Question

In: Statistics and Probability

Problem 6-23 (Algorithmic) The medical community unanimously agrees on the health benefits of regular exercise, but...

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 44.5

3 47.6

4 44.7

5 46.6

6 48

7 48.2

8 51.3

9 49.6

10 49.2

11 49.7

12 53

13 51

14 54.8

15 52.9

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 =

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) %

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)

Solutions

Expert Solution

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 (SSX) = 280
Sum of products (SP) = 193.4

Regression Equation = ŷ = bX + a

b = SP/SSX = 193.4/280 = 0.69071

a = MY - bMX = 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


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