The data salary.txt collects the salary data from a company. We want to study the relationship of

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Question:

The data “salary.txt” collects the salary data from a company. We want to study the relationship of the salary (salary) and the number of years of working experience (years). The response variable is salary, and the explanatory variable is years. Use R to answer the following questions.

years salary
1 41504
1 32619
1 44322
2 40038
2 46147
2 38447
2 38163
3 42104
3 25597
3 39599
3 55698
4 47220
4 65929
4 55794
4 45959
5 52460
5 60308
5 61458
5 56951
6 56174
6 59363
6 57642
6 69792
7 59321
7 66379
7 64282
7 48901
8 100711
8 59324
8 54752
8 73619
9 65382
9 58823
9 65717
9 92816
9 72550
10 71365
10 88888
10 62969
10 45298
11 111292
11 91491
11 106345
11 99009
12 73981
12 72547
12 74991
12 139249
13 119948
13 128962
13 98112
13 97159
14 125246
14 89694
14 73333
14 108710
15 97567
15 90359
15 119806
15 101343
16 147406
16 153020
16 143200
16 97327
17 184807
17 146263
17 127925
17 159785
17 174822
18 177610
18 210984
18 160044
18 137044
19 182996
19 184183
19 168666
19 121350
20 193627
20 142611
20 170131
20 134140
21 129446
21 201469
21 202104
21 220556
22 166419
22 149044
22 247017
22 247730
23 252917
23 235517
23 241276
23 197229
24 175879
24 253682
24 262578
24 207715
25 221179
25 212028
25 312549

(a) Fit an SLRM and print its summary.

(b) Create a scatter plot of the data with the fitted regression line. Does the regression line appear to be a good fit to the data?

(c) Create two residual plots: (i) a scatter plot of the studentized residuals versus the fitted values with three horizontal reference lines: -3, 0, 3, (ii) a QQ plot of the residuals. Are these two plots show any evidence indicating violations of the assumptions for an SLRM? If yes, please explain.

(d) Is there any significant evidence indicating the error variance is not constant? Perform the Brown-Forsythe test at the significance level 0.05. For this test, create two groups of the employees: the employees in one group have less than 13 years of experience, and the employees in the other group have at least 13 years of experience.

(e) Perform the F test to determine whether or not a linear regression function is a good fit for the relationship between the mean salary and the number of years of working experience. Perform the test at the significance level 0.05. State the null and alternative hypotheses (using either plain English or mathematical expression), F test statistic value, p-value, and conclusion.

(f) [5 pts] Find an appropriate transformation of the response variable via the Box-Cox procedure using the R function boxcox. Consider five candidate λ values, λ = −1, −0.5, 0, 0.5, 2. Print out the log-likelihood for each λ value. Choose the optimal λ value which has the maximum log-likelihood.

(g) [5 pts] Perform a natural logarithm transformation on the salary, and fit an SLRM in which the response variable is the logarithm transformed salary, and the explanatory variable is the number of years of experience. (i) Print out the summary of the fitted SLRM. (ii) Create a scatter plot of the transformed salary versus the number of years of experience with the fitted regression line. Does the regression line appear to be a good fit to the transformed data?

(h) [5 pts] Create two residual plots for the SLRM obtained in part (g): (i) a scatter plot of the studentized residuals versus the fitted values with three horizontal reference lines: -3, 0, 3, (ii) a QQ plot of the residuals. Are these two plots show any evidence indicating violations to the assumptions of an SLRM? If yes, please explain.

(i) [5 pts] Obtain Bonferroni simultaneous confidence intervals for the two regression coefficients in the SLRM fitted in part (g) at a 90% family confidence.

(j) [5 pts] Using the SLRM fitted in part (g), we want to estimate the average log salary when the number of years of working experience is 5, 10, 15, and 20. Provide a point estimate for each number of years and Bonferroni simultaneous confidence intervals at a 90% family confidence.