In exams: Automatic Generation of Exams in R

## DATA GENERATION
n <- sample(120:250, 1)
mu <- sample(c(125, 200, 250, 500, 1000), 1)
y <- rnorm(n, mean = mu * runif(1, min = 0.9, max = 1.1), 
           sd = mu * runif(1, min = 0.02, max = 0.06))

## QUESTION/ANSWER GENERATION
Mean <- round(mean(y), digits = 1)
Var <- round(var(y), digits = 2)
tstat <- round((Mean - mu)/sqrt(Var/n), digits = 3)

## TRANSFORM TO SINGLE CHOICE
questions <- tstat
while(length(unique(questions)) < 5) {
  fuzz <- c(0, runif(4, 0.02, 2 * sqrt(Var)))
  sign <- c(sign(tstat), sample(c(-1, 1), 4, replace = TRUE))
  fact <- sample(c(-1, 1), 5, replace = TRUE)
  questions <- round(sign * abs(tstat + fact * fuzz), digits = 3)
}
questions <- paste("$", gsub("^ +", "", fmt(questions, 3)), "$", sep = "")
solutions <- c(TRUE, rep(FALSE, 4))

o <- sample(1:5)
questions <- questions[o]
solutions <- solutions[o]

Question

A machine fills milk into $r mu$ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint $\mu_0 = r mu$. A sample of $r n$ packages filled by the machine are collected. The sample mean $\bar{y}$ is equal to $r Mean$ and the sample variance $s^2_{n-1}$ is equal to $r Var$.

Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the value of the t-test statistic?

answerlist(questions, markup = "markdown")

Solution

The t-test statistic is calculated by: $$ \begin{aligned} t & = & \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{r Mean - r mu}{\sqrt{\frac{r Var}{r n}}} = r tstat. \end{aligned} $$ The t-test statistic is thus equal to $r fmt(tstat, 3)$.

answerlist(ifelse(solutions, "True", "False"), markup = "markdown")

Meta-information

extype: schoice exsolution: r mchoice2string(solutions, single = TRUE) exname: t statistic

exams documentation built on Nov. 14, 2022, 3:02 p.m.