Hypotheses
0 Hypotheses are working models that we adopt temporarily.
0 Our starting hypothesis is called the null hypothesis.
0 The null hypothesis, that we denote by H0, specifies a
population model parameter of interest and proposes a value
for that parameter.
0 We usually write down the null hypothesis in the form H0:
parameter = hypothesized value.
0 The alternative hypothesis, which we denote by HA, contains
the values of the parameter that we consider plausible if we
reject the null hypothesis.
Hypotheses (cont.)
0 The null hypothesis, specifies a population model
parameter of interest and proposes a value for that
parameter.
0 We might have, for example, H0: p = 0.20.
0 We want to compare our data to what we would expect
given that H0 is true.
0 We can do this by finding out how many standard deviations
away from the proposed value we are.
0 We then ask how likely it is to get results like we did if the
null hypothesis were true.
A Trial as a Hypothesis Test
0 Think about the logic of jury trials:
0 To prove someone is guilty, we start by assuming they are
innocent.
0 We retain that hypothesis until the facts make it unlikely
beyond a reasonable doubt.
0 Then, and only then, we reject the hypothesis of innocence
and declare the person guilty.
A Trial as a Hypothesis Test
(cont.)
0 The same logic used in jury trials is used in statistical
tests of hypotheses:
0 We begin by assuming that a hypothesis is true.
0 Next we consider whether the data are consistent with the
hypothesis.
0 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.
P-Values
0 The statistical twist is that we can quantify our level
of doubt.
0 We can use the model proposed by our hypothesis to
calculate the probability that the event we’ve
witnessed could happen.
0 That’s just the probability we’re looking for—it
quantifies exactly how surprised we are to see our
results.
0 This probability is called a P-value.
P-Values (cont.)
0 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.
0 In that case, we have to “retain” the null hypothesis we
started with.
0 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.
0 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.
What to Do with an
“Innocent” Defendant
0 If the evidence is not strong enough to reject the
presumption of innocent, the jury returns with a
verdict of “not guilty.”
0 The jury does not say that the defendant is innocent.
0 All it says is that there is not enough evidence to
convict, to reject innocence.
0 The defendant may, in fact, be innocent, but the jury
has no way to be sure.
What to Do with an
“Innocent” Defendant (cont.)
0 Said statistically, we will fail to reject the null
hypothesis.
0 We never declare the null hypothesis to be true,
because we simply do not know whether it’s true or
not.
0 Sometimes in this case we say that the null hypothesis
has been retained.
What to Do with an
“Innocent” Defendant (cont.)
0 In a trial, the burden of proof is on the prosecution.
0 In a hypothesis test, the burden of proof is on the
unusual claim.
0 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).
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?
The Reasoning of Hypothesis
Testing
0
There are four basic parts to a hypothesis test:
1. Hypotheses
2. Model
3. Mechanics
4. Conclusion
0
Let’s look at these parts in detail…
The Reasoning of Hypothesis
Testing (cont.)
1.
Hypotheses
0
The null hypothesis: To perform a hypothesis test,
we must first translate our question of interest into a
statement about model parameters.
0
0
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.
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.
The Reasoning of Hypothesis
Testing (cont.)
2.
Model
0
To plan a statistical hypothesis test, specify the model you will use
to test the null hypothesis and the parameter of interest.
0
All models require assumptions, so state the assumptions and
check any corresponding conditions.
0
Your model step should end with a statement such
0
Because the conditions are satisfied, I can model the sampling
distribution of the proportion with a Normal model.
0
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.
The Reasoning of Hypothesis
Testing (cont.)
2.
Model (cont.)
0
Each test we discuss in the book has a name that you
should include in your report.
0
The test about proportions is called a one-proportion ztest.
One-Proportion z-Test
0 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
0 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.
The Reasoning of Hypothesis
Testing (cont.)
3.
Mechanics
0
Under “mechanics” we place the actual calculation of our
test statistic from the data.
0
Different tests will have different formulas and different
test statistics.
0
Usually, the mechanics are handled by a statistics
program or calculator, but it’s good to know the formulas.
The Reasoning of Hypothesis
Testing (cont.)
3.
Mechanics (continued)
0
The ultimate goal of the calculation is to obtain a P-value.
0
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.
0
If the P-value is small enough, we’ll reject the null
hypothesis.
0
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.
P-value Example
0 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!
The Reasoning of Hypothesis
Testing (cont.)
4.
Conclusion
0
The conclusion in a hypothesis test is always a statement
about the null hypothesis.
0
The conclusion must state either that we reject or that
we fail to reject the null hypothesis.
0
And, as always, the conclusion should be stated in
context.
The Reasoning of Hypothesis
Testing (cont.)
4.
Conclusion
0
Your conclusion about the null hypothesis should never
be the end of a testing procedure.
0
Often there are actions to take or policies to change.
Conclusion Example
0 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?
Alternatives Hypotheses
0 There are three possible alternative
hypotheses:
0 HA: parameter < hypothesized value
0 HA: parameter ≠ hypothesized value
0 HA: parameter > hypothesized value
Alternatives Hypotheses (cont.)
0 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.
0 For two-sided alternatives, the P-value is the probability of deviating in
either direction from the null hypothesis value.
Alternatives Hypotheses (cont.)
0 The other two alternative hypotheses are called one-sided alternatives.
0 A one-sided alternative focuses on deviations from the null hypothesis
value in only one direction.
0 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.
Steps for Hypothesis Testing for OneProportion z-Tests
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:
0 𝒏𝒑𝟎 ≥ 𝟏𝟎 and
𝒏𝒒𝟎 ≥ 𝟏𝟎
2. State the test you are about to conduct
0 Ex) One-proportion z-test
3. Set up your hypotheses
0
0
H0 :
HA:
Steps for Hypothesis Testing for
One-Proportion z-Tests (cont.)
4. Calculate your test statistic
0
𝒛=
𝒑−𝒑𝟎
𝒑𝟎 𝒒𝟎
𝒏
5. Draw a picture of your desired area under the Normal
model, and calculate your P-value.
6. Make your conclusion.
0
When your P-value is small enough (or below α, if given), reject the
null hypothesis.
0
When your P-value is not small enough, fail to reject the null
hypothesis.
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?
Graphing Calculator Shortcuts
0 One Proportion Z-Test:
0 Stat  TESTS
0 5: 1-Prop ZTest
0 Po = hypothesized proportion
0 x = number of successes
0 n = sample size
0 Determine the tail
0 Calculate
P-Values and Decisions: What to Tell About a
Hypothesis Test
0 How small should the P-value be in order for you to reject the
null hypothesis?
0 It turns out that our decision criterion is context-dependent.
0 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).
0 A longstanding hypothesis, believed by many to be true,
needs stronger evidence (and a correspondingly small Pvalue) to reject it.
0 Another factor in choosing a P-value is the importance of the
issue being tested.
P-Values and Decisions (cont.)
0 Your conclusion about any null hypothesis should be
accompanied by the P-value of the test.
0 If possible, it should also include a confidence interval for the
parameter of interest.
0 Don’t just declare the null hypothesis rejected or not rejected.
0 Report the P-value to show the strength of the evidence
against the hypothesis.
0 This will let each reader decide whether or not to reject the
null hypothesis.
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?
What Can Go Wrong?
0 Hypothesis tests are so widely used—and so widely
misused—that the issues involved are addressed in
their own chapter (Chapter 21).
0 There are a few issues that we can talk about already,
though:
What Can Go Wrong? (cont.)
0 Don’t base your null hypothesis on what you see in
the data.
0 Think about the situation you are investigating and
develop your null hypothesis appropriately.
0 Don’t base your alternative hypothesis on the data,
either.
0 Again, you need to Think about the situation.
What Can Go Wrong? (cont.)
0 Don’t accept the null hypothesis.
0 If you fail to reject the null hypothesis, don’t
think a bigger sample would be more likely
to lead to rejection.
0 Each sample is different, and a larger
sample won’t necessarily duplicate your
current observations.
Recap
0 We can use what we see in a random sample to test a
particular hypothesis about the world.
0 Hypothesis testing complements our use of confidence
intervals.
0 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.
0 We do this by finding a P-value—the probability that
data like ours could have occurred if the model is
correct.
Recap (cont.)
0 We’ve learned:
0 Start with a null hypothesis.
0 Alternative hypothesis can be one- or two-sided.
0 Check assumptions and conditions.
0 Data are out of line with H0, small P-value, reject the
null hypothesis.
0 Data are consistent with H0, large P-value, don’t reject
the null hypothesis.
0 State the conclusion in the context of the original
question.
Recap (cont.)
0 We know that confidence intervals and hypothesis
tests go hand in hand in helping us think about
models.
0 A hypothesis test makes a yes/no decision about the
plausibility of a parameter value.
0 A confidence interval shows us the range of plausible
values for the parameter.
Assignments: pp. 476 – 479
0 Day 1: # 1-3, 9, 11
0 Day 2: # 4, 5, 12, 14, 18
0 Day 3: # 16, 20, 22