{r data generation, echo = FALSE, results = "hide"} n <- sample(8:15, 1) y <- rnorm(n, runif(1, 4940, 4990), runif(1, 30, 50)) alpha <- sample(c(0.1, 0.05, 0.01), 1) Mean <- round(mean(y), digits = 1) Var <- round(var(y), digits = 1) sd <- sqrt(Var/n) fact <- round(qt(1 - alpha/2, df = n - 1), digits = 4) facn <- round(qnorm(1 - alpha/2), digits = 4) LBt <- round(Mean - fact * sd, digits = 3) UBt <- round(Mean + fact * sd, digits = 3) LBn <- round(Mean - facn * sd, digits = 3) UBn <- round(Mean + facn * sd, digits = 3) ## use extended Moodle processing to award 100% for correct solution based on ## t quantiles and 50% for the solution based on normal quantiles ## ## this can be handled as a "verbatim" solution, directly including Moodles ## cloze type: ## ":NUMERICAL:=2.228:0.01~%50%1.960:0.01#Normal-based instead of t-based interval." ## where 2.228 is the correct and 1.960 the partially correct solution, ## the tolerance is 0.01 in both cases, and a comment is supplied at the end. ## More details: https://docs.moodle.org/35/en/Embedded_Answers_(Cloze)_question_type ## solution template (note: % have to be escaped as %% for sprintf) sol <- ":NUMERICAL:=%s:0.1~%%50%%%s:0.1#Normal-based instead of t-based interval; for small samples, intervals based on the normal approximation are too narrow." ## insert correct and partially correct solutions sol <- sprintf(sol, c(LBt, UBt), c(LBn, UBn))  Question ======== It is suspected that a supplier systematically underfills 5 l canisters of detergent. The filled volumes are assumed to be normally distributed. A small sample of $r n$ canisters is measured exactly. This shows that the canisters contain on average $r Mean$ ml. The sample variance $s^2_{n-1}$ is equal to $r Var$. Determine a $r 100 * (1 - alpha)\%$ confidence interval for the average content of a canister (in ml). Answerlist ---------- * What is the lower confidence bound? * What is the upper confidence bound? Solution ======== The $r 100 * (1 - alpha)\%$ confidence interval for the average content $\mu$ in ml is given by: \begin{aligned} & \left[\bar{y} \, - \, t_{n-1;r 1-alpha/2}\sqrt{\frac{s_{n-1}^2}{n}}, \; \bar{y} \, + \, t_{n-1;r 1-alpha/2}\sqrt{\frac{s_{n-1}^2}{n}}\right] \\ &= \left[ r Mean \, - \, r fact\sqrt{\frac{r Var}{r n}}, \; r Mean \, + \, r fact\sqrt{\frac{r Var}{r n}}\right] \\ &= \left[r LBt, \, r UBt\right]. \end{aligned} Answerlist ---------- * The lower confidence bound is $r LBt$. * The upper confidence bound is $r UBt$. Meta-information ============ extype: cloze exclozetype: verbatim|verbatim exsolution: r sol[1]|r sol[2] exname: Confidence interval extol: 0.01