EXERCISE #1 Small Is Beautiful: The relationship between Height and Longevity. T
ID: 384548 • Letter: E
Question
EXERCISE #1 Small Is Beautiful: The relationship between Height and Longevity. The data below shows the President of USA, his age at time of death, and his height. Age at Time Height President of Death (inches) Madison 85 64 Van Buren 79 66 B. Harrison 67 66 J, Adams 90 67 J.Q. Adams 80 67 Jackson 78 73 Washington 67 74 Arthur 56 74 F. Roosevelt 63 74 L. Johnson 64 74 Jefferson 83 74.5. A simple linear regression of the model: LONGEVITY = b + b HEIGHT was run and the computer output results are shown below. Use the output to respond to the following questions:
(a) What sort of relationship exists between the height of the president and the longevity of the president?
(b) Interpret the meaning behind the values of the estimated b and b .
(c) Predict the expected longevity of a president who is: (i) 48 inches, (ii) 60 inches, (iii) 70 inches, (iv) 80 inches tall.
(d) At a level of significance, = 0.05, test for the significance of the relationship between height and longevity. What is your conclusion? What comments do you have on the predictions you made in part (c)?
(e) Give an interpretation to the R-Square value in the computer output.
Explanation / Answer
LONGEVITY = 174.424 - 1.430722 HEIGHT
R-Squared = 0.309636
Adjusted R-Squared = 0.232929
Standard error of estimate = 9.504369
Number of cases used = 11
Analysis of Variance
p-value
Source SS df MS F Value Sig Prob
Regression 364.63900 1 364.63900 4.03661 0.075434
Residual 812.99730 9 90.33304
Total 1177.63600 10
LONGEVITY AND ONE'S HEIGHT
REGRESSION COEFFICIENTS FOR LONGEVITY
Two-Sided p-value
Variable Coefficient Std Error t Value Sig Prob
Constant 174.42400 50.15620 3.47761 0.006965
HEIGHT -1.43072 0.71211 -2.00913 0.075434 *
Standard error of estimate = 9.504369
Durbin-Watson statistic = 1.900782
ANSWER
Model is LONGEVITY = b + bHEIGHT
Estimated regression is: LONGEVITY = 174.424 - 1.430722 HEIGHT
b = - 1.430722 is negative, therefore relationship is INVERSE
ANSWER b = 174.424 is the intercept = expected longevity of a president of zero height
b = - 1.430722 the slope, for every one inch increment in height, longevity drops by 1.430722 years
(c)
ANSWERS
Estimated regression is: LONGEVITY = 174.424 - 1.430722 HEIGHT
(d) ANSWERS Model is LONGEVITY = b + bHEIGHT
H: b= 0 is the null hypothesis, and
H: b 0 is the alternative hypothesis
Using the F-test
Analysis of Variance
p-value
Source SS df MS F Value Sig Prob
Regression 364.63900 1 364.63900 4.03661 0.075434
Residual 812.99730 9 90.33304
Total 1177.63600 10
The p-value = 0.075434 > = 0.05
DO NOT REJECT NULL:
H: b= 0 is the null hypothesis
Therefore Null is correct; the two variables are not related
USING THE t-TEST
LONGEVITY AND ONE'S HEIGHT
REGRESSION COEFFICIENTS FOR LONGEVITY
Two-Sided p-value
Variable Coefficient Std Error t Value Sig Prob
Constant 174.42400 50.15620 3.47761 0.006965
HEIGHT -1.43072 0.71211 -2.00913 0.075434 *
p-value = 0.075434 * > = 0.05
The p-value = 0.075434 > = 0.05
DO NOT REJECT NULL:
H: b= 0 is the null hypothesis
Therefore Null is correct; the two variables are not related
What comments do you have on the predictions you made in part (c)?
Since the the relationship between the two variables is not statistically significant, our predictions in part (c) are not reliable.
(e) ANSWER
R-Squared = 0.309636 = 30.9636%
Height accounts for 30.9636% of variation in longevity
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