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

year. percentage of adults who exercise 30 minutes or more minutes at least three times per week

1 41.9

2 45.5

3 47.3

4 45.8

5 46.7

6 44.7

7 47.9

8 50.4

9 48.4

10 49.3

11 49.6

12 52.6

13 50.6

14 55.1

15 52.4

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) Yes

(b) y-intercept, b₀ = 43.0267
Slope, b₁ = 0.69
MSE = 2.0966

(c) 54.0667

(d) 55.4467

The calculations are:

Period	Demand (y)	Period(x)		Forecast	Error	Absolute	Squared	Abs Pct Err
Period 1	41.9	1		43.71667	-1.81667	1.816667	3.300278	04.34%
Period 2	45.5	2		44.40667	1.093333	1.093333	1.195378	02.40%
Period 3	47.3	3		45.09667	2.203333	2.203333	4.854678	04.66%
Period 4	45.8	4		45.78667	0.013333	0.013333	0.000178	00.03%
Period 5	46.7	5		46.47667	0.223333	0.223333	0.049878	00.48%
Period 6	44.7	6		47.16667	-2.46667	2.466667	6.084444	05.52%
Period 7	47.9	7		47.85667	0.043333	0.043333	0.001878	00.09%
Period 8	50.4	8		48.54667	1.853333	1.853333	3.434844	03.68%
Period 9	48.4	9		49.23667	-0.83667	0.836667	0.700011	01.73%
Period 10	49.3	10		49.92667	-0.62667	0.626667	0.392711	01.27%
Period 11	49.6	11		50.61667	-1.01667	1.016667	1.033611	02.05%
Period 12	52.6	12		51.30667	1.293333	1.293333	1.672711	02.46%
Period 13	50.6	13		51.99667	-1.39667	1.396667	1.950678	02.76%
Period 14	55.1	14		52.68667	2.413333	2.413333	5.824178	04.38%
Period 15	52.4	15		53.37667	-0.97667	0.976667	0.953878	01.86%
				Total	1.28E-13	18.27333	31.44933	37.70%
Intercept	43.0267			Average	8.53E-15	1.218222	2.0966	02.51%
Slope	0.69				Bias	MAD	MSE	MAPE
						SE	1.555371
Forecast	54.0667	16
	55.4467	18				Correlation	0.899509
						Coefficient of determination	0.809117