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Cooling coffee: two models compared with residuals

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Q2: Modeling a Non-Periodic Context · Part A: graphing calculator allowed · 6 points

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Name: ______________________   Date: ____________   Cooling coffee: two models compared with residuals (6 points)

Table

Temperature of a cup of coffee

A cup of coffee is left on a table in a room kept at 2222 degrees Celsius. Its temperature is measured every 55 minutes. Two models are proposed: a linear model LL, found by linear regression on the data, and the model EE given by E(t)=22+70(0.952)tE(t) = 22 + 70(0.952)^t, where tt is in minutes and both models give degrees Celsius.
Temperature of the coffee, in degrees Celsius, t minutes after it is poured
tt (minutes)05101520
Temperature (degrees Celsius)92.076.565.055.348.4

  1. (A(i)) Use linear regression on the data to find L(t)=mt+cL(t) = mt + c. Give mm and cc as decimal approximations. [1 point]

  2. (A(ii)) A residual is the actual value minus the predicted value. Find the residual of LL and the residual of EE at t=10t = 10. [1 point]

  3. (B(i)) The residuals of LL at t=0,5,10,15,20t = 0, 5, 10, 15, 20 are, in order, positive, negative, negative, negative, positive. The residuals of EE are all within 0.50.5 of 00 with no pattern. Which model is more appropriate for the data? Give a reason for your answer. [1 point]

  4. (B(ii)) Use the model EE to predict the temperature of the coffee at t=30t = 30 minutes. [1 point]

  5. (C) (i) Use EE to find the time when the coffee is 4040 degrees Celsius. (ii) Explain why the model LL should not be used to predict the temperature an hour after the coffee is poured. [2 points]