Download 10-3 Notes - morgansmathmarvels

Ch. 10 Correlation and
Regression
10-3 Notes
Inferences for Correlation and
Regression
Focus Points
- Test the ______________________________.
- Use sample data to compute the _____________
________________________________________.
- Find a ____________________ for the value of y
predicted for specified value of x.
- Test the ____________ of the least-squares line.
- Find a ___________________ for the slope β of the
least squares line and interpret its meaning.
Population Correlation Coefficient (ρ) is typically unknown, just like μ.
In statistics we use a random sample from the population and calculate the correlation
for the sample.
If the correlation is strong enough for the sample, then we may conclude that the
population has a correlation.
Steps
1
Establish H0 and H1.
H0: (always)
H1: (choose one)
2
Find the critical region. Use t-Distribution where d.f. = ______ where n =
____________________________________.
3
Use _________________ to find your sample statistic.
4
Draw conclusion
a) ________; we are ?% confident that H1, therefore there is a (+, –, or
*either way) correlation between x and y.
b________________; at α = ?% the evidence is not strong enough to imply H1,
therefore there is no significant (+, –, or *either way) correlation between x
and y.
Ex. 1
A medical research team is studying the effect of a new drug on red blood cells.
Let x be a random variable representing milligrams of the drug given to a patient. Let y be
a random variable representing red blood cells per cubic milliliter of whole blood. A
random sample of n = 7 volunteer patients gave the following results. Use α = 0.05 to test
for any correlation between the drug and red blood cell count.
Steps
1
x
9.2
10.1
9.0
12.5
8.8
9.1
9.5
y
5.0
4.8
4.5
5.7
5.1
4.6
4.2
2
H0:
H1:
t0 =
3
t≈
4
Conclusion:
(different from 0)
See Table 7 using d.f. =
, α = 0.05, and 2-tailed test (≠)
Residual – the difference between an ________
value of y and the corresponding ____________
value.
Standard error of estimate SE – the standard
deviation of the _____________.
To find SE, use __________ and arrow down to s.
s = SE
Confidence Intervals for Prediction
Recall from 10-2 Notes the following example.
Ex. 1 The number of workers on an assembly line varies due to the level of absenteeism
on any given day. In a random sample of production output from several days of work,
the following data were obtained, where x = number of workers absent from the
assembly line and y = number of defects coming off of the line.
x
3
5
0
2
1
y
16
20
9
12
10
# of defects
25
y = 8.257 + 2.338x
20
15
10
5
0
0
1
2
3
4
# w orkers absent
5
6
When making a prediction for y using the least squares line, the prediction lies
on the least squares line. However, we know that not all values (if any) lie on
the line.
Therefore, we build an interval around our prediction that allows us to be a
certain percent confident that the result will be within our interval.
# of defects
25
20
15
10
5
0
0
1
2
3
4
5
6
# w orkers absent
Formula:
yp  E
where E  tc S E
1
n( x  x ) 2
1 
n nx 2  ( x) 2
for tc use d.f. = n – 2 where n = # of ordered pairs
x , Σx2, and Σx can
all be found by doing
1-Var Stats
Recall:
d) On a day when 4 workers are absent from the assembly line, what would the
least-squares line predict for the number of defects coming off the line?
yp = 8.257 + 2.338 (4) =
Ex. 1 Find a 90% confidence interval for the forecast y value in part d.
yp  E
where
E  tc S E
1
n( x  x ) 2
1 
n nx 2  ( x) 2
Using 1Var Stat
E
x =
Σx2,=
E
90% CI 
Σx =
n=
Assignment
Day 1
p. 543 #1, 7, 9, 10
For 7, 9, 10 do parts a-e. For e, also explain its
meaning in context.
Testing the slope β of the least squares line is
the same as for correlation coefficient ρ except:
For Ho and for H1 we use ___ instead of ___.
Ex. 3 How fast do puppies grow? That depends on the puppy.
How about male wolf pups in the Helsinki Zoo (Finland)? Let x =
age in weeks and y = weight in kilograms for a random sample of
male wolf pups. The following data are based on the article
Studies of the Wolf in Finland Canis lupus L by E. Pulliainen,
University of Helsinki.
x
8
10
14
20
28
40
45
y
7
13
17
23
30
34
35
a) Use α = 1% to test the claim that β ≠ 0, and interpret the
results in the context of this application.
Finding a Confidence Interval for β is similar to
finding a Confidence Interval for yp except that
the formula for E varies slightly.
b E
where
E
tc S E
2
(
x
)
x 2 
n
b) Still using the data from ex. 3, compute an
80% Confidence Interval for β and interpret
the results in the context of this application.
Assignment
Day 1
p. 544 #7, 9, 10 f & g; also do #11 all