# Quizbank/Cost-benefit analysis

Suppose a lottery ticket has one chance in a hundred of winning $100. The "expected value" of that ticket is$1 because the person who buys a large number of such tickets is expected, on average, to break even. The purchase of a large number of these tickets for 75¢ will almost certainly result in profit. I believe this is how we need to look at reducing the rising costs of higher education. The following calculation is not a "mathematical proof", but a physicist's plausibility argument about how events might turn out. Begin by defining,

${\displaystyle C}$ to be the cost of higher education,
${\displaystyle F}$ to be the fraction of this cost that this reform will eliminate, and
${\displaystyle P}$ to be the probability that the proposed reform will succeed.

In order to investigate the utility of an open bank of exam questions, we need to perform a preliminary investigation. The cost, ${\displaystyle I}$, of this investigation will be favorable if:

${\displaystyle I

A suitable value for the cost of higher education is difficult to calculate or even define. For the sake of argument take the following estimate: Multiply half the current US undergraduate enrollment by the average tuition at public 2-year colleges. Details are shown below at #Appendix. I get an annual cost of $10 billion, which I believe to be a conservative estimate: ${\displaystyle C\approx {\frac {1}{2}}\cdot \left(20\times 10^{6}{\text{ students }}\right)\cdot \left(10^{4}\;\{\text{/student}}\right)=10^{11}\}$ Suppose, for example, that there is a 1% chance that an experimental pilot program would reduce the cost of higher education by 1%. The United States could plausibly spend$1 million dollars on this idea, since:

${\displaystyle P=F=.01}$ ${\displaystyle \Rightarrow I<10^{7}\=\{\text{10 million}}}$.

But the potential reward is even greater because our "reward" was only the saving associated with a single academic year. If we assume that the reform will accrue benefits over more than one year, we have:

${\displaystyle \langle I\rangle \equiv NCPF,}$

where ${\displaystyle N}$ is the number of years we can expect the reform to accrue savings. Taking ${\displaystyle N}$ to be 10 years and ${\displaystyle C}$ to be ${\displaystyle 10^{11}\}$, we come to the astonishing conclusion that a proposed reform implemented for 10 years, with a 10% probability of reducing the cost of higher education by 10%, has a present value of ${\displaystyle 10^{10}\=}$ \$10 billion dollars.

## The "flipped semester" with undergraduate prelims

Following link represents one of many possible cost-saving proposals that should be explored:

## The economies of scale

The economies of scale play a crucial role in understanding the value of an open exam bank. Existing exam banks are either written and maintained by individual instructors, or are provided by vendors of a textbook and/or Learning Management System. This creates something akin to the cottage industries that prevailed before the industrial revolution: Many individuals are all doing the same thing on a small scale. In contrast, OER and numerous other online sites such as the Khan Academy provide instructional material to a vast audience at zero cost.

This focus on standardized tests for accreditation raises serious pedagogical questions that cannot be dismissed. But partial answers to these objections can take one of four forms:

1. Not doing something about the cost of higher education has serious consequences.
2. An enlightened and sparing use of the bank can help focus educational resources on the students who need it. Instead of excluding students from a course if they fail to pass the preliminary exam, open up small sections for those who need help. Offer alternative assessments such as essays for those not adept at standardized tests.
3. A zero-cost OER bank will exert downward pressure on what the commercial vendors charge.
4. Problems and exam questions are so abundant on the internet that the open exam bank effectively exists right now. Just a few minutes of searching the internet uncovers ample evidence that "the cat is out of the bag":
These are the legal banks. A GRE practice test and a 600 page exam bank from a well-known college physics book are currently available online. I believe both are copyright violations. The question isn't whether there should be a large public bank of exam questions and solved problems (it already exists), but whether this bank should be structured so that instructors can conveniently use it.

## Appendix and footnotes

Here we attempt to estimate the cost of higher education, based only on what we charge students at 2-year public colleges.

By considering only tuition, textbook and living costs are excluded. The tuition associated with 2-year public colleges was chosen because the higher tuition at 4-year colleges could be used to subsidize research and the post-graduate education. To obtain a lower bound on this cost we neglect the contributions by local and federal governments. The reduction by a factor of 2 is an attempt to account for part time enrollments (whereby students don't pay the full tuition). As a check on this estimate, we calculate the cost to grant 30% of US citizens a four-year degree (See Educational Attainment in the United States 2009.png.[1]) The Appendix contains a chart suggesting that about 22 million US residents are currently in a five year age slot where they might be in college. Take 30% of that number and we conclude that at least 6.6 million students are currently paying for an education. This is reasonably close to the estimate of 10 million full-time students used for our calculation of ${\displaystyle C}$.

The following figures and table were used to estimate the annual cost of higher education in the United States:

Year Fall enrollment Degree-granting institutions
(all postsecondary) (degree-granting) (total) (4-year) (2-year)
2010-11 21,591,742 21,019,438 4,599 2,870 1,729
2011-12 21,573,798 21,010,590 4,706 2,968 1,738
2012-13 21,148,181 20,644,478 4,726 3,026 1,700
2013-14 20,848,050 20,376,677 4,724 3,039 1,685
2014-15 20,664,180 20,209,092 4,627 3,011 1,616
2015-16 20,400,164 19,988,204 4,583 3,004 1,579
2016-17 20,224,069 19,841,014 4,360 2,832 1,528

Footnotes
• https://www.census.gov/prod/2012pubs/p20-566.pdf