Question

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	41.8
2	45.4
3	47.2
4	45.7
5	46.8
6	44.8
7	47.8
8	50.5
9	48.3
10	49.3
11	49.5
12	52.5
13	50.7
14	55
15	52.3

1. Does a linear trend appear to be present?
   
   
   
2. 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 =
   
3. 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)
   
   %
   
4. 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)

From the above plot, it shows increasing trend in the % of reporting adults as the year advances

b)

Excel output

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.901667336
R Square	0.813003986
Adjusted R Square	0.798619677
Standard Error	1.537360137
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	1	133.5841429	133.5841429	56.52019826	4.38568E-06
Residual	13	30.72519048	2.36347619
Total	14	164.3093333
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	42.98095238	0.835337183	51.45341699	2.06998E-16	41.17631612	44.78558864	41.17631612	44.78558864
Year	0.690714286	0.091874841	7.517991637	4.38568E-06	0.492230759	0.889197812	0.492230759	0.889197812

Percentage of Adults Who Exercise 30 or more minutes at least three times per week = 0.69071 * Year + 42.98095

Normal calculations :

Sum of X = 120
Sum of Y = 727.6
Mean X = 8
Mean Y = 48.5067
Sum of squares (SS_X) = 280
Sum of products (SP) = 193.4

Regression Equation = ŷ = bX + a

b = SP/SS_X = 193.4/280 = 0.69071

a = M_Y - bM_X = 48.51 - (0.69*8) = 42.98095

ŷ = 0.69071X + 42.98095

Year	Percentage of Adults Who Exercise 30 or more minutes at least three times per week	X - Mx	Y - My	(X - Mx)2	(X - Mx)(Y - My)
1	41.8	-7	-6.7067	49	46.9467
2	45.4	-6	-3.1067	36	18.64
3	47.2	-5	-1.3067	25	6.5333
4	45.7	-4	-2.8067	16	11.2267
5	46.8	-3	-1.7067	9	5.12
6	44.8	-2	-3.7067	4	7.4133
7	47.8	-1	-0.7067	1	0.7067
8	50.5	0	1.9933	0	0
9	48.3	1	-0.2067	1	-0.2067
10	49.3	2	0.7933	4	1.5867
11	49.5	3	0.9933	9	2.98
12	52.5	4	3.9933	16	15.9733
13	50.7	5	2.1933	25	10.9667
14	55	6	6.4933	36	38.96
15	52.3	7	3.7933	49	26.5533
				SS: 280	SP: 193.4

ŷ = 0.69071X + 42.98095

c)

at year = 16 means x = 16

ŷ = 0.69071 * 16 + 42.98095 = 54.0323

In Year 16, 54.03% of adults will report that they exercise for 30 or more minutes at least three times per week

d)

at year 18 means x = 18

ŷ = 0.69071 * 18 + 42.98095 = 55.4137

In Year 18, 55.41% of adults will report that they exercise for 30 or more minutes at least three times per week