## 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)
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.
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}
extype: num
exsolution: r fmt(sol, 3)
exname: Prediction
extol: 0.01
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