Transcript Announcements - University of Pennsylvania

Announcements
• Extra office hours this week: Thursday, 12-12:45.
• The midterm will cover through Section 13.4.
• I will spend half of Thursday’s class going over remaining
material in Section 13.4 and half of Thursday’s class
reviewing.
• Instead of a review session on Sunday, I will hold office
hours from 2-5.
• On Monday, I will hold office hours from 9:00-11:30. I
also should be in my office most of the afternoon (after
1:30).
• Practice exam and practice problems posted. Answers will
be posted Thursday night.
Lecture 7
• Inference for a difference between means
from independent samples
• Observational vs. experimental data
• Differences between means in matched
pairs experiments
Inference about m1 – m2: Equal variances
• Construct the t-statistic as follows:
( x1  x2 )  ( m1  m2 )0
t
1 1
s (  )
n1 n2
2
p
(n1  1) s1  (n2  1) s2
2
d . f .  n1  n2  2, s p 
n1  n2  2
2
• Perform a hypothesis test
H0: m1  m2 = 0
H1: m1  m2 > 0
or < 0
or
0
2
Build a confidence interval
( x1  x 2 )  t  2
1 1
sp (  )
n1 n2
2
w here 1  is the confidencelev el.
Inference about the Difference
between Two Means
• Example 13.1
– Do people who eat high-fiber cereal for breakfast consume,
on average, fewer calories for lunch than people who do not
eat high-fiber cereal for breakfast?
– A sample of 150 people was randomly drawn. Each person
was identified as an eater or non-eater of high fiber cereal.
– For each person the number of calories consumed at lunch
was recorded. There were 43 high-fiber eaters who had a
mean of 604.02 calories for lunch with s=64.05. There were
107 non-eaters who had a mean of 633.23 calories for lunch
with s=103.29.
Inference about m1 – m2: Unequal variances
t
( x1  x2 )  ( m1  m2 )0
2
1
2
2
s
s
(  )
n1 n2
d.f. 
( s12 n1  s22 / n2 ) 2
2
1
2
2
2
( s n1 ) ( s n2 )

n1  1
n2  1
2
Inference about m1 – m2: Unequal variances
Conduct a hypothesis test
as needed, or,
build a confidence interval
Confidenceinterval
s12 s22
( x1  x2 )  t 2 (

)
n1 n2
where 1   is the confidencelevel
What is common (cont.)
• p-values:
  ( x1  x2 )  ( m1  m2 )0 
right

 - sided : Pdf  t  

left 

stderr
  

2 - sided :

( x1  x2 )  ( m1  m2 )0
Pdf  | t | 
stderr





What’s not common
• The way the stderr is estimated:
– Equal variances: pooled sdev estimate
– Separate variances: separate sdev estimates
• The degrees of freedom:
– Equal variances: simple exact dfs
– Separate variances: complex approximate dfs
Which case to use:
Equal variance or unequal
variance?
• Whenever there is insufficient evidence that
the variances are unequal, it is preferable to
perform the equal variances t-test.
• This is so, because for any two given
samples
The number of degrees
of freedom for the equal
variances case

The number of degrees
of freedom for the unequal
variances case
Equal/different Vars in Practice
• First step: obtain sdev or variance estimates
for each group. If drastically different, use
the unequal-variance test.
• Later we will see a test for equality of
variances.
Example 13.1 continued
• Test the scientist’s claim about high-fiber cereal
eaters consuming less calories than non-high fiber
cereal eaters assuming unequal variances at the
5% significance level.
• There were 43 high-fiber eaters who had a mean
of 604.02 calories for lunch with s=64.05. There
were 107 non-eaters who had a mean of 633.23
calories for lunch with s=103.29.
x1  604.02,

x2  633.23 s12  4,103, s 22  10,670
(4103 43  10670107) 2
4103 43
2
43  1

10670107

2
 122.6  123
107  1
–The rejection region is t < -t, = -t.05,123 1.658
t
( x1  x2 )  ( m1  m 2 )
s12 s 22

n1 n2

(604.02  633.23)  (0)
4103 10670

43
107
 -2.09
Additional Example-Problem 13.49
Tire manufacturers are constantly researching ways to produce
tires that last longer and new tires are tested by both professional
drivers and ordinary drivers on racetracks.
Suppose that to determine whether a new steel-belted radial tire
lasts longer than the company’s current model, two new-design
tires were installed on the rear wheels of 20 randomly selected cars
and two existing-design tires were installed on the rear wheels of
another 20 cars. All drivers were told to drive in their usual way
until the tires wore out. The number of miles(in 1,000s) was
recorded(Xr13-49). Can the company infer that the new tire will
last longer than the existing tire?
From 2-sided to right/left sided
• Given a 2-sided p-value, how do we get a 1sided p-value (JMP gives only the former)?
• Right-sided:
– if xbar-difference > mu-difference:
right-sided p-value = 2-sided p-value /2 (!!)
– If xbar-difference < mu-difference:
right-sided p-value > 0.5, so can’t reject …
P-values: 2-sided, right-sided
Observational vs. Experimental
Data
• Observational data: The researcher observes
individuals and measures variables of interest but
does not control which group each individual
(unit) is assigned to
• Experimental data: The researcher controls which
group each individual is assigned to. A common
procedure in statistical studies is to use random
assignment
• Example 13.1 (high fiber cereal example) –
observational or experimental data?
• Problem 13.49 (tire problem) – observational or
experimental data?
Interpretation of Experimental vs.
Observational Data
• For both observational and experimental data obtained by
random sampling, we can use statistical inference to assess
the evidence that there is a difference between the two
groups.
• A well controlled experiment (e.g., a randomized
experiment) provides evidence of the effect of the
treatment (high-fiber cereal) on the outcome (calories eaten
for lunch).
• But for observational data, we cannot conclude that a
difference between groups is due to the treatment. The
difference could be due to a confounding factor, e.g., highfiber cereal eaters are more health conscious.
13.4 Matched Pairs Experiments
• What is a matched pair experiment?
• Why matched pairs experiments are needed?
• How do we deal with data produced in this way?
The following example demonstrates a situation
where a matched pair experiment is the correct
approach to testing the difference between two
population means.
13.4 Matched Pairs Experiment
Example 13.3
– To investigate the job offers obtained by MBA
graduates, a study focusing on salaries was conducted.
– Particularly, the salaries offered to finance majors were
compared to those offered to marketing majors.
– Two random samples of 25 graduates in each discipline
were selected, and the highest salary offer was recorded
for each one. The data are stored in file Xm13-03.
– Can we infer that finance majors obtain higher salary
offers than do marketing majors among MBAs?.
What’s happening here?
• Question
– The difference between the sample means is
65624 – 60423 = 5,201.
– So, why could we not reject H0 and favor H1
where
(m1 – m2 > 0)?
The effect of a large sample
variability
• Answer:
– Sp2 is large (because the sample variances are
large) Sp2 = 311,330,926.
– A large variance reduces the value of the t
statistic and it becomes more difficult to reject
H0.
( x1  x 2 )  (m1  m 2 )
t
1
2 1
sp (  )
n1 n2
Reducing the variability
The range of observations
sample A
The values each sample consists of might markedly vary...
The range of observations
sample B
Reducing the variability
Differences
...but the differences between pairs of observations
might be quite close to one another, resulting in a small
The range of the
variability of the differences.
differences
0
The matched pairs experiment
• Example 13.4
– It was suspected that salary offers were affected by
students’ GPA, (which caused S12 and S22 to
increase).
– To reduce this variability, the following procedure
was used:
• 25 ranges of GPAs were predetermined.
• Students from each major were randomly selected, one
from each GPA range.
• The highest salary offer for each student was recorded.
– From the data presented can we conclude that
Finance majors are offered higher salaries?
Matched Pairs => One-Sample Test
• After taking differences of observations
within each pair, continue with a onesample test with H0 : m  0
Additional Example-Problem 13.75
Tire example contd. (Problem 13.49) Suppose
now we redo the experiment: On 20 randomly
selected cars, one of each type of tire is
installed on the rear wheels and, as before, the
cars are driven until the tires wear out. The
number of miles(in 1000s) is stored in Xr1375. Can we conclude that the new tire is superior?
Practice Problems
• 13.40,13.56,13.68,13.76