<>= ## 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}