Transcript Slide 1- 3 - Department of Economics

HOW MUCH DID YOU DRINK THIS
WEEKEND?
1.
2.
3.
4.
5.
0 drinks
1-2 drinks
3-4 drinks
5-6 drinks
>6
17%
17%
17%
17%
17%
17%
Slide
1- 1
1
2
3
4
5
6
UPCOMING WORK

HW #9 due Sunday

Part 3 of Data Project due October 28th

Quiz #5 in class next Wednesday
EXAMPLES OF ONE-PROPORTION TEST


Everyone (100%) believes in ghosts
More than 10% of the population believes in
ghosts

Less than 2% of the population has been to jail

90% of the population wears contacts
Slide
1- 3
EXAMPLES OF TWO-PROPORTION TESTS

Women believe in ghosts more than men

Blacks believe in ghost more than whites

People who have been to jail believe in ghosts more
than people who haven’t been to jail

Women smoke more than men

Women use facebook in the bathroom more than men
Slide
1- 4
EXAMPLES OF ONE-SAMPLE T-TEST

All Priuses have fuel economy > 50 mpg

Ford Focuses get 5 mpg on average


The average starting salary for ISU graduates
>$100,000
The average cholesterol level for a person with
diabetes is 240.
Slide
1- 5
EXAMPLES OF TWO-SAMPLE T-TEST



The MPG for the Prius is greater than the MPG
for the Ford Focus
ISU male graduates have a greater starting
salary than women
The cholesterol levels are the same for people
with and without diabetes.
Slide
1- 6
MARGIN OF ERROR: CERTAINTY VS. PRECISION

The more confident we want to be, the larger our
z* has to be

But to be more precise (i.e. have a smaller ME
and interval), we need a larger sample size, n.

We can claim, with 95% confidence, that the
interval pˆ  2SE( pˆ ) contains the true population
proportion.


The extent of the interval on either side of
the margin of error (ME).
p̂ is called
In general, confidence intervals have the form
estimate ± ME.
Slide
1- 7
MARGIN OF ERROR PROBLEM



It’s believed that as many as 22% of adults over
50 never graduated from high school.
We wish to see if this percentage is the same
among the 25 to 30 age group.
What sample size would allow us to increase our
confidence level to 95% while recuding the margn
of error to only 4%.
Slide
1- 8
CHAPTERS 17
Testing Hypotheses About Proportions
ISU – STATISTICS 2011 SURVEY RESULTS




55.5% of ISU students reported binge drinking in
the previous two weeks
Sample size = 417
Compared to other campuses 69.1%, believe the
alcohol use at ISU is about the same.
Other campuses…

National results average about 32.2%
Slide
1- 10
THE REASONING OF HYPOTHESIS TESTING

There are four basic parts to a hypothesis test:
1. Hypotheses
2. Model
3. Mechanics

4. Conclusion
Let’s look at these parts in detail…
Slide
1- 11
1. HYPOTHESES
The null hypothesis: To perform a hypothesis test,
we must first translate our question of interest
into a statement about model parameters.


In general, we have
H0: parameter = hypothesized value.
The alternative hypothesis: The alternative
hypothesis, HA, contains the values of the
parameter we consider plausible if we reject the
null.



We can only reject or fail to reject the null hypothesis.
If we reject the null hypothesis, this suggests the
alternative is true.
Slide
1- 12
POSSIBLE HYPOTHESES





Two-tailed test
Ho: parameter = hypothesized value
HA: parameter ≠ hypothesized value
One-tailed test
Ho: parameter = hypothesized value
HA: parameter < hypothesized value
Ho: parameter = hypothesized value
HA: parameter > hypothesized value
Slide
1- 13
IS THE COIN IN MY HAND A FAIR?
1.
2.
3.
Ho p=0.5
Ha p>0.5
Ho p=0.5
Ha p<0.5
Ho p=0.5
Ha p≠0.5
33%
33%
33%
Slide
1- 14
1
2
3
IN THE 1980S ONLY ABOUT 14% OF THE
POPULATION ATTAINED A BACHELOR’S DEGREE.
HAS THE PERCENTAGE CHANGED?
1.
2.
3.
Ho p=0.14
Ha p>0.14
Ho p=0.14
Ha p<0.14
Ho p=0.14
Ha p≠0.14
33%
33%
33%
Slide
1- 15
1
2
3
LAST YEAR RECYCLING RATES WERE AT 25%.
THE TOWN OF TRASHVILLE CLAIMS THAT THE
NEW MANDATE, REQUIRING EVERYONE TO
33%
33%
33%
RECYCLE, HAS INCREASED THE RECYCLING
RATE.
1.
2.
3.
Ho p=0.25
Ha p>0.25
Ho p=0.25
Ha p<0.25
Ho p=0.25
Ha p≠0.25
Slide
1- 16
1
2
3
ACCORDING TO A CENSUS, 16% OF PEOPLE IN THE US ARE
HISPANIC. ONE COUNTY SUPERVISOR BELIEVES HER
COUNTY HAS A SMALLER PROPORTION OF HISPANICS. SHE
SURVEYS THE 493 PEOPLE IN HER COUNTY AND FINDS 41
ARE HISPANIC. STATE THE HYPOTHESIS
25%
25%
25%
25%
1.
Ho p=0.16
Ha p<0.16
2.
Ho p=0.16
Ha p≠0.16
3.
Ho p=0.08
Ha p<0.08
4.
Ho p=0.08
Ha p≠0.08
Slide
1- 17
1
2
3
4
TESTING HYPOTHESES


The null hypothesis specifies a population model
parameter of interest and proposes a value for
that parameter.
We want to compare our data to what we would
expect given that H0 is true.

We can do this by finding out how many standard
deviations away from the proposed value we are.

We then ask how likely it is to get results like we
did if the null hypothesis were true.
Slide
1- 18
ONE-PROPORTION Z-TEST

The conditions for the one-proportion z-test are the
same as for the one proportion z-interval. We test the
hypothesis
H 0: p = p 0
using the statistic
pˆ  p0 

z
SD  pˆ 
where

SD  pˆ  
p0 q0
n
When the conditions are met and the null hypothesis is
true, this statistic follows the standard Normal model,
so we can use that model to obtain a P-value.
2. MODEL



All models require assumptions, so
state the assumptions and check any
corresponding conditions.
Assumptions you will test

Independence

Randomization

10% condition

Success/Failure
Determine Alpha Level
3. MECHANICS
The ultimate goal of the calculation is to
obtain a P-value.

The P-value is the probability that the
observed statistic value could occur if the
null model were correct.

If the P-value is small enough, we’ll reject
the null hypothesis.
 We can define “rare event” arbitrarily by setting
a threshold for our P-value.

The threshold is called an alpha level, denoted by .
 If our P-value falls below that point, we’ll reject H0.

p-value < alpha level  reject null
 p-value > alpha level  fail to reject null

Slide
1- 21
ACCORDING TO A CENSUS, 16% OF PEOPLE IN THE US ARE
HISPANIC. ONE COUNTY SUPERVISOR BELIEVES HER
COUNTY HAS A SMALLER PROPORTION OF HISPANICS. SHE
SURVEYS THE 493 PEOPLE IN HER COUNTY AND FINDS 41
25%
25%
25%
25%
ARE HISPANIC.
FIND THE P-VALUE OF YOUR TEST.
1.
2.
3.
4.
p=0.0668
p=1-0.0668
p=-4.66
p=0.000
Slide
1- 22
1
2
3
4
P-VALUES


The percentile associated with our z-value is
called the p-value.
A p-value is a conditional probability

The probability of the observed statistic given that
the null hypothesis is true.
The P-value is NOT the probability that the null hypothesis
is true.
 It’s not even the conditional probability that null hypothesis
is true given the data.

Slide
1- 23
ACCORDING TO A CENSUS, 16% OF PEOPLE IN THE US ARE
HISPANIC. ONE COUNTY SUPERVISOR BELIEVES HER
COUNTY HAS A SMALLER PROPORTION OF HISPANICS. SHE
SURVEYS THE 493 PEOPLE IN HER COUNTY AND FINDS 41
ARE HISPANIC.
Ho p=0.16
Ha p<0.16
 Two possible conclusions:

1.
2.
Fail to reject null hypothesis at the 5% level. We find no
evidence that suggests the local man finds water better
than simply drilling
Reject the null hypothesis at the 5% level, suggesting the
local man finds water better than simply drilling.
ALPHA LEVELS

Result:


We do not prove or disprove hypotheses.
We only suggest that the likelihood of a
hypothesis being true is very very low or
high.


Null is probably true.
How rare is rare? 1%, 5%, 10% chance?
Common alpha levels are 0.01, 0.05, and 0.1.
 The alpha level is also called the significance
level.


When we reject the null hypothesis, we say that the
test is “significant at that level.”
Slide
1- 25
INTERPRET THE P-VALUE 0.000, FROM OUR
PREVIOUS EXAMPLE OF CENSUS DATA
1.
Because the p-value is so low, there is NOT sufficient
evidence that the Hispanic population in this county
differs from the nation
2.
Because the p-value is so low, there is sufficient
evidence that the Hispanic population in this county
differs from the nation
3.
Because the p-value is so high, there is sufficient
evidence that the Hispanic population in this county
differs from the nation
Slide
1- 26
4. CONCLUSIONS



We can only reject or fail to reject the
null hypothesis.
If we reject the null, there is enough
evidence to suggest the alternative is
true, b/c the p-value is very very small.
If we fail to reject the null, there is NOT
enough evidence to suggest the
alternative is true, b/c the p-value is still
large.
FAILING TO REJECT THE NULL

You should say that “The data have failed to provide
sufficient evidence to reject the null hypothesis.”

Don’t say that you “accept the null hypothesis.”

In a jury trial, if we do not find the defendant guilty,
we say the defendant is “not guilty”—we don’t say
that the defendant is “innocent.”
Slide
1- 28
HW 9 _ PROBLEM 11



An airline’s public relations department says the
airline rarely loses passengers’ luggage.
Claim: When luggage is lost, 85% is recovered
and delivered to its owner with 24 hrs.
Survey of Air Travelers: 114 of 194 people who
lost their luggage on that airline were reunited
with the missing items by the next day
Slide
1- 29
WHAT IS THE CORRECT HYPOTHESIS TEST, IF
WE WANT TO SHOW THAT THE AIRLINE’S RATE
IS WORSE THAN THEY CLAIM?
1.
2.
3.
4.
5.
6.
Ho p=.588 Ha p>.588
Ho p=.588 Ha p<.588
Ho p=.588 Ha p≠.588
Ho p=.85 Ha p>.85
Ho p=.85 Ha p<.85
Ho p=.85 Ha p≠.85
17%
17%
17%
17%
17%
17%
Slide
1- 30
1
2
3
4
5
6
DO THE RESULTS OF THE SURVEY CAST
DOUBT ON THE AIRLINE’S CLAIM OF 85%?
1.
2.
3.
4.
No, because we do not reject the null hypothesis
Yes, because we reject the null hypothesis.
Yes, because we do not reject the null
hypothesis
No, because we reject the null hypothesis.
25%
25%
25%
25%
Slide
1- 31
1
2
3
4
UPCOMING WORK

HW #9 due Sunday

Part 3 of Data Project due October 28th

Quiz #5 in class next Wednesday