Transcript the null hypothesis is true

Chapter 20
Testing Hypotheses
About Proportions
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Hypotheses
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Hypotheses are working models that we adopt
temporarily.
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Our starting hypothesis is called the null hypothesis.
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The null hypothesis, that we denote by H0, specifies a
population model parameter of interest and proposes a
value for that parameter.
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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 if we reject the null hypothesis.
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Hypotheses (cont.)

The null hypothesis, specifies a population model
parameter of interest and proposes a value for
that parameter.
 We might have, for example, H0: p = 0.20.

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.
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A Trial as a Hypothesis Test

Think about the logic of jury trials:
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To prove someone is guilty, we start by
assuming they are innocent.
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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.
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A Trial as a Hypothesis Test (cont.)

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.
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P-Values

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.

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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.
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P-Values (cont.)

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.
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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.
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What to Do with an “Innocent” Defendant

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.
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What to Do with an “Innocent” Defendant (cont.)

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.
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What to Do with an “Innocent” Defendant (cont.)

In a trial, the burden of proof is on the
prosecution.
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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).
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Examples:
1.
A research team wants to know if aspirin helps to thin blood. The
null hypothesis says that it doesn’t. They test 12 patients, observe
the proportion with thinner blood, and get a P-value of 0.32. They
proclaim that aspirin doesn’t work. What would you say?
2.
An allergy drug has been tested and found to give relief to 75% of
the patients in a large clinical trial. Now the scientists want to see if
the new, improved version works even better. What would the null
hypothesis be?
3.
The new drug is tested and the P-value is 0.0001. What would you
conclude about the new drug?
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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…
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The Reasoning of Hypothesis Testing (cont.)
1. Hypotheses

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 when we
reject the null.
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Hypothesis Example:
A 1996 report from the U.S. Consumer Product Safety Commission
claimed that at least 90% of all American homes have at least one
smoke detector. A city’s fire department has been running a public safety
campaign about smoke detectors consisting of posters, billboards, and
ads on radio and TV and in the newspaper. The city wonders if this
concerted effort has raised the local level above the 90% national rate.
Building inspectors visit 400 randomly selected homes and find that 376
have smoke detectors. Is this strong evidence that the local rate is higher
than the national rate? Set up the hypotheses.
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The Reasoning of Hypothesis Testing (cont.)
2. Model

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 model step should end with a statement such
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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.
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The Reasoning of Hypothesis Testing (cont.)
2. Model

Each test we discuss in the book has a name
that you should include in your report.

The test about proportions is called a oneproportion z-test.
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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 H0: p = p0
using the statistic
where SD  p̂  

p̂  p0 

z
SD  p̂ 
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.
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The Reasoning of Hypothesis Testing (cont.)
3. Mechanics

Under “mechanics” we place the actual
calculation of our test statistic from the data.

Different tests will have different formulas and
different test statistics.

Usually, the mechanics are handled by a
statistics program or calculator, but it’s good
to know the formulas.
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The Reasoning of Hypothesis Testing (cont.)
3. Mechanics (continued)
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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.
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P-value Example:
A large city’s DMV claimed that 80% of candidates pass driving tests, but
a survey of 90 randomly selected local teens who had taken the test
found only 61 who passed. Does this finding suggest that the passing
rate for teenagers is lower than the DMV reported? What is the P-value
for the one-proportion z-test? Don’t forget to check the conditions for
inference!
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The Reasoning of Hypothesis Testing (cont.)
4. Conclusion

The conclusion in a hypothesis test is always
a statement about the null hypothesis.

The conclusion must state either that we
reject or that we fail to reject the null
hypothesis.

And, as always, the conclusion should be
stated in context.
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The Reasoning of Hypothesis Testing (cont.)
4. Conclusion

Your conclusion about the null hypothesis
should never be the end of a testing
procedure.

Often there are actions to take or policies to
change.
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Conclusion Example:

Recap: A large city’s DMV claimed that 80% of candidates pass driving
tests. Data from a reporter’s survey of randomly selected local teens
who had taken the test produced a P-value of 0.002. What can the
reporter conclude? And how might the reporter explain what the P-value
means for the newspaper story?
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STEPS FOR HYPOTHESIS TESTING
1. Check Conditions and show that you have checked these!
• Random Sample: Can we assume this?
• 10% Condition: Do you believe that your sample size is less than
10% of the population size?
• Success/Failure:
𝒏𝒑𝟎 ≥ 𝟏𝟎
and
𝒏𝒒𝟎 ≥ 𝟏𝟎
2. State the test you are about to conduct
Ex) One-proportion z-test
3. Set up your hypotheses
H 0:
H :
STEPS FOR HYPOTHESIS TESTING
(CONT.)
4. Calculate your test statistic
𝒛=
𝒑−𝒑𝟎
𝒑 𝟎 𝒒𝟎
𝒏
5. Draw a picture of your desired area under the t-model, and
calculate your P-value.
6. Make your conclusion.
When your P-value is small enough (or below α, if given), reject the
null hypothesis.
When your P-value is not small enough, fail to reject the null
hypothesis.
Alternative Alternatives

There are three possible alternative
hypotheses:

HA: parameter < hypothesized value

HA: parameter ≠ hypothesized value

HA: parameter > hypothesized value
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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.
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Alternative Alternatives (cont.)

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.
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Testing a Hypothesis Example:
Home field advantage –teams tend to win more often when the play
at home. Or do they?
If there were no home field advantage, the home teams would win
about half of all games played. In the 2007 Major League Baseball season,
there were 2431 regular-season games. It turns out that the home team won
1319 of the 2431 games, or 54.26% of the time.
Could this deviation from 50% be explained from natural sampling
variability, or is it evidence to suggest that there really is a home field
advantage, at least in professional baseball?
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Graphing Calculator Shortcuts:
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One Proportion Z-Test:
 Stat  TESTS
 5: 1-Prop ZTest
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Po = hypothesized proportion
x = number of successes
n = sample size
Determine the tail
Calculate
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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?
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It turns out that our decision criterion is context-dependent.
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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 (0.10).
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.
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P-Values and Decisions (cont.)

Your conclusion about any null hypothesis should be
accompanied by the P-value of the test.
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
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.
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Examples:
1.
A bank is testing a new method for getting delinquent customers to pay
their past-due credit card bills. The standard way was to send a letter
(costing about $0.40) asking the customer to pay. That worked 30% of
the time. They want top test a new method that involves sending a DVD
to customers encouraging them to contact the bank and set up a
payment plan. Developing and sending the video costs about $10 per
customer. What is the parameter of interest? What are the null and
alternative hypotheses?
2.
The bank sets up an experiment to test the effectiveness of the DVD.
They mail it out to several randomly selected delinquent customers and
keep track of how many actually do contact the bank to arrange
payments. The bank’s statistician calculates a P-value of 0.003. What
does this P-value suggest about the DVD?
3.
The statistician tells the bank’s management that the results are clear
and they should switch to the DVD method. Do you agree? What else
might you want to know?
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What Can Go Wrong?

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:
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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.
Don’t base your alternative hypothesis on the
data, either.

Again, you need to Think about the situation.
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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.
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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.
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What Can Go Wrong? (cont.)
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Don’t accept the null hypothesis.
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.
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What have we learned?

We can use what we see in a random sample to test a
particular hypothesis about the world.
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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 © 2010 Pearson Education, Inc.
What have we learned? (cont.)

We’ve learned:
 Start with a null hypothesis.
 Alternative hypothesis can be one- or two-sided.
 Check assumptions and conditions.
 Data are out of line with H0, small P-value, reject
the null hypothesis.
 Data are consistent with H0, large P-value, don’t
reject the null hypothesis.
 State the conclusion in the context of the original
question.
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What have we learned? (cont.)

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.
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Assignments: 476 – 479
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Day 1: # 1-3, 9, 11
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Day 2: # 4, 5, 12,14,18
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Day 3: # 16, 20, 22
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