Homework 6

Linear models

For questions 3 and 4, save your Excel files after you have added the xy scatter chart and added the trendline.  Homework 6 will use the charts and trendlines, so saving them now will save you time over the weekend. 

1.    For each of these  described setting, give a formula for the corresponding linear model and use the formula to answer the question. 

a.     A helicopter takes off from the roof of a building that is 200 feet about the ground.  The altitude of the helicopter increases by 150 feet each minute.  Express the height of the helicopter as a function of minutes after it starts rising.  How high will the helicopter be after 6 minutes of flight?

Y = 200 + 150X

Y = 150x + 200

b. A car rental agency charges $29.00 plus 6 cents per mile for a one day rental.   Express the cost of renting the car as a function of miles driven.  How much will you spend to drive the car 200 miles in one day?

Y = $29.00 + .06x

Y = .06x + $29.00

2.      Stores such as Sam's Club and Price-Costco charge an annual membership fee of approximately $40.  As a member you can purchase products at lower prices, but typically in larger quantities than one buys at ordinary stores. Savings vary from product to product, but for this question, let us assume you can save $20 for every $100 dollars you spend at one of these stores compared to shopping elsewhere.

a.  Suppose you spend $100 dollars in one year at such a store.  Do you come out ahead? How much do you save (or lose?) compared to not having membership and paying full price?

Net loss of $20.   Loss of $40 for membership plus $20 savings on $100 spent. 

b.  Create a table in Excel showing the amount you spend in the first column with the corresponding overall savings in the second column.  In the spend column, start with zero dollars and go up to $600 in increments of $100.  Paste the completed table in your Word document.

Spending	Savings
0.00	0.00
100.00	20.00
200.00	40.00
300.00	60.00
400.00	80.00
500.00	100.00
600.00	120.00

 

c.  How much do you have to spend at the store for it to be worthwhile for you to buy membership?
 

$200

d. The amount you save (or lose) by buying membership is a linear function of how much you spend.  Explain why.

Because the absolute amount saved rises by a constant amount as the amount spent increases. 

e.  What are the two variables in this function?  Assign each variable a letter. Write the linear equation for your savings as a function of how much you spend at the store.

Amount spent = X

Amount saved = Y

Y = .2X - 40

f.  From the analysis above, one of the factors one should consider when deciding whether or not to join a store such Sam's Club or Price-Costco is whether or not you are going to going to spend a certain minimal amount at the store in a year. What other factors should you consider when deciding whether to become a member of Sam's Club or Price-Costco?

There are many possible factors depending on the individual values and preferences of the person answering the question.  In purely economic terms, one should consider the costs of travel to get to one of those places, compared to closer stores, both the direct costs (gas, maintenance) and indirect costs (value of time spent driving).

3.      Open the file USTobacco_MostRecent.xls which contains data on the total amount of tobacco produced in the US.

a. Make an XY scatter plot of the data.  Add a linear trendline to your plot, including the equation and the R-squared value. [Note: We have not yet covered R-squared in class, so I do not expect you to understand what it means.  It will be explained Thursday.]  Include it in your document.

b. Use the equation method and extended trendline method to predict the amount of tobacco produced in 2012.  Show both results in your Word document.

.361

c. How much confidence do you have in your prediction? Why?  (Write a short paragraph using the language we learned in class.)

I would have considerable confidence, as this is only a 4 year projection and I don't see what would cause the long-term pattern to change dramatically in the short run.  The prediction most likely will not be exactly correct; there is fluctuation around the trendline, indicating that other short term factors are at work.  But the amount should be close to the prediction. 

d. Save the file with the chart. 

4.      Open the file Smoking_MostRecent.xls which contains data on the percent of the US adult population that smokes cigarettes.

a. Make X,Y scatter graph of the years and the total percentages (not male and female) and add a trendline.  Include the equation and R-squared value for the trendline.  Paste the resulting chart into your Word document.

b. Using your model (equation method) and the extended trendline method, predict the percentage of the total population that smokes in 2010.  Show both results in your Word document.

Y = 898.94 - .438* 2010

Y = 18.8 . . . 18.6 percent of the population will be smokers

c.  How much faith do you have in your prediction?  (Write a short paragraph using the language we learned in class.)

I would be fairly confident of the results.  The forecast is only 4 years past the domain of the data.  Smoking behavior is fairly stable, so dramatic short term changes are unlikely and longer term changes would not have much effect between 2006 and 2010. 

d. Use your model to estimate what percentage of the total population smoked in 1953.  Do you have faith in this prediction?  Explain.

Y = 898.94 - .438* 1953

43.5%

I would not have much confidence in this prediction.  Between 1953 and 1983, important things had changed, such as the scientific determination that smoking is bad for one's health, causing, among other things, cancers and heart disease.  I expect that smoking was much more common than 43.5% in 1953 and the rate of decline in some of the period before 1983 was much faster (that is, the slope was larger in a negative direction).

e. Use your model to estimate when 100% of the US population smoked.  How much faith do you have in this prediction?  Explain.

100 = 898.94 - .438X

100-898.94 = -.438X

(100-989.94)/-.438 = -X

(100-.989.94)/.438 = 1824

I do not believe this "forecast" for one second.  First of all, it never happens that 100% of a group has a given characteristic.  Second, too many other things have changed in the 150 since that year.  One would be women's liberation.  Back then, women would not have been allowed to smoke.  Another is the general increase in national wealth.  Back then, too many people would have been too poor to smoke. 

f. Save your file with the chart. 

5.      Read the article "Obesity Study Looks Thin: Results Warn That Everyone in US Will Be Overweight by the Year 2048," by Carl Bialik (Wall Street Journal, Aug. 15, 2008, A11) and respond to the following questions.

a. In a brief paragraph, summarize the main point of the article.

Projections of the percentage of the population who will be obese people is quite dubious if not totally bogus. 

b. In a brief paragraph, explain how the article relates the material on linear modeling that we have studied, giving at least one specific example from class.

It makes exactly the same point about projections far into the future, as we considered with regard to the number of motor vehicles and  consumption of coal.