Transcript Slide 1- 20 - Department of Economics

HOW MANY DAYS UNTIL THANKSGIVING
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UPCOMING IN CLASS
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Quiz #6 this Wednesday
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HW #12 due Sunday
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Exam #2 next Wednesday

Data Project Due by 5pm Thursday December 5th
via email or my department mailbox.
CHAPTER 20
Comparing Means
COMPARING TWO MEANS
Comparing two means is not very different from
comparing two proportions.
 This time the parameter of interest is the
difference between the two means, 1 – 2.


Examples,
Height of black vs. height of whites
 SAT scores of men vs SAT scores of women
 Sugar content in Children’s cereal vs. Sugar content
in Adult’s cereal

TWO-SAMPLE T-INTERVAL
When the conditions are met, we are ready to find the confidence
interval for the difference between means of two independent
groups.
The confidence interval is
 y1  y2   t

df
 SE  y1  y2 
where the standard error of the difference of the means is
SE  y1  y2  
s12 s22

n1 n2
The critical value depends on the particular confidence level, C, that you
specify and on the number of degrees of freedom, which we get from the
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sample sizes and a special formula.
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DEGREES OF FREEDOM

The special formula for the degrees of freedom for
our t critical value is a bear:
2
 s12 s22 
  
 n1 n2 
df 
2
2
1  s12 
1  s22 
  
 
n1  1  n1  n2  1  n2 

Because of this, we will let technology calculate
degrees of freedom for us!
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ROUTE TO SCHOOL
A student takes two routes to class. Route A and Route
B. Each day she randomly selects a route until she has
walked each route 20 times.
 Route A

Mean = 44
 St.D.= 5
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Route B
Mean =47
 St. D. 4
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Create a 95% confidence interval for the difference
between the routes, and interpret it.
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WE ARE 95% CONFIDENT THE TRUE DIFFERENCE
BETWEEN ROUTE A AND ROUTE B IS IN THE
INTERVAL (0.09, 5.91). WHICH ROUTE IS FASTER?
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Route A
Route B
Our data shows no
difference.
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TESTING THE DIFFERENCE BETWEEN
TWO MEANS
We test the hypothesis H0:1 – 2 = 0, where the
hypothesized difference, 0, is almost always 0, using
the statistic
y1  y2   0

t
SE  y1  y2 
The standard error is
SE  y1  y2  
s12 s22

n1 n2
When the conditions are met and the null hypothesis is true, this
statistic can be closely modeled by a Student’s t-model with a
number of degrees of freedom given by a special formula. We use
that model to obtain a P-value.
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COMPARING DIFFERENT TEACHING
METHODS
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IS THE NEW METHOD BETTER THAN THE
TRADITIONAL METHOD? WHAT IS THE
APPROPRIATE HYPOTHESIS TEST?
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Ho: μ1-μ2=0
Ha: μ1-μ2≠0
Ho: μ1-μ2=0
Ha: μ1-μ2>0
Ho: μ1-μ2=0
Ha: μ1-μ2<0
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WHAT IS YOUR CONCLUSION BASED ON
THE DATA?
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Reject null. There is sufficient evidence that
the new activities are better
Reject null. There is NOT sufficient evidence.
Fail to reject null. There is NOT sufficient
evidence.
Fail to reject null. There is sufficient evidence
that the new activities are better.
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FIND THE P-VALUE AND COMPARE YOUR
TEST RESULTS
http://www.stat.tamu.edu/~west/applets/tdemo.ht
ml
 http://www.tutorhomework.com/statistics_tables/statistics_tables.
html

Calculators
 http://economics.illinoisstate.edu/aohler/eco138/d
ocuments/ComparingMeans_001.pdf

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COMPARING SPORTS LEAGUES
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INTERPRET YOUR INTERVAL. WE ARE 90%
CONFIDENT THAT…
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the points scored per game in both leagues will
fall in the interval
the amount by which the points scored in
League 2 games exceed the points scored in
League 1 games will fall in the interval
the amount by which the points scored in
League 1 games exceed the points scored in
League 2 games will fall in the interval
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DOES THE INTERVAL SUGGEST THAT THE TWO
LEAGUES DIFFER IN AVERAGE NUMBER OF
POINTS SCORED PER GAME?
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No, because the interval
does not contain zero
Yes, because the interval
contains zero
No, because the interval
contains zero
Yes, because the interval
does not contain zero
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COMPARING TV PROGRAMS
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WHAT IS THE BEST CONCLUSION? TEST
USING A 95% CI.
Viewer’s memory are different, since we reject
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the null
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Viewer’s memory are not different, since we
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Viewer’s memory are different, since we do not
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Viewer’s memory are not different, since we do
not reject the null
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BASED ON OUR DATA AND CI, WHAT COULD
YOU SAY ABOUT THE TWO PROGRAMS.
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Program A helps viewers remember
commercials better than Program B.
Program B helps viewers remember
commercials better than Program A.
There is no statistical difference between
Program A and Program B. Viewers remember
the commercials just the same.
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RUNNER’S PROBLEM
In a certain running event, preliminary heats are
determined by random draw, so it would be
expected that the abilities of runners in the
various heats are about the same, on average.
 There are 7 runners in each race, but due to an
outlier in heat 2, we only have 6 observations for
heat 2.
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RUNNER’S PROBLEM

The statistics for heats 2 and 5 are below.

Heat 2
Mean: 52.135 seconds
 SD: 0.635
 N=6
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Heat 5
Mean: 52.333 seconds
 SD: 0.961
 N=7
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IS THERE ANY EVIDENCE THAT THE MEAN TIME
TO FINISH IS DIFFERENT FOR THE HEATS?
WHAT IS THE APPROPRIATE HYPOTHESIS TEST?
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Ho: μ2-μ5=0 Ha: μ2-μ5≠0
Ho: μ2-μ5=0 Ha: μ2-μ5>0
Ho: μ2-μ5=0 Ha: μ2-μ5<0
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DETERMINE THE TEST STATISTICS
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-0.198
0.435
-0.45
0.662
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AT THE 0.05 SIGNIFICANCE LEVEL, TEST THE
HYPOTHESIS THAT THE HEATS HAVE DIFFERENT
AVERAGE TIMES.
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Do reject the null hypothesis. There is not sufficient
evidence to support the claim that the mean running
times in heat 2 and 5 are different.
Do reject the null hypothesis. There is sufficient
evidence to support the claim that the mean running
times in heat 2 and 5 are different.
Do not reject the null hypothesis. There is not
sufficient evidence to support the claim that the mean
running times in heat 2 and 5 are different.
Do not reject the null hypothesis. There is sufficient
evidence to support the claim that the mean running
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times in heat 2 and 5 are different.
UPCOMING IN CLASS

Quiz #6 this Wednesday

HW #12 due Sunday

Exam #2 next Wednesday

Data Project Due by 5pm Thursday December 5th
via email or my department mailbox.