A post using AP exam results to teach probability.
AP Exam Performance and Batting Averages
The table below has the probability distribution for grades
on four popular AP exams. I use the data for some questions on probability. The issues faced here are similar to issues I faced in two of my baseball probability posts.
Probability Distribution
for Grades on Four Popular AP Exams


5

4

3

2

1


French Language

0.167

0.256

0.336

0.189

0.052

1.000

Chemistry

0.084

0.152

0.281

0.255

0.228

1.000

Calculus AB

0.213

0.171

0.187

0.105

0.324

1.000

United States History

0.093

0.178

0.236

0.250

0.243

1.000

Source: https://www.totalregistration.net/APExamRegistrationService/2015APExamScoreDistributions.php
Questions:
Based on these probability distribution functions for AP
exam grades what is the probability a person who takes all five exams gets a
five on each test, assuming the probability of passing each test is independent
and identically distributed?
What is the probability a person gets a one on all four
tests again assuming probabilities are independent and identically distributed?
Do you suspect that the actual observed probability of
extreme outcomes  all fives or all ones  among people who take these four
exams will be higher or lower than the estimated values based on the assumption
that tests are independent and identically distributed?
Analysis:
Based on these
probability distribution functions for AP exam grades what is the probability a
person who takes all five exams gets a 5 on each test, assuming the probability
of passing each test is independent and identically distributed?
The probability of a 5 on all four exams given they are
independent and identically distributed is 0.0167 * 0.084 * 0.213 * 0.093 = 0.00028.
What is the
probability a person gets a 1on all four tests again assuming probabilities are
independent and identically distributed?
The probability of a 1 on all four exams is 0.052 * 0.228 *
0.324 * 0.243 = 0.0009.
Do you suspect that
the probability of extreme outcomes all fives and all ones among people who
take these five exams will be higher or lower than the estimated values based
on the assumption that tests are independent and identically distributed?
I believe the assumption that tests and test takers are
independent and identically distributed is incorrect. People differ in their extent to which they
are prepared to take multiple exams in a short period of time. These differences will lead to a larger
number of people doing extremely well or extremely poorly on all tests.
We saw a similar outcome when studying the performance of
hitters
The post below shows that the number of 10game hitting
streaks that actually occur in baseball is lower than what would expect if at
bats were actually independent and identically distributed.
The post below shows that even great hitters have more
extreme daily batting outcomes (0 for N or N for N) than one would expect if
the assumption that at bats were independent and identically distributed was
appropriate.
Authors Note: I want to conduct a statistical analysis of
the impact of AP exam load and other student characteristics on AP exam
performance. One question that I want to
address is whether, all else equal, do students who take fewer AP exams have a
higher pass rate than students who take a larger number of exams. A second question involves the impact of AP
exam loads on the number of AP exams passed.
A third question involves the impact of AP schedule on the
pass rate of particular exams given the previous academic performance of the
student.
I believe that econometric models of the impact of course
work load and student attributes can help student enroll in class schedules
that will maximize their learning, improve their performance on exams, and
increase the likelihood they will receive college credit.
I need data to conduct this study. Grant financing would also be very
useful. Please contact me at Bernstein.book1958@gmail.com if
you can help with this endeavor.
Below is a discussion of some previous research on AP exam
performance.
Interesting that the skew in the Math AP is likened to the skew in politics. A student of mine recently did a project in which she demonstrated that minorities od all sorts are more likely to be democratic and stay that way. But there is no good explanation for the discrepancy in this last election between white college grads and white nongrads. The gap is so much greater by, I think a factor of 4, than it has been in past elections. There are a lot of avenues to be examined here, but I can't believe a huge number of noncollege whites actually liked everything Donald Trump had to say. My student discovered from exit polls, that many people voting for Trump thought he was doing some political blustering, and that he would never follow through on the examples he provided. Sadly, Some of Hitler's voters felt the same way. History repeats itself.
ReplyDelete