```{r data generation, echo = FALSE, results = "hide"}
## 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 centre", "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 contains ", 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], "do not give 100.")
## 2. interprete total/row/column percentage
if (runif(1) > 0.5) {
questions[2] <- "The percentage table gives the location distribution for each level of satisfaction."
solutions[2] <- variant == 1
explanations[2] <- if (solutions[2]) "The row sums give 100 (except for possible rounding errors)." else
"The row sums do not give 100."
} else {
questions[2] <- "The percentage table gives the satisfaction distribution for each location type."
solutions[2] <- variant == 2
explanations[2] <- if (solutions[2]) "The column sums give 100 (except for possible rounding errors)." else
"The column sums do not give 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 centre.") 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 of row percentages." else
paste("This is an interpretation of row percentages, but the table gives ",
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 of column percentages." else
paste("This is an interpretation of column percentages, but the table gives ",
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 center and evaluated the ",
"public tranportation as very bad.")
solutions[3] <- variant == 3
explanations[3] <- if (solutions[3]) "This is the correct interpretation of total percentages." else
paste("This is an interpretation of total percentages, but the table gives ",
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 centre 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 of column percentages."
else paste("This is an interpretation of column percentages, but the table gives ", 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 of row percentages."
else paste("This is an interpretation of row percentaes, but the table gives ",
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 centre and evaluated",
"the public transportation as bad.")
solutions[4] <- variant == 3
explanations[4] <- if (solutions[4]) "This is the correct interpretation of total percentages." else
paste("This is an interpretation of total percentages, but the table gives ", 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 evaluation gives row percentages.", ifelse(!solutions[5], paste("But the table gives ",
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 evaluation gives column percentages.",
ifelse(!solutions[5], paste("But the table gives ",
percentage[variant], "percentages."), ""))
} else {
questions[5] <- paste("The percentage table can be easily constructed from the original contingency table:",
"Each value is related to the total sample size.")
solutions[5] <- variant == 3
explanations[5] <- paste("This evaluation gives total percentages.",
ifelse(!solutions[5], paste("But the table gives ",
percentage[variant], "percentages."), ""))
}
## permute order of solutions/questions
o <- sample(1:5)
questions <- questions[o]
solutions <- solutions[o]
explanations <- explanations[o]
```
Question
========
In a small city the satisfaction with the local public
transportation is evaluated. One question of interest is whether
inhabitants of the city centre are more satisfied with public
transportation compared to those living in the suburbs.
A survey with 250 respondents gave the following contingency table:
```{r, echo = FALSE, comment = NA}
public
```
The following table of percentages was constructed:
```{r, echo = FALSE, comment = NA}
print(format(prop_public, nsmall = 1, justify = "right",
width = max(nchar(colnames(prop_public)))), quote = FALSE)
```
Which of the following statements are correct?
```{r questionlist, echo = FALSE, results = "asis"}
answerlist(questions, markup = "markdown")
```
Solution
========
```{r, echo = FALSE}
interpretation <- c("conditional proportions for location given satisfaction level.",
"conditional proportions for satisfaction level given location type.",
"the proportions for each location type and satisfaction level.")
```
In the percentage table, the `r rowsums[variant]` are about 100
(except for possible rounding errors). Hence, the table provides
`r percentage[variant]` percentages, i.e.,
`r interpretation[variant]`
```{r solutionlist, echo = FALSE, results = "asis"}
answerlist(
ifelse(solutions, "True", "False"),
explanations, markup = "markdown")
```
Meta-information
================
extype: mchoice
exsolution: `r mchoice2string(solutions)`
exname: Relative frequencies