<<echo=FALSE, results=hide>>=
ok <- FALSE
while(!ok) {
## DATA GENERATION
create_table <- function() {
  city <- rmultinom(1, 100, prob=c(0.2, 0.4, 0.3, 0.1))
  suburb <- rmultinom(1, 150, prob=c(0.1, 0.2, 0.4, 0.3))
  matrix(c(city, suburb), nrow = 4,
         dimnames = list(Evaluation = c("Very good", "Good", "Bad", "Very bad"),
           Location = c("City", "Suburbs")))
}
public <- create_table()
variant <- sample(1:3, 1)
margins <- list(1, 2, NULL)
prop_public <- round(100 * prop.table(public, margins[[variant]]), digits = 1)

while (length(prop_public) != length(unique(prop_public))) {
  public <- create_table()
  prop_public <- round(100 * prop.table(public, margins[[variant]]), digits = 1)
}

## QUESTION/ANSWER GENERATION
questions <- character(5)
solutions <- logical(5)
explanations <- character(5)

## 1. total/rows/columns percentage
percentage <- c("row", "column", "total")
rowsums <- c("row sums", "column sums", "total sums")
variant1 <- sample(1:3, 1)
questions[1] <- paste("The percentage table provides ", percentage[variant1], "percentages.")
solutions[1] <- variant1 == variant
explanations[1] <- if (variant1 == variant) paste("The ", rowsums[variant1],
                         " are about equal to 100 (except for possible rounding errors).") else
paste("The ", rowsums[variant1], "are not equal to 100.")

## 2. interprete total/row/column percentage
if (runif(1) > 0.5) {
  questions[2] <- "The percentage table provides the location distribution for each level of satisfaction."
  solutions[2] <- variant == 1
  explanations[2] <- if (solutions[2]) "The row sums are equal to 100 (except for possible rounding errors)." else
                     "The row sums are not equal to 100."
} else {
  questions[2] <- "The percentage table provides the satisfaction distribution for each location type."
  solutions[2] <- variant == 2
  explanations[2] <- if (solutions[2]) "The column sums are equal to 100 (except for possible rounding errors)." else
                     "The column sums are not equal to 100."
}

## 3. Interpretation row percentage
question <- sample(1:3, 1)
if (question == 1) {
  questions[3] <- if (runif(1) > 0.5) paste("The value in row~1 and column~1 in the percentage table indicates:", prop_public[1,1],
  "percent of those, who evaluated the public transportation as very good,",
  "live in the city.") else paste("The value in row 4 and column~2 in the percentage table indicates:", prop_public[4,2],
  "percent of those, who evaluated the public transportation as very bad,",
  "live in the suburbs.")
  solutions[3] <- variant == 1
  explanations[3] <- if (solutions[3]) "This is the correct interpretation for row percentages." else
  paste("This is an interpretation for row percentages, but the table provides ",
        percentage[variant], "percentages.")
} else if (question == 2) {
  questions[3] <- paste("The value in row~1 and column~2 in the percentage table indicates:", prop_public[1,2],
                        "percent of those living in the suburbs evaluated the public transportation ",
                        "as very good.")
  solutions[3] <- variant == 2
  explanations[3] <- if (solutions[3]) "This is the correct interpretation for column percentages." else
  paste("This is an interpretation for column percentages, but the table provides ",
        percentage[variant], "percentages.")
} else {
  questions[3] <- paste("The value in row 4 and column~1 in the percentage table indicates:", prop_public[4,1],
                         "percent of the respondents live in the city and evaluated the ",
                        "public tranportation as very bad.")
  solutions[3] <- variant == 3
  explanations[3] <- if (solutions[3]) "This is the correct interpretation for total percentages." else
  paste("This is an interpretation for total percentages, but the table provides ",
        percentage[variant], "percentages.")
}

## 4. Interpretation column percentages
question <- sample(1:3, 1)
if (question == 1) {
  if (runif(1) > 0.5) {
    questions[4] <- paste("The value in row~2 and column~1 in the percentage table indicates:", prop_public[2,1],
                          "percent of the respondents in the city evaluated the public transportation as good.")
  } else {
    questions[4] <- paste("The value in row~3 and column~2 in the percentage table indicates:", prop_public[3,2],
                          "percent of the respondents in the suburbs evaluated the public transportation ",
                          "as bad.")
  }
  solutions[4] <- variant == 2
  explanations[4] <- if (solutions[4]) "This is the correct interpretation for column percentages."
  else paste("This is an interpretation for column percentages, but the table provides ", percentage[variant],
             "percentages.")

} else if (question == 2) {
  questions[4] <- paste("The value in row~2 and column~2 in the percentage table indicates:", prop_public[2,2],
                        "percentage of those, who evaluated the public transportation as good",
                        "live in the suburbs.")
  solutions[4] <- variant == 1
  explanations[4] <- if (solutions[4]) "This is the correct interpretation for row percentages."
  else paste("This is an interpretation for row percentages, but the table provides ",
             percentage[variant], "percentages.")
} else {
  questions[4] <- paste("The value in row~3 and column~1 in the percentage table indicates:", prop_public[3,1],
                        "percent of the respondents lived in the city and evaluated",
                        "the public transportation as bad.")
  solutions[4] <- variant == 3
  explanations[4] <- if (solutions[4]) "This is the correct interpretation for total percentages." else
  paste("This is an interpretation for total percentages, but the table provides ", percentage[variant], "percentages.")
}

## 5. Calculation row/column percentages
question <- sample(1:3, 1)
if (question == 1) {
  questions[5] <- paste("The percentage table can be easily constructed from the original contingency table:",
                        "percentages are calculated for each row.")
  solutions[5] <- variant == 1
  explanations[5] <- paste("This calculation yields row percentages.", ifelse(!solutions[5], paste("But the table provides ",
                     percentage[variant], "percentages."), ""))
} else if (question == 2) {
  questions[5] <- paste("The percentage table can be easily constructed from the original contingency table:",
                        "percentages are calculated for each column.")
  solutions[5] <- variant == 2
  explanations[5] <- paste("This calculation yields column percentages.",
                           ifelse(!solutions[5], paste("But the table provides ",
                     percentage[variant], "percentages."), ""))
} else {
  questions[5] <- paste("The percentage table can be easily constructed from the original contingency table:",
                        "Each value is in relation to the total sample size.")
  solutions[5] <- variant == 3
  explanations[5] <- paste("This calculation yields total percentages.",
                           ifelse(!solutions[5], paste("But the table provides ",
                     percentage[variant], "percentages."), ""))
}

ok <- any(solutions) && any(!solutions)
}

## permute order of solutions/questions
o <- sample(1:5)
questions <- questions[o]
solutions <- solutions[o]
explanations <- explanations[o]
@

\begin{question}
In a small city the satisfaction with the local public
transportation is evaluated. One question of interest is whether
inhabitants of the city are more satisfied with public
transportation compared to those living in the suburbs.

A survey with 250 respondents gave the following contingency table:

<<echo=FALSE>>=
public
@

The following table of percentages was constructed:

<<echo=FALSE>>=
print(format(prop_public, nsmall = 1, justify = "right",
             width = max(nchar(colnames(prop_public)))), quote = FALSE)
@

Which of the following statements are correct?

<<echo=FALSE, results=tex>>=
answerlist(questions)
@

\end{question}

\begin{solution}
<<echo=FALSE>>=
interpretation <- c(
  "conditional relative frequencies for location given satisfaction level.",
  "conditional relative frequencies for satisfaction level given location type.",
  "the relative frequencies for each combination of location type and satisfaction level."
)
@

In the percentage table, the \Sexpr{rowsums[variant]} are about 100 (except for possible rounding errors).
Hence, the table provides \Sexpr{percentage[variant]} percentages, i.e., \Sexpr{interpretation[variant]}

<<echo=FALSE, results=tex>>=
answerlist(ifelse(solutions, "True", "False"), explanations)
@

\end{solution}

%% META-INFORMATION
%% \extype{mchoice}
%% \exsolution{\Sexpr{mchoice2string(solutions)}}
%% \exname{Relative frequencies}
