2

  Ingredients of Change: Non-Linear Models


Project 2.1: Compulsory School Laws

 

Setting

In 1852, Massachusetts became the first state to enact a compulsory school attendance law. Sixty-six years later, in 1918, Mississippi became the last state to enact a compulsory attendance law. Table 2.72 lists the first 48 states to enact such laws and the year each state enacted its first compulsory school law.

TABLE 2.72

Source: Richardson, "Variation in Date of Enactment of Compulsory School Attendance Laws," Sociology of Education vol. 53 (July 1980), pp. 153&endash;163.

6. Examine scatter plots for the two data sets in Task 5. Do you believe that logistic models are appropriate for these data sets? Explain.

7. Find logistic models for each set of data in Task 5. Comment on how well each model fits the data.

8. It appears that the northern and western states were slow to follow the lead established by Massachusetts and New York. What historical event may have been responsible for the time lag?

9. One way to reduce the impact of unusual behavior in a data set (such as that discussed in Task 8) is to group the data in a different way. Tabulate the cumulative northern and western state totals for the following 10-year periods:

1852-1861

1862-1871

1872-1881

1882-1891

1892-1901

1902-1911

10. Fit a logistic curve to the data in Task 9. Comment on how well the model fits the data. Compare models for the data grouped in 10-year intervals and the data grouped in 5-year intervals (Task 6). Does grouping the data differently significantly affect how well the model fits? Explain.

11. Find a model that fits the data for the southern states better than the logistic model. Explain your reasoning.


Tasks

1. Tabulate the cumulative number of states with compulsory school laws for the following 5-year periods:

1852-1856

1887-1891

1857-1861

1892-1896

1862-1866

1897-1901

1867-1871

1902-1906

1872-1876

1907-1911

1877-1881

1912-1916

1882-1886

1917-1921

2. Examine a scatter plot of the data in Task 1. Do you believe a logistic model is appropriate? Explain.

3. Fit a logistic model to the data in Task 1.

4. What do most states in the third column of the original data have in common? Why would these states be the last to enact compulsory education laws?

5. The seventeen states considered to be southern states (below the Mason-Dixon Line) are AL, AR, DE, FL, GA, KY, LA, MD, MO, MS, NC, OK, SC, TN, TX, VA, and WV. Tabulate cumulative totals for the southern states and the northern/western states for these dates:

 

Northern/Western States

Southern States

Years

Years

1852-1856

1891-1895

1857-1861

1896-1900

1862-1866

1901-1905

1867-1871

1906-1910

1872-1876

1911-1915

1877-1881

1916-1920

1882-1886

1887-1891

1892-1896

1897-1901

1902-1906


Reporting

1. Prepare a written report of your work. Include scatter plots, models, and graphs. Include discussions of each of the tasks in this project.

2. (Optional) Prepare a brief (15-minute) presentation on your work.

 

Project 2.2: Fund-Raising Campaign

 

Setting

In order to raise funds, the mathematics department in your college or university is planning to sell T-shirts before next year's football game against the school's biggest rival. Your team has volunteered to conduct the fund raiser. Because several other student groups have also volunteered to head this project, your team is to present its proposal for the fund drive, as well as your predictions about its outcome, to a panel of mathematics faculty.

Note: This project is also used as a portion of Project 5.2 on page 319.>>

Create equations for revenue, total cost, and profit as functions of price. (Revenue, total cost, and profit may not be one of the basic models that were discussed in class. They are sums and/or products of the demand function with other functions.)

 

Tasks

1. Getting Started Develop a slogan and a design for the T-shirt. Keep in mind that good taste is a concern. Decide on a target market, and determine a strategy to survey (at random) at least 100 students who represent a cross section of the target market to determine the demand for T-shirts (as a function of price) within that market. It is important that your sample survey group properly represent your target market. If, for example, you polled only near campus dining facilities at lunch time, your sample would be biased toward students who eat lunch at such facilities.

The question you should ask is "How much would you be willing to spend on a T-shirt promoting the big football rivalry? $14, $13, $12, $11, $10, $9, $8, $7, $6, or not interested?" Keep an accurate tally of the number of students who answer in each category. In your report on the results of your poll, you should include information such as your target market; where, when, and how you polled within that market; and why you believe that your polled sample is likely to be a representative cross section of the market.

2. Modeling the Demand Function

a. From the data you have gathered, determine how many students from your sample survey group would buy a T-shirt at $6, $7, $8, and so on.

b. Devise a marketing strategy, and determine how many students within your target market you can reasonably expect to reach. Assuming that your poll is an accurate indicator of your target population, determine the number of students from your target market who will buy a T-shirt at each of the given prices.

c. Taking into account the results of your poll and your projected target market, develop a model for demand as a function of price. Keep in mind that your model must make sense for all possible input values.

3. Modeling Other Functions Estimate the cost that you will incur per T-shirt from the partial price listing in Table 2.73. Use the demand function from Task 2 to create equations for revenue, total cost, and profit as functions of price. (Revenue, total cost, and profit may be one of the basic models that were discussed in class. They are sums and/or products of the demand function with other functions.)


Reporting

1. Prepare a written report summarizing your survey and modeling. The report should include your slogan and design, your target market, your marketing strategy, the results from your poll (as well as the specifics of how you conducted your poll), a discussion of how and why you chose the model of the demand function, a discussion of the accuracy of your demand model, and graphs and equations for all of your models. Attach your questionnaire and data to the report as an appendix.

2. Prepare a 15-minute oral presentation of your survey, modeling, and marketing strategy to be delivered before a panel of mathematics faculty. You will be expected to have overhead transparencies of all graphs and equations as well as any other information that you consider appropriate as a visual aid. Remember that you are trying to sell the mathematics department on your campaign idea.


TABLE 2.73 The T-Shirt Company: Partial Price Listing

Source: Based on data compiled from 1993 prices at Tigertown Graphics, Inc., Clemson, SC.

 






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