Question

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 Reported That They Exercise for 30 or More Minutes at Least Three Times per Week
1	41.0
2	44.9
3	47.1
4	45.7
5	46.6
6	44.5
7	47.6
8	49.8
9	48.1
10	48.9
11	49.9
12	52.1
13	50.6
14	54.6
15	52.4

a time series plot. does a linear trend appear to be present?

simple linear regression to find the parameters for the line that minimizes MSE for this time series.

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.

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?

Answer 1

(a) To check if there is a linear relation between time and % of people exercising we can use a scatter plot for it.

The x-axis will have the year values and the y-axis will have the %.

We can see that the trends follow a linear pattern along the years. The points are also close to each other indicating that there is a linear relation between time and percent of people exercisng.

Sr. No.	Year (x)	Percentage (y)					*
1	1	41	-7	-7.253	49.000	52.611	50.773
2	2	44.9	-6	-3.353	36.000	11.245	20.12
3	3	47.1	-5	-1.153	25.000	1.330	5.7667
4	4	45.7	-4	-2.553	16.000	6.520	10.213
5	5	46.6	-3	-1.653	9.000	2.734	4.96
6	6	44.5	-2	-3.753	4.000	14.088	7.5067
7	7	47.6	-1	-0.653	1.000	0.427	0.6533
8	8	49.8	0	1.547	0.000	2.392	0
9	9	48.1	1	-0.153	1.000	0.024	-0.153
10	10	48.9	2	0.647	4.000	0.418	1.2933
11	11	49.9	3	1.647	9.000	2.712	4.94
12	12	52.1	4	3.847	16.000	14.797	15.387
13	13	50.6	5	2.347	25.000	5.507	11.733
14	14	54.6	6	6.347	36.000	40.280	38.08
15	15	52.4	7	4.147	49.000	17.195	29.027
Total	120	723.8	0	0.000	280.000	172.277	200.3
Mean	8	48.253

The regression equation is given by

Where slope

= 200.3 / 280.00

= 0.715

intercept

= 48.253 - 0.715 * 8

= 42.53

The regression eq therefore is

(b) To forecast percentage of people we just substitute years in place of 'x'

When x = 16 we have

(c) when x = 18

  

(c)