In: Statistics and Probability
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)
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