Question

Forecasting labour costs is a key aspect of hotel revenue management that enables hoteliers to appropriately allocate hotel resources and fix pricing strategies. Mary, the President of Hellenic Hoteliers Federation (HHF) is interested in investigating how labour costs (variable L_COST) relate to the number of rooms in a hotel (variable Total_Rooms). Suppose that HHF has hired you as a business analyst to develop a linear model to predict hotel labour costs based on the total number of rooms per hotel using the data provided.

3.1 Use the least squares method to estimate the regression coefficients b0 and b1

3.2 State the regression equation

3.3 Plot on the same graph, the scatter diagram and the regression line

3.4 Give the interpretation of the regression coefficients b0 and b1 as well as the result of the t-test on the individual variables (assume a significance level of 5%)

3.5 Determine the correlation coefficient of the two variables and provide an interpretation of its meaning in the context of this problem 3.6 Check statistically, at the 0.05 level of significance whether there is any evidence of a linear relationship between labour cost and total number of rooms per hotel

I need only the 3.4 and 3.5 questions.

Total_Rooms   L_COST
412   2.165.000
313   2.214.985
265   1.393.550
204   2.460.634
172   1.151.600
133   801.469
127   1.072.000
322   1.608.013
241   793.009
172   1.383.854
121   494.566
70   437.684
65   83.000
93   626.000
75   37.735
69   256.658
66   230.000
54   200.000
68   199.000
57   11.720
38   59.200
27   130.000
47   255.020
32   3.500
27   20.906
48   284.569
39   107.447
35   64.702
23   6.500
25   156.316
10   15.950
18   722.069
17   6.121
29   30.000
21   5.700
23   50.237
15   19.670
8   7.888
20  
11  
15   3.500
18   112.181
23  
10   30.000
26   3.575
306   2.074.000
240   1.312.601
330   434.237
139   495.000
353   1.511.457
324   1.800.000
276   2.050.000
221   623.117
200   796.026
117   360.000
170   538.848
122   568.536
57   300.000
62   249.205
98   150.000
75   220.000
62   50.302
50   517.729
27   51.000
44   75.704
33   271.724
25   118.049
42  
30   40.000
44  
10   10.000
18   10.000
18  
73   70.000
21   12.000
22   20.000
25   36.277
25   36.277
31   10.450
16   14.300
15   4.296
12  
11  
16   379.498
22   1.520
12   45.000
34   96.619
37   270.000
25   60.000
10   12.500
270   1.934.820
261   3.000.000
219   1.675.995
280   903.000
378   2.429.367
181   1.143.850
166   900.000
119   600.000
174   2.500.000
124   1.103.939
112   363.825
227   1.538.000
161   1.370.968
216   1.339.903
102   173.481
96   210.000
97   441.737
56   96.000
72   177.833
62   252.390
78   377.182
74   111.000
33   238.000
30   45.000
39   50.000
32   40.000
25   61.766
41   166.903
24   116.056
49   41.000
43   195.821
9  
20   96.713
32   6.500
14   5.500
14   4.000
13   15.000
13   9.500
53   48.200
11   3.000
16   27.084
21   30.000
21   20.000
46   43.549
21   10.000

Answer 1

In order to solve this question I used R software.

R codes and output:

First we need to delete the observation with missing values in order to fit regression equation.

> room=scan('clipboard');room
Read 126 items
[1] 412 313 265 204 172 133 127 322 241 172 121 70 65 93 75 69 66 54
[19] 68 57 38 27 47 32 27 48 39 35 23 25 10 18 17 29 21 23
[37] 15 8 15 18 10 26 306 240 330 139 353 324 276 221 200 117 170 122
[55] 57 62 98 75 62 50 27 44 33 25 30 10 18 73 21 22 25 25
[73] 31 16 15 16 22 12 34 37 25 10 270 261 219 280 378 181 166 119
[91] 174 124 112 227 161 216 102 96 97 56 72 62 78 74 33 30 39 32
[109] 25 41 24 49 43 20 32 14 14 13 13 53 11 16 21 21 46 21
> cost=scan('clipboard');cost
Read 126 items
[1] 2165.000 2214.985 1393.550 2460.634 1151.600 801.469 1072.000 1608.013
[9] 793.009 1383.854 494.566 437.684 83.000 626.000 37.735 256.658
[17] 230.000 200.000 199.000 11.720 59.200 130.000 255.020 3.500
[25] 20.906 284.569 107.447 64.702 6.500 156.316 15.950 722.069
[33] 6.121 30.000 5.700 50.237 19.670 7.888 3.500 112.181
[41] 30.000 3.575 2074.000 1312.601 434.237 495.000 1511.457 1800.000
[49] 2050.000 623.117 796.026 360.000 538.848 568.536 300.000 249.205
[57] 150.000 220.000 50.302 517.729 51.000 75.704 271.724 118.049
[65] 40.000 10.000 10.000 70.000 12.000 20.000 36.277 36.277
[73] 10.450 14.300 4.296 379.498 1.520 45.000 96.619 270.000
[81] 60.000 12.500 1934.820 3000.000 1675.995 903.000 2429.367 1143.850
[89] 900.000 600.000 2500.000 1103.939 363.825 1538.000 1370.968 1339.903
[97] 173.481 210.000 441.737 96.000 177.833 252.390 377.182 111.000
[105] 238.000 45.000 50.000 40.000 61.766 166.903 116.056 41.000
[113] 195.821 96.713 6.500 5.500 4.000 15.000 9.500 48.200
[121] 3.000 27.084 30.000 20.000 43.549 10.000
> fit=lm(cost~room)
> summary(fit)

Call:
lm(formula = cost ~ room)

Residuals:
Min 1Q Median 3Q Max
-1485.63 -103.31 -24.63 56.92 1528.86

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -87.050 41.885 -2.078 0.0397 *
room 6.082 0.315 19.307 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 340.4 on 124 degrees of freedom
Multiple R-squared: 0.7504, Adjusted R-squared: 0.7484
F-statistic: 372.8 on 1 and 124 DF, p-value: < 2.2e-16

> plot(room,cost)
> abline(fit)
> cor.test(room,cost)

Pearson's product-moment correlation

data: room and cost
t = 19.307, df = 124, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.8147969 0.9041651
sample estimates:
cor
0.8662512

Que.1

Regression coefficient b0 = -87.050

b1 = 6.082

Que.2

Regression equation:

Labour cost = -87.050 + 6.082 Total rooms

Que.3

Scatter plot:

Que.4

Interpretation of b1 :

When number of rooms in the hotel are increased by 1 then labour cost increased by 6.082  units.

Interpretation bo :

When a hotel have zero room then labour cost is -87.050

The value of individual t test for b0 is -2.078 and p-value is 0.0397 which is less than 0.05, hence b0 is statistically significant.

The value of individual t test for b1 is 19.307 and p-value is 0.0000 which is less than 0.05, hence b1 is statistically significant.

Que.5

The correlation coefficient between total rooms per hotel and labour cost is 0.8663. Which is high degree positive correlation. Which indicates that if we increase number of room per hotel then labour cost also increases and voice-a-versa.

Que.6

The p-value for testing correlation coefficient is 2.2e-16 which is less than 0.05, hence at 5% level of significance we reject null hypothesis. And conclude that their exist linear relationship between labour cost and total number of rooms per hotel.