In: Statistics and Probability
A researcher was interested in investigating a relationship between the age (independent variable) of a driver and the distance the driver can see. For this purpose, he collected data on some drives. The data is provided in Appendix ‘1”.
To help you, partial summary analysis is provided below: SSxx= 13,752 SSxy=- 41,350 SSyy=193,667 ∑ x = ∑ Age = 1,530; ∑ y = ∑(Distance the driver can see)=12,700
Age Distance
18 510
20 590
22 560
23 510
23 460
25 490
27 560
28 510
29 460
32 410
37 420
41 460
46 450
49 380
53 460
55 420
63 350
65 420
66 300
67 410
68 300
70 390
71 320
72 370
73 280
74 420
75 460
77 360
79 310
82 360
a) Write the estimated regression line
b) Is the relationship meaningful (significant at α=0.05)? (3 pts) – make sure to state the null and alternative hypothesis first.
c) What is the strength of the relation? It is significant? (3 pts)
d) What is the coefficient of determination? (2 pts)
e) John is 61 years old. What is the expected driving distance for him? What is the 95% prediction interval for John? (4 pts)
Result:
a) Write the estimated regression line
Distance = 576.682-3.007*Age
b) Is the relationship meaningful (significant at α=0.05)? (3 pts) – make sure to state the null and alternative hypothesis first.
Ho: There is no linear relation between distance and age
H1: There is a linear relation between distance and age
Calculated F=50.21, P=0.000 which is < 0.05 level of significance. Ho is rejected. we conclude that there is a linear relation between distance and age.
c) What is the strength of the relation? It is significant? (3 pts)
strength of the relation r = -0.8012.
yes, it is significant, P<0.05.
d) What is the coefficient of determination? (2 pts)
coefficient of determination R square = 0.642.
64.2% of variance in distance is explained by age.
e) John is 61 years old. What is the expected driving distance for him? What is the 95% prediction interval for John? (4 pts)
predicted driving distance = 393.26.
95% prediction interval for John = (289.28, 497.25).
Excel Addon Megastat used.
Menu used: correlation/Regression ---- Regression Analysis.
Regression Analysis |
|||||||
r² |
0.6420 |
n |
30 |
||||
r |
-0.8012 |
k |
1 |
||||
Std. Error of Estimate |
49.7616 |
Dep. Var. |
Distance |
||||
Regression output |
confidence interval |
||||||
variables |
coefficients |
std. error |
t (df=28) |
p-value |
95% lower |
95% upper |
|
Intercept |
a = |
576.682 |
23.471 |
24.570 |
0.0000 |
528.604 |
624.760 |
Age |
b = |
-3.007 |
0.424 |
-7.086 |
0.0000 |
-3.876 |
-2.138 |
ANOVA table |
|||||||
Source |
SS |
df |
MS |
F |
p-value |
||
Regression |
124,332.643 |
1 |
124,332.643 |
50.21 |
0.0000 |
||
Residual |
69,334.024 |
28 |
2,476.215 |
||||
Total |
193,666.667 |
29 |
|||||
Predicted values for: Distance |
|||||||
95% Confidence Interval |
95% Prediction Interval |
||||||
Age |
Predicted |
lower |
upper |
lower |
upper |
Leverage |
|
61 |
393.26 |
372.72 |
413.80 |
289.28 |
497.25 |
0.041 |
|