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Quadratic Regression (QR)

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Question;Data: On a particular day in April;the outdoor temperature was recorded at 8 times of the day, and the following;table was compiled.;Time of day;(hour);x;Temperature;(degrees F.);y;6 47;8 62;10 68;12 71;13 73;16 74;19 71;22 56;REMARKS: The times are the hours since midnight. For instance, 8 means 8 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) Use the quadratic polynomial to make an outdoor temperature estimate.;Each class member will compute a temperature estimate for a different time of;day assigned by your instructor. See the spreadsheet in this module. Be sure to;use the quadratic regression model to make the estimate (not the values in the;data table). State your results clearly -- the time of day and the;corresponding outdoor temperature estimate.;(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.;(QR-6) Use the quadratic polynomial together with algebra to estimate the;time(s) of day when the outdoor temperature is a specific target temperature.;Each class member will work with a different target temperature, assigned by;your instructor. See the spreadsheet in this module. Report the time(s) to the;nearest quarter hour. Be sure to use the quadratic model to make the time;estimates (not values in the data table). Show work. State your results clearly;-- the target temperature and the associated time(s).;Please see the Technology Tips topic for additional information about;generating the scatterplot and quadratic polynomial.;Exponential Regression (ER);Data: A cup of hot coffee was placed in a room maintained at a constant;temperature of 69 degrees.;Thermometer Measuring Coffee Temperature;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) Use the exponential function to make a coffee temperature estimate. Each;class member will compute a temperature estimate for a different elapsed time x;assigned by your instructor. See the spreadsheet in this module. Substitute;your x value into your exponential function to get y, the corresponding;temperature difference between the coffee temperature and the room temperature.;Since y = T - 69, we have coffee temperature T = y + 69. Take your y estimate;and add 69 degrees to get the coffee temperature estimate. State your results;clearly -- the elapsed time and the corresponding estimate of the coffee;temperature.;(ER-5) Use the exponential function together with algebra to estimate the;elapsed time when the coffee arrived at a particular target temperature. Report;the elapsed time to the nearest tenth of a minute. Each class member will work;with a different target coffee temperature T assigned by your instructor. See;the spreadsheet in this module.;Given your target temperature T, then y = T - 69 is your target temperature;difference between the coffee and room temperatures. Use your exponential model;y = A e-bx. Substitute your target temperature difference for y and solve the;equation y = A e-bx for elapsed time x. Show algebraic work in solving your;equation. State your results clearly -- your target temperature and the;estimated elapsed time, to the nearest tenth of a minute.;For instance, if the target coffee temperature T = 150 degrees, then y = 150 -;69 = 81 degrees is the temperature difference between the coffee and the room;what we are calling y. So, for this particular target coffee temperature of 150;degrees, the goal is finding how long it took for the temperature difference y;to arrive at 81 degrees, that is, solving the equation 81 = A e- bx for x.;Please see the Technology Tips topic for additional information about;generating the scatterplot and exponential function.

 

Paper#60411 | Written in 18-Jul-2015

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