<>=
## DATA GENERATION
n <- 40 + sample(1:12, 2) * 3
Waiting <- rnorm(sum(n), sd = sample(30:40, 1)/10) + rep(sample(30:80, 2)/10, n)
Waiting[Waiting < 0] <- 0
dat <- data.frame(Waiting = Waiting,
Supermarket = factor(rep(1:2, c(n[1], n[2])),
levels = 1:2, labels = c("Sparag", "Consumo")))
## QUESTION/ANSWER GENERATION
questions <- character(5)
solutions <- logical(5)
explanations <- character(5)
tt <- t.test(Waiting ~ Supermarket, data = dat, var.equal = TRUE,
alternative = sample(c("less", "greater", "two.sided"), 1))
questions[1] <- "The absolute value of the test statistic is larger than 1.96."
solutions[1] <- abs(tt$statistic) > 1.96
explanations[1] <- paste("The absolute value of the test statistic is equal to ",
round(abs(tt$statistic), digits = 3), ".", sep = "")
questions[2] <- "A one-sided alternative was tested."
solutions[2] <- tt$alternative != "two.sided"
explanations[2] <- paste("The test aims at showing that the difference of means is ",
switch(tt$alternative, "two.sided" = "unequal to", "less" = "smaller than",
"greater" = "larger than"), "$0$.")
questions[3] <- "The $p$~value is larger than $0.05$."
solutions[3] <- tt$p.value > 0.05
explanations[3] <- paste("The $p$~value is equal to $", format.pval(tt$p.value, digits = 3),
"$.", sep = "")
questions[4] <- paste("The test shows that the waiting time is longer at Sparag ",
"than at Consumo.")
solutions[4] <- tt$p.value < 0.05 & tt$alternative != "less" & diff(tt$estimate) < 0
explanations[4] <- if (solutions[4]) paste("The test result is significant ($p < 0.05$)",
"and hence the alternative is shown that the difference of means is",
ifelse(tt$alternative == "two.sided", "unequal to ", "larger than"), "0.") else
paste(ifelse(tt$alternative != "less", "", paste("The test aims at showing",
"that the alternative that the waiting time is shorter at Sparag than at Consumo.")),
ifelse(tt$p.value < 0.05, "", "The test result is not significant ($p \\ge 0.05$)."))
questions[5] <- paste("The test shows that the waiting time is shorter at Sparag than at Consumo.")
solutions[5] <- tt$p.value < 0.05 & tt$alternative != "greater" & diff(tt$estimate) > 0
explanations[5] <- if (solutions[5]) paste("The test result is significant ($p < 0.05$)",
"and hence the alternative is shown, that the difference of means are",
ifelse(tt$alternative == "two.sided", "unequal to", "smaller than"),
"$0$.") else paste(ifelse(tt$alternative != "greater", "",
paste("The test aims at showing that the waiting time at Sparag is longer than at Consumo.")),
ifelse(tt$p.value < 0.05, "", "The test result ist not significant ($p \\ge 0.05$)."))
@
\begin{question}
The waiting time (in minutes) at the cashier of two supermarket
chains with different cashier systems is compared. The following
statistical test was performed:
<>=
print(tt)
@
Which of the following statements are correct? (Significance level $5\%$)
<>=
answerlist(questions)
@
\end{question}
\begin{solution}
<>=
answerlist(ifelse(solutions, "True", "False"), explanations)
@
\end{solution}
%% META-INFORMATION
%% \extype{mchoice}
%% \exsolution{\Sexpr{mchoice2string(solutions)}}
%% \exname{2-sample t-test}