1. Upload the "salary_data.csv" data set.

2. Build the plot to look at the relationship between the variables. What will be the dependent
variable (outcome), what will be the independent variable (predictor)?

3. Perform linear regression analysis (fit a simple linear regression model between the variables).
Draw the best-fit regression line.

4. Check the main assumptions of the model, use the four main plots for checking:

Plot 1.Linearity of the data, independence of residuals

Plot 2.Normality of residuals using Q-Q plot

Plot 3.Constant variance of residuals

Plot 4. No influential outliers

5. Check the assumption "Normality of residuals" using histogram and normality tests; and "Zero
mean of residuals". Don't forget to look at the Q-Q plot from the previous question.

6. Obtain parameters of the regression line (the intercept, the slope of the line); check the
significance. Fill up the check list.

7. Obtain criteria for the model evaluation (Adjusted R-squared, RSE, AIC, the 95% confidence
intervals). Fill up the check list.

8. After checking all the assumptions, what conclusion can you make?

9. Take away the outlier (number 5 on the previous plots) that has a high influence on the
regression line.

To identify the outlier, first, look at histograms of the variables.

10. Delete the outlier using the tidyverse package.

11. Now, when you have data without the outlier, fit an adjusted simple linear regression model
(repeat steps 2-7).

12. Fill up the check list, compare the models, choose a better model and draw your conclusions.


Check list

                                          Model_version_1

                                          Model_version_2

                               Assumptions after Linear regression:

                     Plot 1: Linearity of the data, independence of residuals



                                  Plot 2: Normality of residuals

                                   +histogram + normality tests



                                      Zero mean of residuals



                              Plot 3: Constant variance of residuals



                                  Plot 4: No influential outliers



                           Results interpretation and model evaluation:

                                   Parameters of the regression:

                                          - intercept (α)

                                      - slope of the line (β)



                                  Significance of β and the model



                                Criteria for the model evaluation:

                                  Adjusted R^2; RSE; 95% CI; AIC



                               Conclusion based on the chosen model:

The assumptions are _______________; the model and the independent variable (_______________) are
_________________ (p____________).

The _______________ variable explains ________% of the ________________________ variability, RSE
equals ______________.

The estimate of the β-coefficient equals ________________

(95% CI [_______________________]), the intercept α equals _______.

 Y(____________)=__________________________________(for each one-unit shift of ___________________
   ____________________increases by ___________).



Check list

                                          Model_version_1

                                          Model_version_2

                               Assumptions after Linear regression:

                     Plot 1: Linearity of the data, independence of residuals

                                                met

                                                met

                                  Plot 2: Normality of residuals

                                   +histogram + normality tests

                                              not met

                                                met

                                      Zero mean of residuals

                                                met

                                                met

                              Plot 3: Constant variance of residuals

                                                met

                                                met

                                  Plot 4: No influential outliers

                                              not met

                                                met

                           Results interpretation and model evaluation:

                                   Parameters of the regression:

                                          - intercept (α)

                                      - slope of the line (β)

                                        α= -28.63,  β=0.62

                                         Y(productivity)=

                                       -28.63+0.62*X(salary)


                                        α= -41.75,  β=0.71

                                         Y(productivity)=

                                       -41.75+0.71*X(salary)


                                  Significance of β and the model

                                              p<0.001

                                              p<0.001

                                Criteria for the model evaluation:

                                  Adjusted R^2; RSE; 95% CI; AIC

         R^2[adj.]=33%, RSE=7.95 (thousand dollars per year), 95%CI [0.44;0.79], AIC=702.5

         R^2[adj.]=51%, RSE=6.22 (thousand dollars per year), 95%CI [0.57;0.85], AIC=646.7

                               Conclusion based on the chosen model:

The assumptions are met the model and the independent variable (salary) are significant (p<0.001).

The salary variable explains 51% of the productivity variability, RSE equals 6.22 thousand dollars
per year.

The estimate of the β-coefficient equals 0.71 (95% CI [0.57;0.85]), the intercept α equals -41.75.

  Y(productivity)= -41.75+0.71*X(salary) (for each one-unit shift of salary productivity increases
by 0.71).