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Chapter 20
Testing Hypotheses About
Proportions
Copyright © 2009 Pearson Education, Inc.
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Main topics of this chapter
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The reasoning behind hypothesis tests
The null hypothesis
P-values
The one-proportion z-test
One and two sided alternatives
Things to watch out for when testing hypotheses
Copyright © 2009 Pearson Education, Inc.
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Division of Mathematics, HCC
Course Objectives for Chapter 20
After studying this chapter, the student will be able
to:
 Perform a one-proportion z-test, to include:
 writing appropriate hypotheses,
 checking the necessary assumptions,
 drawing an appropriate diagram,
 computing the P-value,
 making a decision, and
 interpreting the results in the context of the
problem.
Copyright © 2009 Pearson Education, Inc.
What we’re doing
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I flip a coin 3 times and get 3 heads.
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I flip a 4th time and get heads.
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This could have happened by chance (p = 0.0625).
The coin could be fair?
I flip a 5th time and get heads.
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This could have happened by chance (p = 0.125).
The coin could be fair.
This could have happened by chance (p = 0.03125).
The coin could be fair (??).
I flip a 6th time and get heads.
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This could have happened by chance (p = 0.015625).
Do you still think that this coin is fair?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 6
What we’re doing
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Starting out: The coin is fair.
 Burden of proof is on us to show that it is not.
After 3rd flip: Some evidence, but it still could be
fair.
After 4th flip: More evidence – still could be fair,
but we’re beginning to wonder.
After 5th flip: More evidence - Hmmmmmm!
After 6th flip: Lots of evidence – probably not fair.
What do we mean “probably?”
Copyright © 2009 Pearson Education, Inc.
Slide 1- 7
Electoral College poll from Chapter 19
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62% of Americans say that they would amend the
Constitution to replace the Electoral College system of
electing the President and Vice-President with a Popular
Vote system.
Gallup Poll taken October 6 – 9, 2011; announced last
week.
Gallup sampled 1005 adults 18 or older living in the
continental United States.
Other surveys may have gotten different results.
There is variability among results. We will measure it.
Copyright © 2009 Pearson Education, Inc.
What have we learned?
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We can be 95% confident that the true percentage of Americans who
want the Presidential election decided by popular vote is between
59.0% and 65.0%
We cannot say that “More than 60% of Americans want the
Presidential election decided by popular vote ” and be 95% confident
that we are correct.
The 70% CI is (60.4%,63.58%)
We can say that “More than 60% of Americans want the Presidential
election decided by popular vote” and be 70% confident that we are
correct.
But is 70% really that confident?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 9
Hypotheses
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Hypotheses are working models that we adopt
temporarily.
Our starting hypothesis is called the null hypothesis.
The null hypothesis, that we denote by H0, specifies a
population model parameter of interest and proposes a
value for that parameter.
We usually write down the null hypothesis in the form H0:
parameter = hypothesized value.
The alternative hypothesis, which we denote by HA,
contains the values of the parameter that we consider
plausible when we reject the null hypothesis.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 10
Testing Hypotheses
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The null hypothesis, specifies a population model
parameter of interest and proposes a value for that
parameter.
 H0: p = 0.60 would be appropriate for our Gallup
example
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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 11
A Trial as a Hypothesis Test
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Think about the logic of jury trials:
1
 To prove someone is guilty , we start by
assuming they are innocent.
 We retain that hypothesis until the facts make it
unlikely beyond a reasonable doubt.
 Then, and only then, we reject the hypothesis
of innocence and declare the person guilty.
1Guilty
in a criminal trial; “liable” or “in favor of the
plaintiff” in a civil trial – we’ll use “guilty” for both.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 12
What happens in a jury trial?
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The trial starts by assuming that the defendant is innocent –
recall “innocent until proved guilty.”
During the trial, evidence is gathered (by physical exhibits
and/or testimony.)
The jury examines the evidence.
If it is unlikely that, given innocence, we would not have all of
this evidence just by chance, the jury rejects the hypothesis of
innocence and returns a Guilty verdict.
If the evidence is not strong enough to reject the hypothesis of
no guilt, then the jury returns “Not guilty.”
The jury does not prove guilt or innocence.
Source for next slide: Food and Drug Administration, Center for Food Safety
and Applied Nutrition, Basic Statistics Course (Marc Boyer, Martine Ferguson)
Copyright © 2009 Pearson Education, Inc.
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Step
Trial by Jury
Statistical significance
1.
Start with the presumption that the
defendant is innocent
Start with the presumption that
the null hypothesis is true.
2.
Listen to only the factual evidence
presented in the trial. Ignore
newspaper and television.
Base the conclusion only on data from
this one experiment. Don’t consider any
other data.
3.
Evaluate whether you believe the
witness. Ignore testimony from
unreliable witnesses.
Evaluate whether the experiment was
performed properly.
4.
Think about whether the evidence
is consistent with the assumption of
innocence.
Calculate the p-value
5.
If the evidence is inconsistent with
the assumption, then reject the
assumption of innocence and
declare the defendant guilty.
Otherwise reach a verdict of not
guilty. A juror can’t conclude
“maybe” or ask for more evidence.
If the p-value is less than a preset
threshold (.05 is common), conclude that
the data are inconsistent with the null
hypothesis, and declare the difference to
be statistically significant. Otherwise,
conclude that sufficient evidence does
not exist.
Copyright © 2009 Pearson Education, Inc.
A Trial as a Hypothesis Test (cont.)
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The same logic used in jury trials is used in
statistical tests of hypotheses:
 We begin by assuming that a hypothesis is
true.
 Next we consider whether the data are
consistent with the hypothesis.
 If they are, all we can do is retain the
hypothesis we started with. If they are not, then
like a jury, we ask whether they are unlikely
beyond a reasonable doubt (or
“preponderance of evidence” in a civil trial.)
Copyright © 2009 Pearson Education, Inc.
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P-Values
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The statistical twist is that we can quantify our
level of doubt.
 We can use the model proposed by our
hypothesis to calculate the probability that the
event we’ve witnessed could happen.
 That’s just the probability we’re looking for—it
quantifies exactly how surprised we are to see
our results.
 This probability is called a P-value.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 16
P-Values (cont.)
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When the data are consistent with the model from the null
hypothesis, the P-value is high and we are unable to
reject the null hypothesis.
 In that case, we have to “retain” the null hypothesis we
started with.
 We can’t claim to have proved it; instead we “fail to
reject the null hypothesis” when the data are consistent
with the null hypothesis model and in line with what we
would expect from natural sampling variability.
If the P-value is low enough, we’ll “reject the null
hypothesis,” since what we observed would be very
unlikely were the null model true.
In a jury trial, the null hypothesis is that of innocence.
Copyright © 2009 Pearson Education, Inc.
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“An ode to p-values” 
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P-value low? The null’s gotta go.
P-value high? The null will fly.
Source: Mario F. Triola, “Elementary Statistics using
EXCEL”, 4th edition, © 2010, Pearson Publishing Co.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 18
Testing Hypotheses (reminder)
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The null hypothesis, which we denote H0, specifies a
population model parameter of interest and proposes a
value for that parameter.
 We might have, for example, H0: p = 0.60, as in the
Gallup electoral college poll example.
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.
 In other words, we find the z-score that our
observations give rise to, and then compare it with the
distribution that reflects the null hypothesis.
We then ask how likely it is to get results like we did if the
null hypothesis were true.
Copyright © 2009 Pearson Education, Inc.
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What to Do with an “Innocent” Defendant
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If the evidence is not strong enough to reject the
presumption of innocent, the jury returns with a
verdict of “not guilty.”
 The jury does not say that the defendant is
innocent.
 All it says is that there is not enough evidence
to convict, to reject innocence.
 The defendant may, in fact, be innocent, but
the jury has no way to be sure.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 20
What to Do with an “Innocent” Defendant (cont.)
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Said statistically, we will fail to reject the null
hypothesis.
 We never declare the null hypothesis to be
true, because we simply do not know whether
it’s true or not.
 Sometimes in this case we say that the null
hypothesis has been retained.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 21
What to Do with an “Innocent” Defendant (cont.)
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In a criminal trial, the burden of proof is on the
prosecution. In a civil trial, it is on the plaintiff, or
the party filing the lawsuit.
In a hypothesis test, the burden of proof is on the
unusual claim.
The null hypothesis is the ordinary state of affairs,
so it’s the alternative to the null hypothesis that
we consider unusual (and for which we must
marshal evidence).
Copyright © 2009 Pearson Education, Inc.
Slide 1- 22
Recall the Casey Anthony Trial
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Early July 2011 – Casey Anthony was found Not
Guilty of killing her 2-year old daughter Caylee.
Very emotional reaction nationwide.
Casey Anthony juror Jennifer Ford said that she and
the other jurors cried and were "sick to our stomachs"
after voting to acquit Ms. Anthony
"I did not say she was innocent," said Ford. "I just
said there was not enough evidence. If you cannot
prove what the crime was, you cannot determine
what the punishment should be."
Source: ABC News
Copyright © 2009 Pearson Education, Inc.
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A Comment on the jury trial example.
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There is a law forum on the Internet sponsored
by Martindale-Hubble. Lawyers there are in
agreement that
 A judge or jury would never decide a case
based solely on statistics, no matter how
significant the p-values are.
 If this did happen, the case would certainly be
appealed.
Statistics is a tool, to be used in conjunction with
other tools, to explain phenomena.
Copyright © 2009 Pearson Education, Inc.
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The Reasoning of Hypothesis Testing
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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…
Recall our example – 62% of Americans feel
that the Electoral College should be scrapped.
Can we say that more than 60% do?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 25
The Reasoning of Hypothesis Testing (cont.)
1. Hypotheses
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The null hypothesis: To perform a hypothesis test,
we must first translate our question of interest into a
statement about model parameters.
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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.
In some texts, the alternative hypothesis is denoted
H1. It’s the same thing – only author’s preference
dictates which is used.
I typically use Ho and Ha because they are easier to
type.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 26
Hypothesis – back to our
Electoral College example
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We are interested in whether the number of Americans
that want the Electoral College scrapped is more than
60%.
We take the “devil’s advocate” position – form a
statement saying “no change”. In this case, it is
 HO: p = 0.60
The we formulate the alternative,
 HA: p > 0.60.
As for things turning out lower, we’d abandon the claim
whether the optimism rate is exactly or less than 60%.
So (in this example) we are not interested in whether less
than 60% are optimistic.
Copyright © 2009 Pearson Education, Inc.
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Hypothesis – back to our
Electoral College example
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Remember: We form our hypothesis before collecting
the data.
Therefore, it is incorrect to say Ho: p = 0.62.
We got the 0.62 from the survey.
We do not know about the 0.62 when we form our
hypothesis.
Therefore, Ho: p = 0.62 cannot possibly be correct.
The correct null hypothesis is what is supposed
before collecting the data.
Ho: p = 0.60.
Copyright © 2009 Pearson Education, Inc.
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The Reasoning of Hypothesis Testing (cont.)
2.
Model
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To plan a statistical hypothesis test, specify the
model you will use to test the null hypothesis and the
parameter of interest.
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All models require assumptions, so state the
assumptions and check any corresponding
conditions.
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Your plan should end with a statement like
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Because the conditions are satisfied, I can model the
sampling distribution of the proportion with a Normal model.
Watch out, though. It might be the case that your model step
ends with “Because the conditions are not satisfied, I can’t
proceed with the test.” If that’s the case, stop and
reconsider.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 29
The Reasoning of Hypothesis Testing (cont.)
2. Model
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Each test we discuss in the book has a name
that you should include in your report.
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The test about proportions is called a oneproportion z-test.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 30
Checking assumptions – our example
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Independence: Gallup has expertise in assuring
that their respondents are independent.
Representative sample? Yes.
Sample size: npo = 1005 * 0.62 = 623 > 10
nqo = 1005 * 0.38 = 382 > 10
10% condition: 1005 is a small portion of the
American population of over 300 million.
Note: If HO were precisely true, there would be
exactly 603 (60% of 1005) in our sample who
want the Electoral College scrapped.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 31
One-Proportion z-Test
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The conditions for the one-proportion z-test are the same
as for the one proportion z-interval. We test the
hypothesis
H0: p = p0
using the statistic
p̂  p0 

z
SD  p̂ 
where SD  p̂  
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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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 32
The Reasoning of Hypothesis Testing (cont.)
3. Mechanics
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Under “mechanics” we place the actual
calculation of our test statistic from the data.
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Different tests will have different formulas and
different test statistics.
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Usually, the mechanics are handled by a
statistics program or calculator, but it’s good
to know the formulas.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 33
The Reasoning of Hypothesis Testing (cont.)
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In this case, we get a z-score.
Recall that a z-score measures the number of
standard deviations we are above or below the
mean.
A very high or very low z-score indicates a rare
event.
We measure this by looking at the area under the
standard normal curve.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 34
The Reasoning of Hypothesis Testing (cont.)
3. Mechanics
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The ultimate goal of the calculation is to
obtain a P-value.
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The P-value is the probability that the observed
statistic value (or an even more extreme value)
could occur if the null model were correct.
If the P-value is small enough, we’ll reject the null
hypothesis.
Note: The P-value is a conditional probability—it’s
the probability that the observed results could
have happened if the null hypothesis is true.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 35
Mechanics in our example
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pˆ  p0 

z
SD  pˆ 
The formula is
So we need to compute SD, which is
SD  pˆ  
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p0 q0
n
po =0.62,so qo = 0.38. n = 1005
Note: We are rounding po =0.62 for easy
computation. In reality,
623/1020 = 0.6199004975.
This is what the technologies will use.
Copyright © 2009 Pearson Education, Inc.
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Mechanics in our example
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Therefore
 SD = SQRT (0.6 * 0.4 / 1005 = 0.015453
Then z = (0.62 – 0.60) / 0.015453 or + 1.294
What do we do with this?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 37
Mechanics in our example
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Then z = (0.62 – 0.60) / 0.015454 = + 1.294.
This is a z-score, so we compare it with N(0,1).
Normalcdf(1.294,+99999)= 0.098
It turns out that we get a percentile of about
9.8%.
About 9.8% (of the time, we’d see a result this
extreme or more if there is no effect.
Now what do we do with this?
Copyright © 2009 Pearson Education, Inc.
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The Reasoning of Hypothesis Testing (cont.)
4. Conclusion
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The conclusion in a hypothesis test is always
a statement about the null hypothesis.
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The conclusion must state either that we
reject or that we fail to reject the null
hypothesis.
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And, as always, the conclusion should be
stated in context.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 39
The Reasoning of Hypothesis Testing (cont.)
4. Conclusion
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Your conclusion about the null hypothesis
should never be the end of a testing
procedure.
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Often there are actions to take or policies to
change.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 40
Let’s review our example – what did we do?
1.
Set up the Hypothesis
State the null and the alternative hypotheses.
2.
3.
4.
Find the model
State and check the four assumptions
Name the test (1 proportion z-test)
Mechanics: Find the p-value
Conclusion: Say what the p-value tells us.
This is called the 1-sample z-test in a few texts, but
“1-proportion z-test” is more accurate.
Copyright © 2009 Pearson Education, Inc.
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Conclusion in Our
Electoral College Example
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A sample of 1005 Americans showed that 623 want the
Electoral College scrapped. This is above 60%, or 603,
for this sample. The one-proportion z-test of whether
60% of Americans want the Electoral College scrapped
gave a p-value of 0.098. We would see a result this
extreme or more 9.8% of the time, or a hair above one
time in 10, by chance alone.
There is not sufficient evidence to show that this was a
real phenomenon and not just a chance occurrence.
Why? Normally, we consider 1 in 20 (or more) to be
sufficient evidence. More about this later.
Copyright © 2009 Pearson Education, Inc.
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Using the technology
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62% want the Electoral College scrapped.
The technology will not allow us to use 623.1
observations.
We will need to input 623 successes.
Both technologies will therefore work with
623/1005 = 0.6199004975 instead of 0.62.
We will therefore get a slightly different answer.
In most cases, this will not matter.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 43
Hypothesis Testing with the TI
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Press the STAT key.
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Then move to 1-propZTest
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Enter data as on the left and “Calculate.”
See result
at right
Copyright © 2009 Pearson Education, Inc.
Slide 1- 44
Hypothesis Testing with the TI (alternative)
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1.
2.
You can “make a picture” of the test.
Use “Draw” instead of “Calculate.”
Before doing this, make sure of two things:
That all of the equations entered in “Y=“ are
turned off.
That all of the plots in [STAT PLOT] are turned
off.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 45
Hypothesis Testing with the TI (alternative)
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Press the STAT key.
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Then move to 1-propZTest
Copyright © 2009 Pearson Education, Inc.
Slide 1- 46
Hypothesis Testing with the TI (alternative)
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Make sure that PLOTS and Y1 through Y6 are
turned off or these plots may overlay your
drawing!
Choose Draw instead of Calculate
Copyright © 2009 Pearson Education, Inc.
Slide 1- 47
Hypothesis Testing with StatCrunch
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See the screen captures on the next few slides.
It is very similar to the confidence interval.
When the time comes, select “Hypothesis Test”.
Note” “Proportions” is where you want to go, not
“Z-statistics.”
Copyright © 2009 Pearson Education, Inc.
Slide 1- 48
Copyright © 2009 Pearson Education, Inc.
Slide 1- 49
Why select “summary” and not “data”.
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“Data” refers to the 1005 people surveyed.
If you select “data”, then StatCrunch will look for
1005 respondents in the first column and how
they responded in the second!
We do not have the raw data; we have a
summary.
Remember Chapter 3? Same thing here.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 50
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Slide 1- 51
Copyright © 2009 Pearson Education, Inc.
Slide 1- 52
Hypothesis test results:
p : proportion of successes for population
H0 : p = 0.6
HA : p > 0.6
Proportion
Count
Total
Sample Prop.
p
623
1005
0.6199005
Std. Err.
Z-Stat
0.015453348 1.287779
P-value
0.0989
We get p = 0.0989, same as with the TI.
Slide 1- 53
Copyright © 2009 Pearson Education, Inc.
Confidence Interval for our example.
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Recall from Chapter 19:
This is the formula for the 95% confidence interval
ˆ ˆ when we tested
We already computed SE( pˆ )  pq
n
(i.e. computed our hypothesis test.) It is 0.015311.
We also found that the 95% confidence interval is
(0.5899, 0.6499). Note that it contains 0.60. our
hypothesized mean.
If we do the interval in the TI and then do the test (or
vice-versa), our data will be there. We do not have to
re-enter it for the other procedure! In StatCrunch, we
will have to re-enter.
Copyright © 2009 Pearson Education, Inc.
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It appears that
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Hypothesis Testing and Confidence Intervals are
Related Concepts.
They are two sides of the same coin.
In Confidence Intervals, we are estimating a
parameter.
In Hypothesis Testing, we are testing a
hypothesis.
We get the same information in both – we just
tell the story differently.
Copyright © 2009 Pearson Education, Inc.
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One important distribution difference
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Confidence Interval: The normal distribution is
around the 62% because this is what the data
gives.
But in hypothesis testing, the hypothesis is set up
before we even see the data.
Hence the distribution is centered around the null
hypothesis of 60%.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 56
Video example – cracking of ingots
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Cracking is a serious problem for aluminum ingots
weighing 30,000 pounds each.
Cracking of ingots is hovering at about 20% of ingots
made.
The engineers have designed a process that they hope
will reduce the percent of cracking.
Based on data that the engineers have collected on the
performance of 400 randomly examined ingots in which
68 were cracked, has their process worked?
Statistically, we can answer their question.
NOTE: This example comes from the video on this
chapter as presented by Dr. DeVeaux.
Copyright © 2009 Pearson Education, Inc.
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Checking assumptions – our example
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Independence: This is not a simple random sample. But
the company has told us that defects in one ingot will not
affect any other.
Representative sample? Yes.
Sample size: npo = 400 * 0.2 = 80 > 10
nqo = 400 * 0.8 = 320 > 10
10% condition: The ingots being tested are a small
fraction of the total ingots produced.
We can use the normal model.
Note: If HO were precisely true, there would be 80
cracked ingots.
Copyright © 2009 Pearson Education, Inc.
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Cracked ingot example with the TI
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[STAT][TESTS], then
to 5 (1-PropZ-test)
Enter in the
appropriate places:
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What we are testing –
0.2
X: 68, n: 400
One sided test
Calculate
Copyright © 2009 Pearson Education, Inc.
Slide 1- 59
Cracked ingot example with the TI
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Here is our result:
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We could also choose
Draw instead of
Calculate
Copyright © 2009 Pearson Education, Inc.
Slide 1- 60
Conclusion in Our Example
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A sample of 400 ingots showed 68 cracked. This is
below 20%, or 80, for this sample. The one-proportion ztest of whether the new method has reduced cracking
gave a p-value of 0.067. Thus, if the method did not
reduce cracking, we would see a result this extreme or
more 6.7% of the time, or one time in 15, by chance
alone.
While this is promising, it is not sufficient evidence to
show that there was a real change and not just a chance
occurrence.
Why? Normally, we consider 1 in 20 (or more) to be
sufficient evidence. More about this later.
Copyright © 2009 Pearson Education, Inc.
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Alternative Alternatives
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There are three possible alternative
hypotheses:
HA: parameter < hypothesized value
 HA: parameter ≠ hypothesized value
 HA: parameter > hypothesized value
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Copyright © 2009 Pearson Education, Inc.
Slide 1- 62
Alternative Alternatives (cont.)
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HA: parameter ≠ value is known as a two-sided alternative because
we are equally interested in deviations on either side of the null
hypothesis value.
For two-sided alternatives, the P-value is the probability of deviating
in either direction from the null hypothesis value.
Note: The “0.069” should be “0.067”.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 63
Back to our ingot example
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Suppose the metallurgist were interested in what
happens either way. He would test
HO: p = 0.20 against HA: p ≠ 0.20.
He would get the same z-score.
The difference is that he would be interested in
z-scores both below – 1.5 and above 1.5.
 We pick up 6.7% of the area on both sides.
 Thus, the metallurgist would report p = 0.134.
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Copyright © 2009 Pearson Education, Inc.
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Back to our ingot example
Two-sided alternative with the TI
Just pick the two-sided
alternative and do as before.
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Copyright © 2009 Pearson Education, Inc.
Slide 1- 65
Alternative Alternatives (cont.)
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The other two alternative hypotheses are called one-sided
alternatives.
A one-sided alternative focuses on deviations from the null hypothesis
value in only one direction.
Thus, the P-value for one-sided alternatives is the probability of
deviating only in the direction of the alternative away from the null
hypothesis value. Again, this should be 0.067.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 66
One and two sided alternatives
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Note that we got two conclusions:
Report to the manager: If there is really no
improvement in our process, we have a result this
extreme or more 6.7% of the time.
Report to the metallurgist: If there is really no
difference as a result of our method, we have a
result this extreme or more 13.4% of the time.
Both statements are true, and they are consistent
with each other.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 67
Alternative Alternatives (cont.)
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The decision to use a one-sided or two-sided
alternative is rarely a statistical one.
You must decide whether you should test onesided or two-sided
 Before you see the data? Not even that far –
 Before you even do the experiment; i.e. before
that data even exist!
Copyright © 2009 Pearson Education, Inc.
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P-Values and Decisions:
What to Tell About a Hypothesis Test
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How small should the P-value be in order for you to reject
the null hypothesis?
It turns out that our decision criterion is contextdependent.
 When we’re screening for a disease and want to be
sure we treat all those who are sick, we may be willing
to reject the null hypothesis of no disease with a fairly
large P-value.
 A longstanding hypothesis, believed by many to be
true, needs stronger evidence (and a correspondingly
small P-value) to reject it.
Another factor in choosing a P-value is the importance of
the issue being tested.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 69
P-Values and Decisions (cont.)
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Your conclusion about any null hypothesis should be
accompanied by the P-value of the test.
 If possible, it should also include a confidence interval
for the parameter of interest.
Don’t just declare the null hypothesis rejected or not
rejected.
 Report the P-value to show the strength of the
evidence against the hypothesis.
 This will let each reader decide whether or not to reject
the null hypothesis.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 70
What Can Go Wrong?
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Hypothesis tests are so widely used—and so
widely misused—that the issues involved are
addressed in their own chapter (Chapter 21).
There are a few issues that we can talk about
already, though:
Copyright © 2009 Pearson Education, Inc.
Slide 1- 71
What Can Go Wrong? (cont.)
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Don’t base your null hypothesis on what you see
in the data.
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Think about the situation you are investigating and
develop your null hypothesis appropriately.
If you develop the null hypothesis (and the alternative
hypothesis) before running the experiment, you won’t
make this mistake (or the next one.)
Don’t base your alternative hypothesis on the
data, either.
 Again, you need to Think about the situation.
Copyright © 2009 Pearson Education, Inc.
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What Can Go Wrong? (cont.)
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Don’t make your null hypothesis what you want to show
to be true.
 You can reject the null hypothesis, but you can never
“accept” or “prove” the null.
Don’t forget to check the conditions.
 We need randomization, independence, and a
sample that is large enough to justify the use of the
Normal model.
If you fail to reject the null hypothesis, don’t think a
bigger sample would be more likely to lead to rejection.
 Each sample is different, and a larger sample won’t
necessarily duplicate your current observations.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 73
What have we learned?
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We can use what we see in a random sample to test a
particular hypothesis about the world.
 Hypothesis testing complements our use of confidence
intervals.
Testing a hypothesis involves proposing a model, and
seeing whether the data we observe are consistent with
that model or so unusual that we must reject it.
 We do this by finding a P-value—the probability that
data like ours could have occurred if the model is
correct.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 74
What have we learned? (cont.)
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We’ve learned the process of hypothesis testing,
from developing the hypotheses to stating our
conclusion in the context of the original question.
We know that confidence intervals and
hypothesis tests go hand in hand in helping us
think about models.
 A hypothesis test makes a yes/no decision
about the plausibility of a parameter value.
 A confidence interval shows us the range of
plausible values for the parameter.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 75
Main topics of this chapter
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The reasoning behind hypothesis tests
The null hypothesis
P-values
The one-proportion z-test
One and two sided alternatives
Things to watch out for when testing hypotheses
Copyright © 2009 Pearson Education, Inc.
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Division of Mathematics, HCC
Course Objectives for Chapter 20
After studying this chapter, the student will be able
to:
 Perform a one-proportion z-test, to include:
 writing appropriate hypotheses,
 checking the necessary assumptions,
 drawing an appropriate diagram,
 computing the P-value,
 making a decision, and
 interpreting the results in the context of the
problem.
Copyright © 2009 Pearson Education, Inc.
A seasonal example for Halloween
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Do you have children who go trick or treating
Halloween night?
Kid brothers and sisters count!
If so, do you steal candy from their trick or treat
bags?
Go ahead and admit it!!
A majority of parents in the Baltimore-Washington
area do it!
But can we really say that?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 78
A survey about candy-copping parents!
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In early October, Mandala Research took a poll.
Surveys were taken in 20 major US metro areas.
4000 parents were questioned; 200 in each area.
In our area, 51% (say 102) admitted to stealing
candy from their children’s trick or treat bags.
Can we claim this to be a majority?
We’ll use α-level 0.10
Source: http://betterinbulk.net/2011/10/halloween2011-in-washington-d-c.html
Copyright © 2009 Pearson Education, Inc.
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A survey about candy-copping parents!
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Hypotheses:
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Assumptions and Conditions;
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Ho: p = 0.50
Ha: p > 0.50
Independence: We’ll assume that Mandala knows
how to do this
10 successes and failures? 102 and 98 are both
greater than 10
Sampling less than 10% of parents of trick-ortreating age in our area.
The test can proceed.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 80
Our hypothesis test
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Copyright © 2009 Pearson Education, Inc.
p = 0.3886
If Ho is true, we have
a result this extreme or
more 0.3886 (or
38.86%) of the time.
This is perfectly
reasonable.
There is not enough to
reject Ho at α = 0.10.
We fail to reject.
Slide 1- 81
A confidence interval
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Copyright © 2009 Pearson Education, Inc.
The confidence interval
contains proportions on
both sides of 0.50.
We cannot claim that a
majority of parents
steal candy from their
children's’ trick or treat
bags because there are
proportions less than
0.5 in the interval.
Slide 1- 82
Hypothesis Tests and Confidence Intervals
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If the stated proportion is inside an α percent
confidence interval, then a two-sided hypothesis will
be rejected at the (1-α) percent level.
If the stated proportion is inside a 95% confidence
interval, then a two-sided hypothesis will be rejected
at the 0.05 level.
The rule breaks down if
 The test is one-sided, or
 The percent changes; for example, 95%
confidence level and a rejection decision at the
0.01 level.
Copyright © 2009 Pearson Education, Inc.
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