February 26, 2023
laut zwei Stichproben (A und B) mithilfe zweier Boxplots dargestellt. Welche der folgenden Aussagen ist korrekt?
Hinweis: Die Aussagen sind entweder eindeutig richtig oder eindeutig falsch.
Categories:
---
extype: schoice
exsolution: r mchoice2string(solutions, single = TRUE)
exname: boxplots-de1a
expoints: 1
categories:
- vis
- eda
- schoice
date: '2023-02-26'
slug: boxplots-de1a
title: boxplots-de1a
---
```{r data generation, echo = FALSE, results = "hide"}
library(exams)
## convenience functions
SK <- function(x) diff(diff(fivenum(x)[2:4]))/diff(fivenum(x)[c(2, 4)])
trob <- function(a, b)
(median(a) - median(b))/sqrt(var(a)/length(a) + var(b)/length(b))
## DATA GENERATION
## dgp for one sample
dgp <- function(location = 0, scale = 1, skewed = FALSE, outlier = NULL,
n = 10, amount = 0.1)
{
## basic intervals from which equal amounts of observations are drawn
qq <- if (skewed) c(0, 2, 2.2, 6, 10) else c(0, 3, 5, 7, 10)
sim <- function(x) {
rval <- NULL
for(i in 1:(length(x)-1)) rval <- c(rval, runif(n, min = x[i], max = x[i+1]))
rval <- jitter(rval, amount = amount)
rval <- rval/4
rval
}
## draw under restrictions about IQR and SK
rval <- sim(qq)
if (skewed) {
while(IQR(rval) > 1.15 | IQR(rval) < 0.85 | abs(SK(rval)) < 0.7) rval <- sim(qq)
} else {
while(IQR(rval) > 1.15 | IQR(rval) < 0.85 |abs(SK(rval)) > 0.15) rval <- sim(qq)
}
## raw values (location = 0, scale = 1)
rval <- rval - ifelse(skewed, 0.55, 1.25)
## add outliers
rval <- c(rval, outlier)
## scale and shift
rval <- rval * scale + location
return(rval)
}
# initialize number of correct solutions:
n_correct <- 0
# keep on generating results until exactly one answer is true:
while (n_correct != 1){
## generate random parameters for dgp()
scale <- sample(c(1, sample(c(1, 3), 1))) * runif(1, min = 0.5, max = 10) * sample(c(-1, 1), 1)
location <- sample(c(0, sample(c(0, 2), 1))) * scale + runif(1, min = -50, max = 50)
skewed <- if (runif(1) < 2/3) c(FALSE, FALSE) else sample(c(TRUE, sample(c(TRUE, FALSE), 1)))
if (runif(1) < 2/3) {
outlier <- list(NULL, NULL)
} else {
if (any(skewed)) {
outlier <- if (skewed[1]) -sign(scale[1]) * runif(sample(1:2, 1), min = 3, max = 4) else NULL
outlier <- c(list(outlier), if (skewed[2])
list(-sign(scale[1]) * runif(sample((0:1 + !skewed[1]), 1), min = 3, max = 4)) else list(NULL))
} else {
outlier <- runif(sample(1:3, 1), min = 3, max = 4)
outlier <- outlier * sample(c(-1, 1), length(outlier), replace = TRUE)
id <- sample(1:length(outlier), sample(1:length(outlier), 1))
outlier1 <- outlier[id]
outlier2 <- outlier[-id]
outlier <- list(if(length(outlier1) > 0) outlier1 else NULL,
if (length(outlier2) > 0) outlier2 else NULL)
}
}
## call dgp under certain constrains
A <- dgp(location = location[1], scale = scale[1], skewed = skewed[1], outlier = outlier[[1]],
n = sample(8:12, 1))
B <- dgp(location = location[2], scale = scale[2], skewed = skewed[2], outlier = outlier[[2]],
n = sample(8:12, 1))
while((length(unique(location)) < 2 & abs(trob(A, B)) > 0.4) | (abs(SK(A)) > 0.15 & abs(SK(A)) < 0.7) | (abs(SK(B)) > 0.15 & abs(SK(B)) < 0.7)) {
A <- dgp(location = location[1], scale = scale[1], skewed = skewed[1], outlier = outlier[[1]],
n = sample(8:12, 1))
B <- dgp(location = location[2], scale = scale[2], skewed = skewed[2], outlier = outlier[[2]],
n = sample(8:12, 1))
}
SK_A <- ifelse(abs(SK(A)) < 0.2, "symmetric", ifelse(SK(A) >= 0.2, "right-skewed", "left-skewed"))
SK_B <- ifelse(abs(SK(B)) < 0.2, "symmetric", ifelse(SK(B) >= 0.2, "right-skewed", "left-skewed"))
## QUESTION/ANSWER GENERATION
questions <- character(5)
solutions <- logical(5)
explanations <- character(5)
questions[1] <- "Die zentrale Tendenz der Verteilungen ist (etwa) identisch."
solutions[1] <- abs(trob(A, B)) < 0.5
explanations[1] <- if (solutions[1]) "Die Verteilungen haben eine etwa identische zentrale Tendenz." else
paste("Verteilung ", c("A", "B")[(median(A) < median(B)) + 1],
" hat im Durchschnitt höhere Werte als Verteilung ",
c("A", "B")[(median(A) >= median(B)) + 1], ".", sep = "")
outlier <- length(unlist(outlier)) > 0
questions[2] <- "Beide Verteilungen haben keine Ausreißer."
solutions[2] <- !outlier
explanations[2] <- if (solutions[2])
"Beide Verteilungen haben keine Beobachtungen, die mehr als das 1.5-fache der Interquartilsabstands von der Box entfernt sind" else "Es gibt Beobachtungen, die mehr als das 1.5-fache des Interquartilsabstand von der Box entfernt sind."
questions[3] <- "Die Streuung in Stichprobe A ist deutlich größer als in Stichprobe B."
solutions[3] <- IQR(A)/IQR(B) > 1.5
explanations[3] <- paste("Der Interquartilsabstand in Stichprobe A ist",
ifelse(solutions[3], "", "_nicht_"), "deutlich größer als in B.")
questions[4] <- "Die Schiefe der beiden Stichproben ist ähnlich."
solutions[4] <- SK_A == SK_B
explanations[4] <- if (solutions[4]) paste("Die Schiefe der beiden Verteilungen ist ähnlich, beide sind",
ifelse(abs(SK(A)) < 0.2, "etwa symmetrisch.", ifelse(SK(A) >= 0.2, "rechtsschief.", "linksschief."))) else
paste("Die Schiefe der beiden Verteilungen ist unterschiedlich. Stichprobe A ist",
ifelse(abs(SK(A)) < 0.2, "etwa symmtrisch.", ifelse(SK(A) >= 0.2, "rechtsschief.", "linksschief.")),
"Stichprobe B ist", ifelse(abs(SK(B)) < 0.2, "etwa symmetrisch.", ifelse(SK(B) >= 0.2, "rechtsschief.", "linksschief.")))
i <- sample(1:2, 1)
j <- sample(1:3, 1)
questions[5] <- paste("Verteilung ", c("A", "B")[i], " ist ",
c("etwa symmetrisch", "rechtsschief", "linksschief")[j], ".", sep = "")
SK_i <- SK(list(A, B)[[i]])
solutions[5] <- switch(j, abs(SK_i) < 0.2,
SK_i >= 0.2,
SK_i <= -0.2)
explanations[5] <- paste("Verteilung ", c("A", "B")[i], " ist ",
ifelse(abs(SK_i) < 0.2, "etwa symmetrisch.",
ifelse(SK_i >= 0.2, "rechtsschief.", "linksschief.")))
# number of correct solutions:
n_correct <- sum(solutions==TRUE)
}
```
# Aufgabe
laut zwei Stichproben (A und B) mithilfe zweier Boxplots dargestellt.
Welche der folgenden Aussagen ist korrekt?
*Hinweis*:
Die Aussagen sind entweder eindeutig richtig oder eindeutig falsch.
```{r boxplot, echo = FALSE, results = "hide", fig.height = 4, fig.width = 5, fig.path = "", fig.cap = ""}
par(mar = c(2.5, 2, 1, 0.5))
dat <- data.frame(y = c(A, B), x = factor(rep(c("A", "B"), c(length(A), length(B)))))
boxplot(y ~ x, data = dat, xlab = "", ylab = "")
```
```{r questionlist, echo = FALSE, results = "asis"}
answerlist(questions, markup = "markdown")
```
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# Lösung
```{r solutionlist, echo = FALSE, results = "asis"}
answerlist(ifelse(solutions, "Wahr", "Falsch"), explanations, markup = "markdown")
```
---
Categories:
- vis
- eda
- schoice