Part - 1:
June Bugg Electronics, Inc., manufactures and distributes the BuggOff, a solar powered insect repellent device. June Bugg was founded in 1978, and since that time it has enjoyed fairly consistent increases in sales revenue. The firm's sales manager, Cathy Kollogee, feels that more spending on advertising campaigns that emphasize the non-polluting nature of the BuggOff would cause revenues to grow much more rapidly. The firm's owner, June Bugg, is skeptical. She thinks that a better understanding of the relationship between advertising expenditures and sales revenue is necessary before the advertising budget is increased, so she comes to you for help.
The table below lists annual observations (OBS) over the period from 1981 to 1991 on BuggOff sales (Y) in millions of dollars per year and average advertising expenditures (X) in thousands of dollars per month. Use this information to solve Problems 1 through 3.
1. Plot the points that correspond to these observations on the graph that follows. Next, use a ruler to draw an approximate line of best fit on the graph. That is, draw a line that appears (to your eye) to fit the points well enough to be an adequate description of the relationship between the two variables. Label this line VISUAL.
Choose two points on your "visual" line of best fit and use the points to calculate the intercept and slope of the line.
2. Use OLS to calculate the intercept, the slope, the coefficient of determination, and the standard error of the slope.
Plot the OLS regression line on the graph from Problem 1 and label it OLS1. How does this line differ from your visual line of best fit? Evaluate the fit of this line using the calculated values.
After noticing that observation 5 differs substantially from the other observations, you question the sales manager about events that are related to this observation. She explains that late in 1984 a competing firm, the Lingering Death Insecticide Company, used advertisements in which the BuggOff was represented as a danger to children and pets. June Bugg sales revenues fell substantially in 1985, but the firm responded with a major advertising campaign that was effective in returning things to normal by the end of the year.
3. Apply OLS to the 10 observations that remain when observation 5 is omitted from the data set. Plot this second OLS regression line on the graph with OLS1 and VISUAL and label it OLS2. Evaluate the fit of this line and compare it with that of OLS1.
Part - 2:
Tom Acadvipski operates the Mucho Macho Hair Restoration Clinic in Cranium, Arizona. His son, Ollie, recently completed a course in managerial economics and is eager to apply his skills to the family business. Mucho Macho is one of many clinics around the country that are franchised by Hairball International, so Ollie contacts the regional Hairball representative and asks for cross-sectional data on the demand for hair restoration services. After some persuasion, the representative finally coughs up the Hairball data. Ollie collects some additional data for each of the 32 clinic areas and then uses a computer to calculate regression results from the 32 observations.
The variables employed in the analysis are listed below with their average values in parenthesis:
Q = Number of clients per week (269)
I = Average annual income in $1,000s (23)
M = Single men over 35 in 1,000s (292)
P = Price per restoration treatment (584)
W = Single women over 21 in 1,000s (387)
B = Number of singles bars (33)
A = Advertising expenditures in $1,000s per month (4)
D = Divorce rate in percent (45)
The regression results, with t-ratios in parenthesis, are listed below:
SE of regression 37.62
R2
0.91
Q = 187.27 + 2.46I + 0.62M - 0.27P - 0.14W + 1.02B - 3.27A + 0.74D (1.22) (10.33) (-6.89) (-3.60) (1.52) (-0.77) (0.84)
Use these results to solve Problems 4 through 7.
4. Calculate the adjusted coefficient of determination and explain its meaning.
5. Test the overall explanatory power of the regression equation at a 5% level of significance. Explain, in words, what your test result implies.
6. Test each of the slope parameters at a 5% level of significance. Explain, in words, what the results of each of your tests imply. Should any of the independent variables be eliminated from the equation? Which ones?
7. Use the mean values of the variables and the regression results to calculate the income elasticity of demand and the price elasticity of demand. Comment on these elasticities.
After examining the regression results shown above, Ollie decides to re-estimate the equation using only the significant independent variables. The results, with t-ratios in parentheses, are listed below. Use these results to solve Problems 8 through 12.
SE of regression 40.02
R2 0.88
Q = 294.79 + 0.62M - 0.28P - 0.11W
(10.72) (-7.84) (-3.27)
8. Calculate the adjusted coefficient of determination and compare it to the value obtained in Problem 4.
9. Test the regression equation at a 5% level of significance. Explain, in words, what your test result implies.
10. Test each of the slope parameters at a 5% level of significance. Explain, in words, what the results of each of your tests imply.
11. Use the mean values of the variables and the regression results to calculate the price elasticity of demand. Comment on this elasticity. What general advice would you give to Hairball International regarding the prices charged for services at their franchised clinics? Assuming that Hairball International receives a fixed percentage of the total revenue earned by their franchised clinics, what average price per treatment should they recommend?
12. The Cranium area has a population that includes about 110,000 single men over 35 and 170,000 single women over 21. Calculate the demand function for hair restoration in Cranium. Also calculate the corresponding marginal revenue function. Assume that the marginal cost per client per week for Mucho Macho is $250 and calculate marginal cost. Determine what price Mucho Macho should charge in order to maximize profit, how many treatments per week will be sold at that price, and what total revenue per week will be. Also calculate 95% confidence intervals on the number of clients per week and on total revenue per week.
Part - 3:
Ulysses Travel Consulting Services is a firm that specializes in organizing every aspect of unique cruise vacations for discriminating travelers who are seeking adventurous odysseys. The owner and general manager, Helena Troy, has operated Ulysses Travel for over twenty years. In that time, as she likes to say, she has launched a thousand ships. She charges a flat fee per person to organize every part of a vacation. Her fee does not include any of the direct costs of travel. The personal qualities that she brings to her job have inspired a remarkable degree of client loyalty. Many of her patrons come to her every year for travel advice and new clients are generally the result of their recommendations.
Her pricing decisions have always been based on an intuitive understanding of the travel market, but recently Helena has started to wonder if a more quantitative understanding of the demand for her services might help her to end up with a better bottom line. As a consequence, she goes through her records and pulls out the consulting price (PC) in $100s and the number of clients (QC) for each of the past twenty years and uses these data to estimate a demand function. The results of her regression calculations, with the tratios in parenthesis, follow below. Use them to solve Problems 13 through 15.
SE of regression 206.00
R2 0.03
Durbin Watson 0.41
QC = 475.65 - 8.47PC (-0.80)
13. Comment on the regression results obtained by Helena Troy. Identify any problems you see and suggest possible solutions.
Helena Troy is perplexed by the results of her calculations. "I'm sure that the price I charge has some effect on the number of clients I get" she says. When she is asked whether anything else could influence the demand for her services, she suggests that the average income of her clients might have some effect. Consequently, she goes back into her records and, based on her knowledge of her clients, she estimates the average annual income (I) in $1,000s of her clients during each of the past twenty years. She then runs the regression again with both variables in the equation. Her results are shown below.
SE of regression 68.82
R2 0.90
Durbin Watson 1.92
QC = 231.30 - 19.08PC + 4.04I (-5.23) (12.01)
14. Compare these regression results with those obtained previously and comment on any differences between the two.
15. The average values of QC, PC and I are 382.5, 11, and 89.4, respectively. Calculate the price elasticity and income elasticity of demand for Helena Troy's services based on these twenty-year averages. Comment on these elasticities. During the past year, PC and I were equal to 11 and 174, respectively. Calculate the price elasticity of demand under the current conditions and comment on the price that Helena is currently charging. If the marginal cost per client is about $300, what price should Helena charge?