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 R 2 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 Cost of Houses

Sample Graph Using Fathom Housing Costs

Sample Data Table Cost of Houses

Sample Graph Using Graphmatica 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.

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 (e ) –Rate of change: e.0004 (approximately ) Interesting information: –Positive correlation between cost and square footage. –Predicts high cost for an empty lots and lower square footage cost for smaller houses.

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 August Fathom Graphmatica Excel