{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