## Exam 1

1. #### Question

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

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

#### Solution

The presented results describe a semi-logarithmic regression.
```Call:
lm(formula = log(y) ~ x1 + x2, data = d)
Residuals:
Min       1Q   Median       3Q      Max
-2.68802 -0.67816 -0.01803  0.68866  2.35064
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.06802    0.13491  -0.504    0.616
x1           1.37863    0.13351  10.326 9.34e-15 ***
x2          -0.21449    0.13995  -1.533    0.131
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.052 on 58 degrees of freedom
Multiple R-squared:  0.6511,	Adjusted R-squared:  0.6391
F-statistic: 54.12 on 2 and 58 DF,  p-value: 5.472e-14
```
The mean of the response y increases with increasing x1. If x1 increases by 1 unit then a change of y by about 296.94 percent can be expected. Also, the effect of x1 is significant at the 5 percent level.
Variable x2 has no significant influence on the response at 5 percent level.
The R-squared is 0.6511 and thus 65.11 percent of the variance of the response is explained by the regression.
The F-statistic is 54.12.

1. Proportion of variance explained: 65.11 percent.
2. F-statistic: 54.12.
3. Characterization: semi-logarithmic.
4. R code.