In exams: Automatic Generation of Exams in R

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 muml 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).