Question

A suburban hotel derives its revenue 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.

Income	Occupied
1452	30
1361	31
1426	32
1470	32
1456	30
1430	29
1354	31
1442	32
1394	33
1459	33
1399	30
1458	33
1537	32
1425	32
1445	30
1439	33
1348	31
1450	32
1431	30
1446	32
1485	30
1405	29
1461	31
1490	33
1426	30

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

c-1. State the decision rule for 0.025 significance level: H0: ρ ≤ 0; H1: ρ > 0. (Round your answer to 3 decimal places.)

c-2Compute the value of the test statistic. (Round your answer to 2 decimal places.)

c-3. Is it reasonable to conclude that there is a positive relationship between revenue and occupied rooms? Use the 0.025 significance level.

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.)

Answer 1

Based on the given data,

Assuming a linear relationship between the variables Income (Yi) and Occupied (Xi), the pearson's correlation coefficient r can be obtained as:

where,

Substituting the values,

We get, r = 0.224

c-1. To test: H0: ρ ≤ 0; H1: ρ > 0

c-2. Test statistic:

with critical region

Substituting the values:

= 1.102

c-3. The p-value of the test can easily be obtained using the excel function:

We get p-value = 0.149. Since the p-value 0.141>0.025, we fail to Reject H0 at 2.5% level. We may conclude that the data does not provide sufficient evidence to support the claim that there exists a significant positive correlation between the variables Income and Occupied.

Percent of the variation in revenue in the restaurant, accounted for by the number of rooms occupied is called the Coefficient of Determination which is nothing but the square of the correlation coefficient:

This figure imlies that the number of rooms occupied accounts for 10% of the variation in revenue in the restaurant.