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# Multiple Regression Analysis

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The total expenses of a hospital are related to many factors. Two of these factors are the number of beds in the hospital and the number of admissions. Data were collected on the 14 hospitals, as shown in the table:

HOSPITAL NUMBER OF BED's ADMISSSIONS (100s) TOTAL EXPENSES (MILLIONS)
1 215 77 57
2 336 160 127
3 520 230 157
4 135 43 24
5 35 9 14
6 210 155 93
7 140 53 45
8 90 6 6
9 410 159 99
10 50 18 12
11 65 16 11
12 42 29 15
13 110 28 21
14 305 98 63

-------------------------------------

ASSIGNMENT:

* State the linear equation,
* Explain the overall statistical significance of the model,
* Explain the statistical significance for each independent variable in the model
* Interpret the Adjusted R2,
* Is this a good predictive equation(s)? Which variables should be excluded (if any) and why? Explain.

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#### Solution Preview

We're going to look at the relationship between the number of beds in the hospital, the number of admissions, and the total expenses.
The independent variables are beds and admissions. The dependent variable is expenses.
I used Excel to do a linear regression.

State the linear equation,

The linear equation is expenses = 0.023*beds + 0.623*admissions - 0.654.

Explain ...

\$2.19

## Multiple Regression Analysis, Time Series Analysis

See attached data file.

Background:

One day, after reporting the performance of the company to the shareholders, the CEO of A. Fictitious & Co. decided that he would like to quantify the impact of the company's expenditures has on how much sales it generates. In other words, he would like to know if the company increases the amount spent on marketing by one dollar, how large of an increase (or decrease) in sales would be expected? The major categories of expenditures and how much was spent in each are known for the company, as is the total sales generated per quarter for the last five years. He has the data file with the relevant data sent to you, and asks you to do the multiple-regression analysis to find out the answer to his questions. Oh, and he also asks you to do a time-series analysis on the total sales per quarter and forecast the amount of sales expected in the future.

Part I. Multiple Regression:

1. Look over the expenditure categories that the CEO gave you. Check to see if there interaction between the category Capital Equipment and Materials, and the category Salary and Benefits. Namely, do a multiple regression model with quarterly sales as the y-variable and the four expenditure categories given in the Data set as the x-variables. Do a second multiple regression model with four expenditure categories plus an interaction term between the category Capital Equipment and Materials, and the category Salary and Benefits. After comparing the two models, which is the better model? Can you conclude whether the two categories are independent of each other?

2. Based on your analysis in Question 1, write down the best-fit multiple regression equation for this problem with quarterly sales as the y-variable and the expenditure categories as the x-variables; do not forget the interaction term if there is one. Define each variable in the equation.

3. Answer the CEO's question. Namely, for each the four categories of expenditure (marketing, R&D, equipment and supplies, and salaries and benefits), if the CEO increases spending in one category by one million dollars (holding the others fixed), how much increase in sales should he expect? If you found an interaction term, explain its effects as well.

4. Being a very cautious person, you decide to also give the CEO the confidence intervals for the rates of increase your calculated in Question 3. Calculate the 95% confidence intervals for the slopes you calculated in Question 2, including the interaction term if you found one.

5. Being even more cautious - you are reporting the results to the CEO, after all - you decide to do a residual error analysis by applying the F-Test on the entire regression model. Do so, and interpret the results.

Part II. Time-series Analysis

The CEO noticed that he has five years of quarterly sales data in hand, and they form a time series. He decided to also ask you to perform time-series analysis on it, and use it to forecast what future sales are expected to be at the end of 1Q 2009.

6. Plot the quarterly sales as a function of time in your Excel data spreadsheet. From the shape of these graphs, and any analysis that you think is needed, determine what type of trend model is best suitable for this data. Write down the equation for the trend model, and define and explain each of the variables as it applies to this problem.

7. Do a regression analysis on the data for the trend model you decided on in Question 6, and determine the parameters for the model.

8. Answer the CEO's question. Tell him how much sales are expected to be at the end of 1Q 2009. Be careful, and also include the 95% confidence interval for this number.

Part III. Conclusions

9. Write a report to the CEO of your findings. Which expenditure has the largest impact on sales, and which one has the least impact on sales? How fast do you expect sales to increase in time?

10. When tasked to do this analysis, the CEO made a number of assumptions about the data, and what can be extrapolated from it. List down the assumptions he made, and criticize each assumption. Criticize also the conclusions that you drew from your analysis for your report to the CEO. As a starting point, remember that the data covered a period of five years.

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