Kurtosis 0.18 0.01 -0.21 -0.11 Count 31 31 31 31

 Private Universities in California verbal25 verbal75 math25 math75 mean 523.50 627.50 533.65 640.65 STD 65.88 63.65 76.66 69.18 Skewness 0.50 0.41 0.25 0.22 Kurtosis 0.41 0.65 0.16 0.29 Count 20.00 20.00 20.00 20.00

Comment on these statistics:

Mean SAT scores of private schools is higher than mean SAT scores for public schools in California for all four SAT measures.

Standard deviation levels are not very different

Skewness and Kurtosis statistics indicate distribution of public school data is closer to assumption of normality than the distribution of private school data.

Is the difference between SAT scores of private and public schools statistically significant?  Present results for the test of means under the assumption that variances are equal and the Wilcoxon Rank Sum test.

 Hypothesis Test Results SAT Score P value on t-test on means equal variances P Value for Wilcoxon Rank Sum Test Verbal 25 0.0001 0.0005 Verbal 75 0.0009 0.0014 Math 25 0.0051 0.0047 Math 75 0.0265 0.0179

Comment:   Both tests reveal significant differences at the 0.05 level for all SAT scores.   The null hypothesis of no differences between private and public math STAT at the 75th percentile =cannot be rejected.

Calculate the 10th percentile and the 90th percentile of the combined sample of the 51 California Universities.   What is the percent of the private and public universities in the lower tail of the combined distribution?   What is the percent of private and public universities in the upper tail of the universities?  What additional information can we learn by directly examining the tails of the combined private and public sample?

 Number of universities in bottom decile of combined sample Verbal 25 Verbal 75 Math 25 Math 75 Private 8 8 6 6 Public 0 1 1 2 Number of universities in top decile of combined sample Private 2 2 2 3 Public 5 4 4 4

Comment:

The lower tail of the distribution consists almost entirely of private schools.

More private schools are in the upper tail but at least two very large public schools Cal Berkeley and UCLA are in the upper tail.