91Ó°ÊÓ

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. $$ \begin{array}{|rrr|rrr|} \hline \text { Day } & \text { Revenue } & \text { Occupied } & \text { Day } & \text { Revenue } & \text { Occupied } \\ \hline 1 & \$ 1,452 & 23 & 14 & \$ 1,425 & 27 \\ 2 & 1,361 & 47 & 15 & 1,445 & 34 \\ 3 & 1,426 & 21 & 16 & 1,439 & 15 \\ 4 & 1,470 & 39 & 17 & 1,348 & 19 \\ 5 & 1,456 & 37 & 18 & 1,450 & 38 \\ 6 & 1,430 & 29 & 19 & 1,431 & 44 \\ 7 & 1,354 & 23 & 20 & 1,446 & 47 \\ 8 & 1,442 & 44 & 21 & 1,485 & 43 \\ 9 & 1,394 & 45 & 22 & 1,405 & 38 \\ 10 & 1,459 & 16 & 23 & 1,461 & 51 \\ 11 & 1,399 & 30 & 24 & 1,490 & 61 \\ 12 & 1,458 & 42 & 25 & 1,426 & 39 \\ 13 & 1,537 & 54 & & & \\ \hline \end{array} $$ Use a statistical software package to answer the following questions. a. Does the revenue seem to increase as the number of occupied rooms increases? Draw a scatter diagram to support your conclusion. b. Determine the correlation coefficient between the two variables. Interpret the value. c. Is it reasonable to conclude that there is a positive relationship between revenue and occupied rooms? Use the .10 significance level. d. What percent of the variation in revenue in the restaurant is accounted for by the number of rooms occupied?

Step by step solution

01

Organize the Data

List the number of occupied rooms and corresponding restaurant revenue from the dataset. This will help in plotting and calculating correlation. The dataset given already provides this for 25 days.

02

Create a Scatter Plot

Plot the number of occupied rooms on the x-axis and the restaurant revenue on the y-axis. Each point on the scatter diagram represents a day's revenue and the number of occupied rooms. Observe the pattern; a positive slope suggests revenue increases with more occupied rooms.

03

Calculate the Correlation Coefficient

Use statistical software or a calculator to compute the correlation coefficient (r) using the formula: \[ r = \frac{N(\sum xy) - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}} \] where \(x\) is the number of occupied rooms and \(y\) is the revenue. This measurement indicates the strength and direction of the linear relationship.

04

Interpret the Correlation Coefficient

A correlation coefficient close to +1 indicates a strong positive linear relationship. If \(r > 0.5\), it suggests revenue tends to increase as more rooms are occupied.

05

Conduct a Hypothesis Test for Correlation Significance

Apply the hypothesis testing steps to verify if the correlation is significant at the 0.10 level. Null Hypothesis \(H_0: \rho = 0\) (no correlation), Alternative Hypothesis \(H_a: \rho > 0\) (positive correlation exists). Use a t-test: \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \] Compare the calculated t-value with the critical t-value from the t-distribution table for \(n-2\) degrees of freedom. If the calculated t is greater than the table value, reject \(H_0\).

06

Calculate the Coefficient of Determination

Square the correlation coefficient to get the coefficient of determination \(r^2\). This represents the proportion of total variation in revenue that can be explained by the variation in the number of occupied rooms.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Diagram

A scatter diagram is an essential tool to visually examine the relationship between two variables, such as revenue from a restaurant and the number of rooms occupied in a hotel. In this context, we plot each day as a point on a graph, with the number of rooms on the x-axis and the restaurant revenue on the y-axis.
A scatter diagram helps to identify any visible patterns, such as a general upward slope that would suggest revenue increases as more rooms are occupied.
To draw the diagram:

• Place each point based on the day's rooms and revenue.
• Look for trends; a positive pattern indicates a direct relationship.
• Note any outliers, which may skew results or suggest unusual observations.

The scatter diagram acts as a preliminary step in visually assessing the data before diving into more complex calculations, such as calculating correlation coefficients.

Correlation Coefficient

The correlation coefficient, often denoted as \(r\), is a numerical measure that quantifies the strength and direction of a linear relationship between two variables. In our example, this coefficient helps determine how closely the number of rooms occupied relates to the restaurant revenue.
To compute \(r\), you can use statistical formulas or software. The formula is:\[r = \frac{N(\sum xy) - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}}\]Here, \(x\) represents the number of rooms occupied, and \(y\) is the restaurant revenue. The result is a value between -1 and +1.

• If \(r\) is close to +1, there is a strong positive correlation, meaning revenue tends to increase as more rooms are occupied.
• If \(r\) is close to -1, there's a strong negative correlation, indicating that revenue decreases as more rooms are occupied.
• If \(r\) is around 0, it suggests no linear relationship between the variables.

Understanding the correlation coefficient is critical for interpreting the strength of the observed relationship.

Coefficient of Determination

The coefficient of determination, represented as \(r^2\), offers insight into how well the variation in one variable explains the variation in another. In our hotel revenue and room occupancy example, it tells us what percentage of the revenue changes can be attributed to the changes in the number of rooms occupied.
To find \(r^2\), simply square the correlation coefficient \(r\):\[r^2 = (r)^2\]The value of \(r^2\) ranges from 0 to 1.

• An \(r^2\) close to 1 indicates a high explanatory power – most of the revenue variation is explained by room occupancy.
• An \(r^2\) close to 0 suggests that the room occupancy explains very little of the revenue variation, indicating other factors might be influencing the restaurant income.

By determining \(r^2\), we gain a focused view on the strength of the relationship and how predictive it might be for future scenarios.

Hypothesis Testing for Correlation

Hypothesis testing for correlation is used to assess whether the observed relationship between two variables is statistically significant. In our case, we want to decide if the positive relationship between room occupancy and restaurant revenue is not just random.
We start with:

• Null Hypothesis \(H_0\): \(\rho = 0\), indicating no correlation.
• Alternative Hypothesis \(H_a\): \(\rho > 0\), a positive correlation exists.

Using a t-test helps us determine whether to reject \(H_0\). The test statistic is calculated as:\[t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}\]Where \(n\) is the number of observations. Compare this \(t\)-value to the critical \(t\)-value from the t-distribution table for \(n-2\) degrees of freedom.

• If the calculated \(t\)-value is greater than the critical value, we reject \(H_0\), supporting a statistically significant positive correlation.
• A smaller \(t\)-value suggests we cannot reject \(H_0\), meaning the correlation might be due to chance.

This analysis ensures the observed relationship is not just a coincidence but has a basis in statistical significance.