A suburban hotel derives its gross income from its hotel and restaurant operatio

Question

A suburban hotel derives its gross income from its hotel and restaurant operations. The owners are interested in the relationship between the number of rooms occupied on a nightly basis and the revenue per day in the restaurant. Below is a sample of 25 days (Monday through Thursday) from last year showing the restaurant income and number of rooms occupied.

    

Determine the coefficient of correlation between the two variables. (Round your answer to 3 decimal places.)

State the decision rule for 0.1 significance level: H0: 0; H1: > 0. (Round your answer to 3 decimal places.)

Is it reasonable to conclude that there is a positive relationship between revenue and occupied rooms? Use the 0.1 significance level.

H0, There is

What percent of the variation in revenue in the restaurant is accounted for by the number of rooms occupied?(Round your answer to 1 decimal place.)

rev: 08_13_2013_QC_23228, 02_17_2015_QC_CS-1453

Day Income Occupied Day Income Occupied 1 $ 1,452 35 14 $ 1,425 31 2 1,361 20 15 1,446 51 3 1,426 21 16 1,439 62 4 1,470 45 17 1,348 45 5 1,456 70 18 1,450 41 6 1,430 23 19 1,431 62 7 1,354 30 20 1,446 47 8 1,442 21 21 1,485 43 9 1,394 15 22 1,405 38 10 1,469 36 23 1,461 36 11 1,399 41 24 1,490 30 12 1,468 35 25 1,426 65 13 1,637 30

Explanation / Answer

We will use R to solve all the question.

1st take the data in R using the code below:

Code;

income<-c(1452,   1361,   1426,   1470,   1456,   1430,   1354,   1442,   1394,   1469,   1399,   1468,   1637,
   1425,   1446,   1439,   1348,   1450,   1431,   1446,   1485,   1405,   1461,   1490,   1426)
length(income)

occ<-c(35,
20,
21,
45,
70,
23,
30,
21,
15,
36,
41,
35,
30,
31,
51,
62,
45,
41,
62,
47,
43,
38,
36,
30,
65
)
occ

Now write the following codes to solve the questions one by one.

a) code:

cor(occ,income)

Output:

> cor(occ,income)
[1] 0.05811175

So the pearson correlation = 0.058

test of correlation:

Code:

cor.test(income,occ,alternative="greater")

Output:

Pearson's product-moment correlation

data: income and occ
t = 0.2792, df = 23, p-value = 0.3913
alternative hypothesis: true correlation is greater than 0
95 percent confidence interval:
-0.2844403 1.0000000
sample estimates:
cor
0.05811175

So from the output we see that degrees of freedom of t-statistic = 23

so we will reject H0 if t > T.INV(1-0.1,23) = 1.319

From the output the value of the test-statistic = 0.28

c) As the value of t-statistic = 0.28 < crtical value = 1.319, so at 1% level of significance we fail to reject H0 to conclude that there is no positive relationship between revenue and occupied rooms.

d) Percent of variation in the revenue in the restaurant is accounted for by the number of rooms occupied

= (pearson correlation)2 = 0.0582 = 0.004 = 0.4%

so 0.4% of variation in revenue is explained by variation in occupied rooms.

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