Question

1. Problem 6-23 (Algorithmic)

   The medical community unanimously agrees on the health benefits of regular exercise, but are adults listening? During each of the past 15 years, a polling organization has surveyed americans about their exercise habits. In the most recent of these polls, slightly over half of all American adults reported that they exercise for 30 or more minutes at least three times per week. The following data show the percentages of adults who reported that they exercise for 30 or more minutes at least three times per week during each of the 15 years of this study.

   
   

   Year	Percentage of Adults Who Exercise 30 or more minutes at least three times per week
   1	41.5
   2	45.2
   3	47.1
   4	45.6
   5	46.7
   6	44.7
   7	47.9
   8	50
   9	48.2
   10	49.4
   11	50
   12	52.4
   13	51.1
   14	54.9
   15	52.5

   
   

   
   

   
   

1. Choose the correct time series plot.
   


Plot (ii)

Does a linear trend appear to be present?

Possible Positive Linear Trend


1. use simple linear regression to find the parameters for the line that minimizes MSE for this time series. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   y-intercept, b₀ =
   
   Slope, b₁ =
   
   MSE =
   

2. Use the trend equation from part (b) to forecast the percentage of adults next year (year 16 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   %
   

3. Use the trend equation from part (b) to forecast the percentage of adults three years from now (year 18 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   %

   
   

Answer 1

(a) I am not able to see the plots since the plots did not load in my browser. However, the time series plot would look like the diagram below. I have added the data labels and markers (triangles) for clear visualization of the time series plot.

% of adults who exercise 30 or more minutes at least 3 times per week 52.5 51.1 482 49.4 50 52.4 5 0 47. 9 45.2 47.1 45.6 46.7 % of adults who exercise 30 or more minutes at least 3 times per week 1 2 3 4 5 6 7 9 10 11 8 Year 12 13 14 15

(b) y-intercept, b0 = 42.7085714285714 = 42.7086

slope, b1 = 0.721428571428571 = 0.7214

Hence, the estimated equation is: % of adults who exercise 30 or more minutes at least 3 times per week = 42.7086 + 0.7214 * Year

MSE = SSE/(n-k) = 26.8954285714286/(15-2) = 2.0689, where n = number of observations and k = number of parameters, SSE = error sum of square

The calculated values are as follows:

6.9 Year Yt Yt - Y_bar X_t-X_bar (Y_t-Y_bar) (X_t - X_bar) (x_t - X_bar)^2Y_hat (Prediction) (residual)^2 1) 41.5 -6.98 -7.00 48.86 49.00 43.43 3.7249 2 45.2 -3.28 -6.00 19.68 36.00 44.15 1.099502041 31 47.1 -1.38 -5.00 25.00 44.87 4.960165306 b1 0.721428571 41 45.6 -2.88 -4.00 11.52 16.00 45.59 3.26531E-05 bo 42.70857143 5| 46.7 -1.78 -3.00 5.34 9.00 46.32 0.14767551 SSE 26.89542857 6 44.7 -3.78 -2.00 7.56 4.00 47.04 5.462236735 MSE 2.068879121 7 47.9 -0.58 -1.00 0.58 1.00 47.76 0.020002041 850 1.52 0.00 0.00 48.48 2.3104 9| 48.2 -0.28 1.00 -0.28 1.00 49.20 1.002859184 101 49.4 2.00 1.84 4.00 49.92 0.273379592 11 50 3.00 4.56 9.00 50.64 0.415104082 12 52.4 3.92 4.00 15.68 16.00 51.37 1.069746939 13 51.1 5.00 13.1 25.00 52.09 0.97445102 14 54.9 6.42 6.00 38.52 36.00 52.81 4.374073469 15 52.5 4.02 7.00 28.14 49.00 53.531 1.0609 54.25 54.97 55.69 0.92 Average 8.00 48.48 Total 202.00 280.00

Steps of estimation as shown below (the formula view of the excel)

A B Year C Y_t b1 SSE =F23/G23 =C22-L4*B22 =SUM(12:116) =L6/(COUNT(B2:B16)-COUNT(L4:L5)) MSE 41.5 45.2 47.1 45.6 46.7 44.7 47.9 50 48.2 49.4 50 52.4 51.1 154.9 Y_t-Y_bar X_t-X_bar lie_t-Y_bar) =C2-C$22 |=B2-B$22 =D2 E2 =C3-C$22 =B3-B$22 I=D3*E3 =C4-C$22 I=B4-B$22 I=D4E4 =C5-C$22 =B5-B$22 =D5 E5 =C6-C$22 =B6-B$22 =D6°E6 =C7-C$22 =B7-B$22 =D7 E7 =C8-C$22 =B8-B$22 =D8E8 =C9-C$22 =B9-B$22 =D9E9 =C10-C$22 =B10-B$22 =D10*E10 =C11-C$22 =B11-B$22 =D11-E11 =C12-C$22 =B12-B$22 =D12-E12 =C13-C$22 =B13-B$22 =D13E13 =C14-C$22 =B14-B$22 =D14E14 =C15-C$22 =B15-B$22 =D15*E15 =C16-C$22 =B16-B$22 =D16E16 (X_t-X_bar) (X_t-X_bar)^2 =E2^2 =E3^2 =E4^2 =E5^2 =E6^2 =E742 =E8^2 =E942 =E10^2 =E11^2 =E12^2 =E1342 =E1442 =E1542 =E16^2 Y_hat (Prediction)| (residual)^2 =$L$5+$L$4*B2 =(C2-H2)^2 =$L$5+$L$4*B3 =(C3-H3)^2 =$L$5+$L$4*B4 =(C4-H4)^2 =$L$5+$L$4*B5 =(C5-H5)^2 =$L$5+$L$4*B6 =(C6-H6)^2 =$L$5+$L$4*B7 =(C7-H7)^2 =$L$5+$L$4*B8 =(C8-H8)^2 =$L$5+$L$4*B9 =(C9-H9)^2 =$L$5+$L$4*B10 =(C10-H10)^2 =$L$5+$L$4*B11 =(C11-H11)^2 =$L$5+$L$4*B12 =(C12-H12)^2 =$L$5+$L$4*B13 =(C13-H13)^2 =$L$5+$L$4*B14 =(C14-H14)^2 =$L$5+$L$4*B15 =(C15-H15)^2 =$L$5+$L$4*B16 =(C16-H16)^2 =$L$5+$L$4*B17 =$L$5+$L$4*B18 =$L$5+$L$4*B19 52.5 =AVERAGE(B2:316) =AVERAGE(C2:016) 22 Average 23 Total 201 =SUM(F2:F16) |-SUM(G2:616)

(c) The forecast for year 16 = 42.7085714285714 + 0.721428571428571 * 16 = 54.2514 (refer to the yellow highlighted row above with year = 16)

(d) The forecast for year 16 = 42.7085714285714 + 0.721428571428571 * 18 = 55.6943 (refer to the yellow highlighted row above with year = 18)