Question

1. 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

   SOLVE:

   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)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	120	727.2	280	172.6	202.00
mean	8.00	48.48	SSxx	SSyy	SSxy

sample size ,   n =   15          
here, x̅ = Σx / n=   8.00   ,     ȳ = Σy/n =   48.48  
                  
SSxx =    Σ(x-x̅)² =    280.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   202.0          
                  
estimated slope , ß1 = SSxy/SSxx =   202.0   /   280.000   =   0.7214
                  
intercept,   ß0 = y̅-ß1* x̄ =   42.7086          
                  
so, regression line is   Ŷ =   42.7086   +   0.7214   *x

......................

b)

Predicted Y at X=   16   is                  
Ŷ =   42.70857   +   0.721429   *   16   =   54.2514

................

c)

Predicted Y at X=   18   is                  
Ŷ =   42.70857   +   0.721429   *   18   =   55.6943
............................

THANKS

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