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## Idea

Our colleagues over at the economics department became interested in using R/exams for

generating large-scale exams in their introductory economics courses. However, they face the challenge that

so far they had been writing static exercises and modified them *by hand* if they wanted to reuse them in a

different exam in another semester. To let R/exams do this job it is illustrated how a static arithmetic

exercise can be turned into a dynamic exercise template either in `num`

format with a numeric solution or into `schoice`

format with a single-choice solution.

The idea for the exercise is a very basic *price elasticity of demand* task:

Consider the following inverse demand function:
$p(x)=a\u2013b\xb7x$
for the price $p$given the demanded quantity $x$. What is the price elastiticy of demand at a price of $p$? |

The natural candidates for “parameterizing” this exercise are the

price

and the parameters

$a$and

$b$ of the inverse demand function.

Based on these the solution is simply:

First, we obtain the demand function by inverting the inverse demand function:
$x=D(p)=(a\u2013p)/b$
. the price elasticity of demand is $\frac{D\u2018(p)}{D(p)}p=\frac{\u20131/b}{x}p.$ |

## Overview

Below various incarnations of this exercise are provided in both R/Markdown `Rmd`

and R/LaTeX `Rnw`

format.

The following table gives a brief overview of all available versions along with a short description of the idea behind it.

More detailed explanations are provided in the subsequent sections.

# | Exercise templates | Dynamic? | Type | Description |
---|---|---|---|---|

1 | elasticity1.Rmd elasticity1.Rnw |
No | `num` |
Fixed parameters and numeric solution. |

2 | elasticity2.Rmd elasticity2.Rnw |
No | `schoice` |
As in #1 but with single-choice solution (five answer alternatives). |

3 | elasticity3.Rmd elasticity3.Rnw |
Yes | `num` |
Randomly drawn parameters with dynamic computation of correct solution, based on #1. |

4 | elasticity4.Rmd elasticity4.Rnw |
Yes | `schoice` |
Randomly drawn parameters (as in #3) with dynamically-generated single-choice solution (as in #2), computed by `num_to_schoice()` . |

5 | elasticity5.Rmd elasticity5.Rnw |
Yes | `schoice` |
As in #4 but with the last alternative: None of the above. |

## Static numeric

The starting point is a completely static exercise as it had been used in a previous

introductory economics exam. The parameters had been set

to

,

$a=50$,

and

.

This implies that

,

leading to an elasticity of

.

The corresponding R/exams templates simply hard-code these numbers into the question/solution

and wrap everything into R/Markdown (elasticity1.Rmd)

or R/LaTeX (elasticity1.Rnw).

Note that LaTeX is used in either case for the mathematical notation. In case you are unfamiliar

with the R/exams format, please check out the First Steps tutorial.

The meta-information simply sets `extype`

to `num`

, supplies the `exsolution`

with a precision of three digits,

and allows an `extol`

tolerance of 0.01. To see what the result looks like download the files linked above

and run `exams2html()`

and/or `exams2pdf()`

. (The examples below always use the R/Markdown version but the

R/LaTeX version can be used in exactly the same way.)

library("exams") exams2html("elasticity1.Rmd") exams2pdf("elasticity1.Rmd")

## Static single-choice

Single-choice versions of exercises are often desired for use in written exams

because they can be conveniently scanned and automatically evaluated. Thus, we need to come up

with a number of incorrect alternative solutions (or “distractors”). If desired, these could include

typical wrong solutions or a *None of the others* alternative.

The R/exams templates

elasticity2.Rmd and

elasticity2.Rnw are

essentially copies of the static numeric exercise above but:

- Question/solution now contain an answerlist (with five alternatives).
- The
`extype`

has been changed to`schoice`

. - The
`exsolution`

now contains a binary coding of the correct solution. - Furthermore, to obtain some basic randomization we have turned on shuffling by setting
`exshuffle`

to`TRUE`

. (Subsampling more than five alternatives would also be possible and would add some further randomization.)

As above `exams2html()`

and/or `exams2pdf()`

can be used to display the exercise interactively in R/exams.

## Dynamic numeric

Next, the static exercise from above is made dynamic by drawing the parameters from a suitable

data-generating process. In this case, the following works well:

## p = a - b * x p <- sample(5:15, 1) fctr <- sample(c(2, 4, 5, 10), 1) x <- sample(5:15, 1) * fctr b <- sample(1:5, 1) / fctr a <- p + b * x ## elasticity sol <- -1/b * p/x

Note that in order to obtain “nice” numbers a common scaling factor `fctr`

is used for both `x`

and `b`

.

Also, while the examinees are presented with parameters `a`

and `b`

and have to compute `x`

,

the data-generating process actually draws `x`

and `b`

and computes `a`

from that. Again, this makes it

easier to obtain “nice” numbers.

The R/exams templates

elasticity3.Rmd and

elasticity3.Rnw include

the data-generating process above as a code chunk either in R/Markdown or R/LaTeX format.

The parameters are then inserted into question/solution/metainformation using ``r a``

(R/Markdown)

or `\Sexpr{a}`

(R/LaTeX). Sometimes the `fmt()`

function is used for formatting the parameters

with a desired number of digits. (See `?fmt`

for more details.)

As before `exams2html()`

and/or `exams2pdf()`

can be used to display the *random draws* from

the exercise templates. For checking that the meta-information is included correctly, it is often

helpful to run

exams_metainfo(exams2html("elasticity3.Rmd"))

Furthermore, some tweaking is usually required when calibrating the parameter ranges in the

data-generating process. The `stresstest_exercise()`

function draws a large number of random replications

and thus can help to spot errors that occur or to find solutions that are “extreme” (e.g., much larger or smaller than usual). To clean up your

global environment and run the function use something like this:

rm(list = ls()) s <- stresstest_exercise("elasticity3.Rmd", n = 200) plot(s) plot(s, type = "solution")

The latter command plots the correct solution against the (scalar) parameters that were

generated in the exercise. This might show patterns for which parameters the solution becomes

too large or too small etc. See `?stresstest_exercise`

for further features/details.

## Dynamic single-choice

To go from the dynamic numeric exercise to a dynamic single-choice exercise we need to

extend the data-generating process to also produce a number of wrong alternatives.

The function `num_to_schoice()`

helps with this by providing different sampling

mechanisms. It allows to set a range in which the alternatives have to be, a minimum

distance between all alternatives, possibly include typical wrong solutions, etc.

It also shuffles the resulting alternatives and tries to make sure that the correct

solution is *not* a certain order statistic (e.g., almost always the largest or

smallest alternative).

The R/exams templates

elasticity4.Rmd and

elasticity4.Rnw first

wrap the data-generating process into a while loop with `while(sol > -0.11)`

. This makes sure

that there is enough “space” for four wrong alternatives between -0.11 and 0, using a minimum

distance of 0.017. Subsequently, the four wrong alternatives are generated by:

## single-choice incl. typical errors err <- c(1/sol, sol/p, p/sol) err <- err[(err > -5) & (err < -0.2) & abs(err - sol) > 0.01] rng <- c(min(1.5 * sol, -1), -0.01) sc <- num_to_schoice(sol, wrong = err, range = rng, delta = 0.017, method = "delta", digits = 3)

This suggests a number of typical wrong solutions `err`

(provided that they are not too small

or too large) and makes sure that the range `rng`

is large enough. With these arguments

`num_to_schoice()`

is run, see `?num_to_schoice`

for the details. The resulting `sc`

list

contains suitable `$questions`

and `$solutions`

that can be easily embedded in an `answerlist()`

and in the meta-information. Sometimes it is useful to wrap `num_to_schoice()`

into another

`while()`

loop to make sure a valid result is found. See the deriv2

template for illustraion.

As before `exams2html()`

and/or `exams2pdf()`

can be used to display the *random draws* from

this exercise template. And some stress-testing could be carried out by:

rm(list = ls()) s <- stresstest_exercise("elasticity4.Rmd", n = 200) plot(s) plot(s, type = "ordering")

As a final variation we could include *None of the above* as the last alternative. This is

very easy because we can simply replace the fifth element of the question list with the corresponding

string:

sc$questions[5] <- "None of the above."

No matter whether this was actually the correct alternative or one of the incorrect alternatives, the

corresponding solution will stay the same. The R/exams templates

elasticity5.Rmd and

elasticity5.Rnw incorporate

this additional line of code in the data-generating process.

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