In exams: Automatic Generation of Exams in R

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: $$ \begin{aligned} & \left[\bar{y} \, - \, t_{n-1;r 1-alpha/2}\sqrt{\frac{s_{n-1}^2}{n}}, \; \bar{y} \, + \, t_{n-1;r 1-alpha/2}\sqrt{\frac{s_{n-1}^2}{n}}\right] \ &= \left[ r Mean \, - \, r fact\sqrt{\frac{r Var}{r n}}, \; r Mean \, + \, r fact\sqrt{\frac{r Var}{r n}}\right] \ &= \left[r LBt, \, r UBt\right]. \end{aligned} $$