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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.

| Day | Revenue | Occupied | Day | Revenue | Occupied |
|-----|---------|----------|-----|---------|----------|
| 1 | $1,452 | 35 | 14 | $1,425 | 31 |
| 2 | 1,361 | 20 | 15 | 1,445 | 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 | 1,459 | 36 | 23 | 1,461 | 36 |
| 11 | 1,399 | 41 | 24 | 1,490 | 30 |
| 12 | 1,458 | 35 | 25 | 1,426 | 65 |
| 13 | 1,537 | 30 |

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

2. State the decision rule for a 0.02 significance level:

- \( H_0: \rho \leq 0 \)
- \( H_1: \rho > 0 \)

(Round your answer to 3 decimal places.)

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

4. Is it reasonable to conclude that there is a positive relationship between revenue and occupied rooms? Use the 0.02 significance level.

5. 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 :

To determine if a positive relationship exists between number of rooms occupied and daily restaurant revenue, run a linear regression analysis. The p-value received should be less than 0.02 to reject the null hypothesis. The square of the correlation coefficient, the coefficient of determination, will show the percentage of the revenue variation explained by the number of rooms occupied.

To answer your question, you want to test the hypothesis and find out if there is a significant positive relationship between the number of rooms occupied and daily revenue from the restaurant, using the 0.02 significance level. Unfortunately, we cannot calculate the correlation coefficient and run the exact hypothesis tests without an available data set. However, you can proceed by running a linear regression analysis using a software or a scientific calculator, which would provide you with your output like p-value and correlation coefficient.

In your case, the null hypothesis (H0) is that the correlation coefficient (rho) is less than or equal to zero, implying there's no positive correlation between the variables. Thus, the alternative hypothesis (H1) is that rho is greater than zero, which suggests a positive relationship between the variables.

You'll reject H0 if the p-value you get is less than the significance level of 0.02. This means there is enough evidence to suggest a positive correlation between the variables, hence a strong relationship between the number of rooms occupied and restaurant revenue. Else, if the p-value is greater than or equal to 0.02, fail to reject the null hypothesis.

The percentage of variation in revenue that can be explained by the number of rooms occupied can be found by squaring the correlation coefficient, known as the coefficient of determination (r²).

Learn more about coefficient here:

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