In exams: Automatic Generation of Exams in R

## regression parameters
n <- sample(40:90, 1)
b <- sample(c(-1, 1), 1) * runif(1, 1, 2) * sample(c(0.1, 0.5, 1), 1)
s <- sample(c(0.5, 1, 2), 1)

## data and regression
x2_ok <- FALSE
while(!x2_ok) {

d <- data.frame(
  x1 = rnorm(n),
  x2 = rnorm(n),
  err = rnorm(n, sd = s)
)
d$y <- 0 + b * d$x1 + d$err

x2_ok <- summary(lm(y ~ x1 + x2, data = d))$coefficients[3, 4] > 0.1
}

## different types
type <- sample(c("linear", "semi-logarithmic", "log-log"), 1)
if(type == "linear") {
  m <- lm(y ~ x1 + x2, data = d)
  xunit <- "unit" 
  yunit <- "units"
  eff <- round(coef(m)[2], digits = 2)
} else if(type == "semi-logarithmic") {
  d$y <- exp(d$y)
  m <- lm(log(y) ~ x1 + x2, data = d)
  xunit <- "unit" 
  yunit <- "percent"
  eff <- round(100 * exp(coef(m)[2]) - 100, digits = 2)
} else if(type == "log-log") {
  d$y <- exp(d$y)
  d$x1 <- exp(d$x1)
  d$x2 <- exp(d$x2)
  m <- lm(log(y) ~ log(x1) + log(x2), data = d)
  xunit <- "percent" 
  yunit <- "percent"
  eff <- round(100 * exp(0.01 * coef(m)[2]) - 100, digits = 2)
}

## summaries
direct <- if(coef(m)[2] > 0) "increases" else "decreases"
if(summary(m)$coefficients[2, 4] < 0.05) {
  sign1 <- "Also"
  sign2 <- ""
} else {
  sign1 <- "However"
  sign2 <- "_not_"
}

rsq <- summary(m)$r.squared
fstat <- summary(m)$fstatistic["value"]

write.csv(d, "regression.csv", row.names = FALSE, quote = FALSE)

Question

Using the data provided in regression.csv estimate a linear regression of y on x1 and x2. Answer the following questions.

Answerlist

Proportion of variance explained (in percent):
F-statistic:
Characterize in your own words how the response y depends on the regressors x1 and x2.
Upload the R script you used to analyze the data.

Solution

The presented results describe a r type regression.

summary(m)

The mean of the response y r direct with increasing x1. If x1 increases by 1 r xunit then a change of y by about r eff r yunit can be expected. r sign1, the effect of x1 is r sign2 significant at the 5 percent level.

Variable x2 has no significant influence on the response at 5 percent level.

The R-squared is r signif(rsq, 4) and thus r fmt(100 * rsq, 2) percent of the variance of the response is explained by the regression.

The F-statistic is r fmt(fstat, 2).

Answerlist

Proportion of variance explained: r fmt(100 * rsq, 2) percent.
F-statistic: r fmt(fstat, 2).
Characterization: r type.
R code.

Meta-information

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