Question

Consider the natural log transformation (“ln” transformation) of variables labour cost (L_COST), and total number of rooms per hotel (Total_Rooms). 4.1 Use the least squares method to estimate the regression coefficients b0 and b1 for the log-linear model 4.2 State the regression equation 4.3 Give the interpretation of the regression coefficient b1. 4.4 Give an interpretation of the coefficient of determination R2 . Also, test the significance of your model using the F-test. How, does the value of the coefficient of determination affect the outcome of the above test? 4.5 Test whether a 1% increase of the total number of rooms per hotel can increase the labour cost by more than 0.20%? Use the 5% level of significance for this test.

L_COST   Total_Rooms     
2.165.000   412     
2.214.985   313     
1.393.550   265     
2.460.634   204     
1.151.600   172     
801.469   133     
1.072.000   127     
1.608.013   322     
793.009   241     
1.383.854   172     
494.566   121     
437.684   70     
83.000   65     
626.000   93     
37.735   75     
256.658   69     
230.000   66     
200.000   54     
199.000   68     
11.720   57     
59.200   38     
130.000   27     
255.020   47     
3.500   32     
20.906   27     
284.569   48     
107.447   39     
64.702   35     
6.500   23     
156.316   25     
15.950   10     
722.069   18     
6.121   17     
30.000   29     
5.700   21     
50.237   23     
19.670   15     
7.888   8     
3.500   15     
112.181   18     
30.000   10     
3.575   26     
2.074.000   306     
1.312.601   240     
434.237   330     
495.000   139     
1.511.457   353     
1.800.000   324     
2.050.000   276     
623.117   221     
796.026   200     
360.000   117     
538.848   170     
568.536   122     
300.000   57     
249.205   62     
150.000   98     
220.000   75     
50.302   62     
517.729   50     
51.000   27     
75.704   44     
271.724   33     
118.049   25     
40.000   30     
10.000   10     
10.000   18     
70.000   73     
12.000   21     
20.000   22     
36.277   25     
36.277   25     
10.450   31     
14.300   16     
4.296   15     
379.498   16     
1.520   22     
45.000   12     
96.619   34     
270.000   37     
60.000   25     
12.500   10     
1.934.820   270     
3.000.000   261     
1.675.995   219     
903.000   280     
2.429.367   378     
1.143.850   181     
900.000   166     
600.000   119     
2.500.000   174     
1.103.939   124     
363.825   112     
1.538.000   227     
1.370.968   161     
1.339.903   216     
173.481   102     
210.000   96     
441.737   97     
96.000   56     
177.833   72     
252.390   62     
377.182   78     
111.000   74     
238.000   33     
45.000   30     
50.000   39     
40.000   32     
61.766   25     
166.903   41     
116.056   24     
41.000   49     
195.821   43     
96.713   20     
6.500   32     
5.500   14     
4.000   14     
15.000   13     
9.500   13     
48.200   53     
3.000   11     
27.084   16     
30.000   21     
20.000   21     
43.549   46     
10.000   21     

Answer 1

4.1 Use the least squares method to estimate the regression coefficients b0 and b1 for the log-linear model

b0=5.40738

b1(Total_Rooms..)=1.58207

4.2 State the regression equation

y predict(L_COST)=5.40738+1.58207*Total_Rooms

4.3 Give the interpretation of the regression coefficient b1.

When the total rooms increased by 1 unit, The labor cost will be increased by 1.58207 units.

4.4 Give an interpretation of the coefficient of determination R2 . Also, test the significance of your model using the F-test. How, does the value of the coefficient of determination affect the outcome of the above test?

Below is the summary of the model:

Call:
lm(formula = L_COST ~ ., data = wine)

Residuals:
Min 1Q Median 3Q Max
-2.9712 -0.4908 0.0837 0.5022 3.5097

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.40738 0.35904 15.06 <2e-16 ***
Total_Rooms.. 1.58207 0.08699 18.19 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.009 on 124 degrees of freedom
Multiple R-squared: 0.7273,   Adjusted R-squared: 0.7251
F-statistic: 330.8 on 1 and 124 DF, p-value: < 2.2e-16

R^2 is 0.7273, which means Total_Rooms variable is able to explain 72.73% variation in the labor cost.

F-test P-value is close to 0 (< 2.2e-16), so the variable & model is more significant, below is the hypothesis

Null hypothesis - variables does not have any significance.

Alternate hypothesis - variables are having significance, thus important.

F-statistic test: 330.8 which is (explained variance) / (unexplained variance)