Quadratic Regression (QR)

Class Members List
n = 23
Quadratic Regression (QR)
Data: On a particular day in April, 2012, the outdoor temperature was recorded at 8 times of the day, and the following table was compiled. REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.
Time of day
(hour)
x Temperature
(degrees F.)
y
7 35
9 50
11 56
13 59
14 61
17 62
20 59
23 44
REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.
The temperature is low in the morning, reaches a peak in the afternoon, and then decreases.
Tasks for Quadratic Regression Model (QR)
(QR-1) Plot the points (x, y) to obtain a scatterplot. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.
(QR-2) Find the quadratic polynomial of best fit and graph it on the scatterplot. State the formula for the quadratic polynomial.
(QR-3) Find and state the value of r2, the coefficient of determination. Discuss your findings. (r2 is calculated using a different formula than for linear regression. However, just as in the linear case, the closer r2 is to 1, the better the fit. Just work with r2, not r.) Is a parabola a good curve to fit to this data?
(QR-4) My “n” is 23. Use the quadratic polynomial to make an outdoor temperature estimate. Each class member will compute a temperature estimate for a different time of day. Look at the Class Members list, and look for the number “n” next to your name. Take your number n, multiply it by 0.5 and add to 6 to get a time x in hours; that is, compute x = 0.5n + 6. The value of x is the time you will substitute into the quadratic polynomial to make a temperature estimate. For instance, if your n is 15, then x = 0.5(15) + 6 = 13.5 hours (1:30 pm).
(QR-5) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For instance, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a degree. Show work.

Exponential Regression (ER)
Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees.
The temperature of the coffee was recorded periodically, and the following table was compiled.
Table 1:
Time Elapsed
(minutes) Coffee
Temperature
(degrees F.)
x T
0 166.0
10 140.5
20 125.2
30 110.3
40 104.5
50 98.4
60 93.9
REMARKS: Common sense tells us that the coffee will be cooling off and its temperature will decrease and approach the ambient temperature of the room, 69 degrees.
So, the temperature difference between the coffee temperature and the room temperature will decrease to 0.
We will be fitting the data to an exponential curve of the form y = A e- bx. Notice that as x gets large, y will get closer and closer to 0, which is what the temperature difference will do.
So, we want to analyze the data where x = time elapsed and y = T – 69, the temperature difference between the coffee temperature and the room temperature.
Table 2
Time Elapsed
(minutes) Temperature
Difference
(degrees F.)
x y
0 97.0
10 71.5
20 56.2
30 41.3
40 35.5
50 29.4
60 24.9
Tasks for Exponential Regression Model (ER)
(ER-1) Plot the points (x, y) in the second table (Table 2) to obtain a scatterplot. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.
(ER-2) Find the exponential function of best fit and graph it on the scatterplot. State the formula for the exponential function. It should have the form y = A e- bx where software has provided you with the numerical values for A and b.
(ER-3) Find and state the value of r2, the coefficient of determination. Discuss your findings.(r2 is calculated using a different formula than for linear regression. However, just as in the linear case, the closer r2 is to 1, the better the fit.) Is an exponential curve a good curve to fit to this data?
(ER-4) My “n” is 23 – Use the exponential function to make a coffee temperature estimate. Each class member will compute a temperature estimate for a different elapsed time. Look at the Class Members list, and look for the number “n” next to your name. Take your number n, multiply it by 6 to get an elapsed time x; that is, compute x = 6n. The value of x is the time (in minutes) you will substitute into the exponential function to make a temperature estimate. For instance, if your n is 14, then x = 6(14) = 84 minutes, and then you would substitute x = 84 minutes into your exponential function to get y, the corresponding temperature difference between the coffee temperature and the room temperature.
(ER-5) My “n” is 23. Use the exponential function together with algebra to estimate the elapsed time when the coffee arrived at a particular target temperature. Report the time to the nearest tenth of a minute. Each class member will work with a different target temperature. Look at the Class