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

TR=TOTAL ROOMS, L COST =LABOUR COST

TR L_COST       Turnover_per_Room
412       2,165,000       21,519.42
313       2,214,985       21,755.04
265       1,393,550       17,937.91
204       2,460,634       37,400.05
172       1,151,600       31,824.30
133       801,469 19,444.46
127       1,072,000       22,551.18
322       1,608,013       18,205.04
241       793,009 8,793.00
172       1,383,854       25,114.16
121       494,566       14,095.35
70       437,684       22,231.59
65       83,000 5,953.85
93       626,000       18,150.99
75       37,735       3,871.67
69       256,658       11,071.70
66       230,000       8,030.30
54       200,000       10,185.19
68       199,000      
57       11,720       2,982.46
38       59,200       6,342.11
27       130,000       25,185.19
47       255,020       18,223.26
32       3,500 1,000.00
27       20,906 2,384.85
48       284,569       14,264.58
39       107,447       10,478.26
35       64,702       10,811.29
23       6,500 3,478.26
25       156,316       22,231.56
10       15,950       8,150.00
18       722,069       81,556.71
17       6,121 2,151.88
29       30,000       4,068.97
21       5,700 4,142.86
23       50,237       5,113.83
15       19,670       10,037.87
8       7,888 4,849.25
20 3,750.00
11 1,753.91
15       3,500 2,666.67
18       112,181       34,260.90
23              
10       30,000 12,000.00
26       3,575 3,001.81
306       2,074,000       19,803.92
240       1,312,601       15,823.58
330       434,237       4,361.65
139       495,000       17,050.36
353       1,511,457       15,370.22
324       1,800,000       15,432.10
276       2,050,000       22,101.45
221       623,117       9,199.82
200       796,026       18,158.06
117       360,000       11,649.57
170       538,848       10,294.08
122       568,536       17,510.12
57       300,000       15,614.04
62       249,205       9,623.61
98       150,000       6,326.53
75       220,000       6,666.67
62       50,302       2,058.19
50       517,729       20,000.00
27       51,000       16,666.67
44       75,704       7,118.52
33       271,724       40,499.76
25       118,049       9,664.80
42              
30       40,000       4,833.33
44 522.73
10       10,000       7,300.00
18       10,000       5,555.56
18 1,338.22
73       70,000       4,958.90
21       12,000       6,904.76
22       20,000       3,636.36
25       36,277       1,489.72
25       36,277       1,489.72
31       10,450       2,348.39
16       14,300       5,000.00
15       4,296       732.00
12 1,083.33
11 2,000.00
16       379,498      
22       1,520 673.36
12       45,000       58,333.33
34       96,619       18,817.53
37       270,000       21,621.62
25       60,000 10,000.00
10       12,500 9,000.00
270       1,934,820       27,977.57
261       3,000,000       36,781.61
219       1,675,995       17,559.77
280       903,000 15,907.14
378       2,429,367       16,666.67
181       1,143,850       22,352.93
166       900,000 20,180.72
119       600,000       31,932.77
174       2,500,000       32,628.43
124       1,103,939       17,559.77
112       363,825 8,054.72
227       1,538,000       16,173.81
161       1,370,968       23,161.53
216       1,339,903       12,503.53
102       173,481       6,795.40
96       210,000       15,833.33
97       441,737       11,759.43
56       96,000       8,000.00
72       177,833       7,501.82
62       252,390       25,266.45
78       377,182       17,409.35
74       111,000       9,891.89
33       238,000       23,848.48
30       45,000       5,919.30
39       50,000       3,846.15
32       40,000       6,250.00
25       61,766       4,237.28
41       166,903       25,266.46
24       116,056       17,409.33
49       41,000       5,102.04
43       195,821       11,759.42
9              
20       96,713       17,409.35
32       6,500       2,953.13
14       5,500       2,500.00
14       4,000       4,285.71
13       15,000       2,307.69
13       9,500       1,538.46
53       48,200       3,528.30
11       3,000       10,909.09
16       27,084       3,652.44
21       30,000       2,380.95
21       20,000       2,380.95
46       43,549       1,314.04
21       10,000       952.38

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.

> 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(room,cost)
[1] 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.