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Interpreting
and Curve Fitting Temperature Data
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The chart below lists the average monthly temperatures for three
cities in different parts of the world.
Part 1: Plotting the Temperatures
(1) Observe the temperatures for Washington DC and describe
their pattern over one year's time. Do the same for the other two cities.
Make several interpretive statements about the data based on your observations.
(2) Enter all of the data (including the months as 1, 2, 3...)
into a statistical list in your graphing calculator and draw a scatterplot
of the Washington DC data. Reconcile the shape of this scatterplot with your
earlier description and interpretive statements. (3) Draw scatterplots of the data for the other two cities.
Make several interpretative statements that compare and contrast the scatterplots
for all three cities. Relate these statements to the geographical locations
of the cities. (4) Discuss the advantages of a visual representation for interpretation.
Part 2: Determining and Interpreting "best fit" Curves (5) Discuss what type of curve would "best fit" the data. (6) Write the general equation of the sine function [y = Asin(B(x+C))+D], and discuss the effect of each of the coefficients on the graph of the sine function. Discuss how these coefficients can be determined from the data given above. (7) Determine a best fit sine curve for the Washington data incrementally, by determining one coefficient at a time and graphing the resulting curves. At each stage, observe the graph and compare your observations with your answers to Task 6.
Discuss the fit of this curve.
(8) Use your calculator to determine the least squares sine regression for this data. Compare your equation with the one generated by the calculator. (Note: some calculators display sine equations in the form Asin(Bx+C)+D) (9) By referring to the coefficients for the Washington equation, estimate from the scatterplot, the A coefficient for the best fit curve for the Verhoyansk data. Next, do the same for the vertical shift coefficient, the period, and the horizontal shift coefficient. Graph the resulting equation. Discuss the fit of your curve. (10) Revise your equations as needed to obtain a better fit, and discuss your revisions in relation to the scatterplot and numerical data. Graph your revised equation. (11) Use your calculator to derive a least squared regression equation for Verhoyansk, and compare this equation to your revised equation. (12) Repeat tasks 9, 10 and 11 for the Buenos Aires data. (13) Discuss the relationships between the coefficients for the best fit equations for the three cities, and the geographical locations of these cities (e.g. hemispheres, latitude). Extension 1: Oscillatory Data Discuss other natural
phenomena that have associated oscillating variables. Find two or three
sets of oscillating data and determine best fit equations for them. Relate
the coefficients of the best fit equations with physical aspects of the
phenomena. Open up the program Green Globs and play an expert game, using only sine equations. Discuss the differences between using the Asin(Bx+C)+D form and the Asin(B(x+C))+D form of the sine equation. |
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Back to Project Activities | Back to Math Homepage Send questions or comments here. Last modified on August 14, 2001. |
/12],
and graph y = Asin(Bx)+D.