AP Calculus 2007

Mrs. Powell and Ms. Sheehan

For this project you will…

• Investigate a data set from the internet about a topic of your choice

• Your data must have two quantitative variables for you to analyze - categorical variables can be used for extra credit

• Find and discuss two mathematical models that describe your data

• Create a poster/paper or PowerPoint presentation to present your findings

Include the following on Your PowerPoint/Poster

• Graphs – 2 different models– Title your Graphs– Label your x-axes with your

independent variable name– Label your y-axes with your dependent variable name– Clearly mark the scale on your graphs– Scatter plots of your data points– Use your regression tools to find two equations (your

mathematical models) and a graph the models on the same graphs as your scatter plots (You can put each model on one scatter plot or two separate scatter plots)

– Find/write the correlation coefficients of your models, if available (The square root of R2 should equal the correlation coefficient)

On Your Poster/Presentation

Data table(s)– Your data pairs (at least fifteen)– The value predicted by your models for each value of

the independent/explanatory variable– The error between the model values and the true

value (model value – real value for each data pair) (for each model)

Sample Data Table

2644 248500 324,962.80 76,462.802649 349900 325,456.30 24,443.702681 369900 328,614.70 41,285.302690 362900 329,503.00 33,397.002722 335900 332,661.40 3,238.602825 360000 342,827.50 17,172.502916 314799 351,809.20 37,010.202928 349900 352,993.60 3,093.602940 346000 354,178.00 8,178.002997 342500 359,803.90 17,303.903008 320000 360,889.60 40,889.603069 359900 366,910.30 7,010.303074 294900 367,403.80 72,503.80

Cost of Houses

: 98.7 64000Mathematical Model y x

cost

120000

140000

160000

180000

200000

220000

240000

260000

280000

300000

320000

340000

360000

380000

square_feet1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200

cost = 98.7square_feet + 64000; r^2 = 0.49

Sample Graph Using FathomHousing Costs

: 98.7 64000Mathematical Model y x

Sample Data Table

sq ft cost per foot Model Value Error1069 158900 174,137.41 15,237.411145 144900 179,512.48 34,612.481200 129900 183,505.51 53,605.511236 152900 186,167.11 33,267.111300 209900 190,994.52 18,905.481394 154900 198,312.63 43,412.631533 154900 209,651.10 54,751.101554 199900 211,419.59 11,519.591560 249500 211,927.60 37,572.401611 267000 216,295.33 50,704.671685 219900 222,793.36 2,893.361700 174900 224,134.14 49,234.14

Cost of Houses0.0004 11.64 : xMathematical Model y e

Sample Graph Using Graphmatica

0.0004 11.64 : xMathematical Model y e

Housing Costs

Cost of house

Number of Square Feet

Information to include

• Why would this be interesting?

• Who would/could use this information?

• Analyze your data– Directional pattern– Model form– Strength of relationship between variables

(correlation coefficient)– Continuous or discrete?

Information to include

• For each model, – Over what domain values does your model make

sense?– Does it make sense to use the function that you

modeled your data with? Why or why not?– What does the model say about your sample data?– What do the coefficients in your model tell you?– What interesting information does your data reveal?

• Which of your two models fit your data the best?

• Bibliography!!

Why would this be interesting and to who?

• People who are looking to buy a house or to build a house

• Realtors

• Graph shows the relationship between cost and square footage – gives an estimate of cost per square foot

Data Analysis:

• Positively associated• Linear or exponential form• Fairly strong relationship between

variables, both models fit reasonably well• Our data is continuous, because it is

square feet measurements are not discrete – there are infinitely many possible measures between x-values and interpolation is meaningful

Analysis – Linear Model

• The domain must be greater than or equal to 0 because square footage cannot be negative and it causes no problems in the function.

• The model makes sense because if a house has zero square feet there is still a cost for the lot and in many housing markets the price is based on the number of square feet.

• The model shows that as the square footage of a house increases, the cost of the house increases.

• What do the coefficients represent?– Y-intercept: price of lot– Slope: Price per square foot

• Interesting information:– Positive correlation between cost and square footage.– Each additional square foot costs approximately $100.

f x = x+1

: 98.7 64000Mathematical Model y x

Analysis – Exponential Model

• The domain must be greater than or equal to 0 because square footage cannot be negative and it causes no problems in the function.

• The model appears to fit the data well visually. However, the y-intercept is approximately $113,500, which seems very high as the lot price, since some houses cost only slightly more than this amount.

• The model shows that as the square footage of a house increases, the cost of the house increases.

• What do the coefficients represent?– Y-intercept: the price of the lot (e11.64)– Rate of change: e.0004 (approximately 1.0004)

• Interesting information:– Positive correlation between cost and square footage.– Predicts high cost for an empty lots and lower square footage cost

for smaller houses.

0.0004 11.64 11.64 .0004 xxy e e e

Which model fit your data and the real-life situation best?

• The exponential model seemed to fit better, but upon further analysis it did not make sense in terms of the real-life situation – an empty lot probably would not cost $113,550, when a house built on a lot could cost as little as $100,000.

• Thus the linear model better fits our data because it presents a more realistic lot price and price per square foot.

Bibliography• Bloomington Normal Association of

Realtors. 2007. 27 August 2007. <http:// www.bnrealtors.com/>.

• Fathom

• Graphmatica

• Excel