```{r data generation, echo = FALSE, results = "hide"}
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)
)
Mean <- round(mean(y), digits = 1)
Var <- round(var(y), digits = 2)
tstat <- round((Mean - mu)/sqrt(Var/n), digits = 3)
```

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 absolute value of the t-test statistic?

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 absolute value of the t-test statistic is thus equal to
`r fmt(abs(tstat), 3)`.

Meta-information
================
extype: num
exsolution: `r fmt(abs(tstat), 3)`
exname: t statistic
extol: 0.01