In exams: Automatic Generation of Exams in R

## DATA GENERATION
n <- sample(35:65,1)
mx <- runif(1, 40, 60)
my <- runif(1, 200, 280)
sx <- runif(1, 9, 12)
sy <- runif(1, 44, 50)
r <- round(runif(1, 0.5, 0.9), 2)
x <- rnorm(n, mx, sd = sx)
y <- (r * x/sx + rnorm(n, my/sy - r * mx/sx, sqrt(1 - r^2))) * sy

mx <- round(mean(x))
my <- round(mean(y))
r <- round(cor(x, y), digits = 2)
varx <- round(var(x))
vary <- round(var(y))

b <- r * sqrt(vary/varx)
a <- my - b * mx

X <- round(runif(1, -10, 10) + mx)

## QUESTION/ANSWER GENERATION
sol <- round(a + b * X, 3)

Question

For r n firms the number of employees $X$ and the amount of expenses for continuing education $Y$ (in EUR) were recorded. The statistical summary of the data set is given by:

| | Variable $X$ | Variable $Y$ | |:--------:|:------------:|:------------:| | Mean | r mx | r my | | Variance | r varx | r vary |

The correlation between $X$ and $Y$ is equal to r r.

Estimate the expected amount of money spent for continuing education by a firm with r X employees using least squares regression.

Solution

First, the regression line $y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$ is determined. The regression coefficients are given by: \begin{eqnarray} && \hat \beta_1 = r \cdot \frac{s_y}{s_x} = r r \cdot \sqrt{\frac{r vary}{r varx}} = r round(b,5), \ && \hat \beta_0 = \bar y - \hat \beta_1 \cdot \bar x = r my - r round(b,5) \cdot r mx = r round(a,5). \end{eqnarray}

The estimated amount of money spent by a firm with r X employees is then given by: \begin{eqnarray} \hat y = r round(a, 5) + r round(b, 5) \cdot r X = r sol. \end{eqnarray}