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Inferences about the slope of the regression line Section 13.2
Inferences about the slope of the regression line
Section 13.2
Slope
• Average or expected change in the y (dependent) variable associated with a 1-unit increase in the x (independent) variable
• b is the point estimate of
Properties of the Sampling Distribution of b
•
•
• It’s normally distributed
b
2
2 where b xx
xx
xS x
nS
s
So…..
• Estimated standard deviation of b:
• Test Statistic: (df=n -2)
eb
xx
Ss
S
b
bt
s
Confidence Interval for
•
• (small) implies that the interval is narrow which means there’s relatively precise knowledge of
bs
bb t s
Is cardiovascular fitness related to an athlete’s performance.
x 7.7 8.4 8.7 9 9.6 9.6 10 10.2 10.4 11 11.7
y 71 71.4 65 68.7 64.4 69.4 63 64.6 66.9 62.6 61.7
treadmill time
20 km ski time
x
y
2.333b 43.09732.188
2 9resid
e
SSs
n
2.1880.591
3.702e
b
xx
ss
S
2
2 13.705xx
xS x
n
2.333 2.26 0.591
3.671, 0.999
bb t s
95% confident that the true average decrease in ski time associated with a 1 min. increase in treadmill time is between 1 and 3 minutes.
Hypothesis Test
: 0
: , , 0o
A
H
H
b
bt
s
Why do you think we use 0?
x 7.7 8.4 8.7 9 9.6 9.6 10 10.2 10.4 11 11.7
y 71 71.4 65 68.7 64.4 69.4 63 64.6 66.9 62.6 61.7
: 0
: 0o
A
H
H
Is there a negative association between treadmill time and ski time?
2.333 0
0.5913.948
b
bt
s
t
t
( , 3.948,9)
0.00168
p value tcdf
Reject the Ho. There is sufficient evidence to support the claim that there is a negative association between treadmill time & ski time.
Is a runner’s speed related to their steps per second?
90% confidence interval
Speed (ft/s)
Steps per second
x
y
x 15.86 16.88 17.5 18.62 19.97 21.06 22.11
y 3.05 3.12 3.17 3.25 3.36 3.46 3.55
Is there relationship between a runner’s speed & their average # of steps per second?
Leaning Tower of PisaFind a 90% confidence interval.
Year 75 77 78 80 81 82 83 84 85 87
Lean 642 656 667 688 696 698 713 717 725 757
Leaning Tower of PisaTest to see if there’s a positive relationship between the year and the amount of leaning.
Year 75 77 78 80 81 82 83 84 85 87
Lean 642 656 667 688 696 698 713 717 725 757
