A mixture of sucrose and water was heated on a hot plate, and the temperature (i

Question

A mixture of sucrose and water was heated on a hot plate, and the temperature (in degree C) was recorded each minute for 20 minutes by three thermocouples. The results are shown in the following table. a. Compute the least-squares line for estimating the temperature as a function of tune, using T_1 as the value for temperature. b. Compute the least-squares line for estimating the temperature as a function of time, using T_2 as the value for temperature. c. Compute the least-squares line for estimating the temperature as a function of tune, using T_3 as the value for temperature. d. It is desired to compute a single line to estimate temperature as a function of time. One person suggests averaging the three slope estimates to obtain a single slope estimate, and avenging the three intercept estimates to obtain a single intercept estimate. Find the equation of the line that results from this method. e. Someone else suggests averaging the three temperature measurements at each time to obtain T = (T_1 + T_2 + T_3)/3. Compute the least-squares line using T as the value for temperature. f. Are the results of parts (d) and (e) different?

Explanation / Answer

f) Yes, both the estimates in (d) & (e) are same.

Excel formula:

A) Least sq line (linear regression): T1 on time: Time T1 T2 T3 T_Average Slope Intercept 0 20 18 21 19.66667 4.99924812 17.35714 1 18 22 11 17 2 29 22 26 25.66667 B) Least sq line (linear regression): T2 on time: 3 32 25 35 30.66667 Slope Intercept 4 37 37 33 35.66667 4.985714286 16.38571 5 36 46 55 45.66667 6 46 45 44 45 C) Least sq line (linear regression): T3 on time: 7 46 44 43 44.33333 Slope Intercept 8 58 54 63 58.33333 5.056390977 16.61429 9 64 69 62 65 10 72 65 65 67.33333 D) Averaging the coefficients: 11 79 80 80 79.66667 Slope Intercept 12 84 74 75 77.66667 5.013784461 16.78571 13 82 87 78 82.33333 14 87 93 88 89.33333 E) Rgression line with average of temperatures 15 98 90 91 93 Slope Intercept 16 103 100 103 102 5.013784461 16.78571 17 101 98 109 102.6667 18 103 103 107 104.3333 19 102 103 104 103

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