```{r data generation, echo = FALSE, results = "hide"}
## 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?
```{r questionlist, echo = FALSE, results = "asis"}
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
$`r fmt(tstat, 3)`$.
```{r solutionlist, echo = FALSE, results = "asis"}
answerlist(ifelse(solutions, "True", "False"), markup = "markdown")
```
Meta-information
================
extype: schoice
exsolution: `r mchoice2string(solutions, single = TRUE)`
exname: t statistic