Hypothesis Testing

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Statistics
HYPOTHESIS TESTING
TERMS
Null hypothesis– The claim being assessed in a
hypothesis test is called the null hypothesis.
 Usually, the null hypothesis is a statement of
“no change from the traditional value,” “no
effect,” “no difference,” or “no relationship.”
 For a claim to be a testable null hypothesis, it
must specify a value for some population
parameter that can form the basis for
assuming a sampling distribution for a test
statistic.

TERMS
Alternative hypothesis—The alternative
hypothesis proposes what we should conclude
if we find the null hypothesis to be unlikely.
 Two-sided alternative–An alternative hypothesis
is two-sided H A : p  p 0  when we are interested
in deviations in either direction away from the
hypothesized parameter value.


TERMS

One-sided alternative– An alternative
hypothesis is one-sided (e.g., HA: p > p0 or
HA: p < p0) when we are interested in deviations
in only one direction away from the
hypothesized parameter value.
TERMS
P-value– The probability of observing a value for a
test statistic at least as far from the hypothesized
value as the statistic value actually observed if the
null hypothesis is true.
 A small P-value indicates either that the
observation is improbable or that the probability
calculation was based on incorrect assumptions.
 The assumed truth of the null hypothesis is the
assumption under suspicion.

TERMS

One-proportion z-test– A test of the null
hypothesis that the proportion of a single
sample equals a specified value (H0: p = p0) by
pˆ  p
referring the statistic z 
to a Standard
SD  pˆ 
Normal model.
0

HYPOTHESIS TESTING
A hypothesis proposes a model for the world.
 Check the data
 Is the data consistent with the model?

WHAT IS A HYPOTHESIS?
A hypothesis is like a jury trial.
 The jury starts by assuming that the person is
innocent.
 The jury then needs to prove the person guilt
beyond a reasonable doubt.
 Then and only then can the jury reject the
hypothesis of innocence and declare the
person guilty.

HYPOTHESIS TESTING
Statistics is similar except that we quantify the
level of doubt.
 If the data is surprising, but we believe it is
trustworthy, then we doubt our hypothesis.

HYPOTHESIS TESTING
What do we mean by surprising?
 An event that has a low probability of occurring
is a surprise by definition.
 We can look at the probability that the event
could have happened by chance.
 The probability quantifies how surprised we
are.
 This is the P-value.

HYPOTHESIS TESTING

Start by assuming the hypothesis is true.
The null hypothesis denoted by H0, specifies
the population model parameter of interest and
proposes a value for that parameter.
 Written: H0: parameter = hypothesized value.
 The values comes from the Who and What of
the data.

HYPOTHESIS TESTING– COURTROOM EXAMPLE
We are still in the jury trial.
 Suppose the defendant is accused of robbery.
 The data is gathered when the lawyers present
evidence either for or against the defendant.
 The jury has to use “hypothesis testing” to
determine if the defendant is guilty beyond a
reasonable doubt.
 The jury has to determine the degree to which
the evidence contradicts the presumption of
innocence.

GUILTY VS. INNOCENT
A jury is given the duty to decide the level of
innocence if any.
 Upon looking at the evidence, they may decide
that the person is “not guilty.”
 They did not say he was “innocent.”
 Not guilty means there was not enough
evidence to prove him guilty.
 So we “failed to reject” the hypothesis.

REASONING OF HYPOTHESIS TESTING
Hypothesis tests follow a carefully structured
path.
 There are four sections that need to be dealt
with.
 Hypothesis
 Model
 Mechanics
 Conclusion

REASONING OF HYPOTHESIS TESTING
Hypotheses—
 State the null hypothesis—

 Translate
the question into a statement with
parameters so that it can be tested.
 H0: parameter = hypothesized value

State the alternative hypothesis
 Contains
the values of the parameter we accept if
we reject the null hypothesis
 HA represents the alternative hypothesis
REASONING OF HYPOTHESIS TESTING
Model—
 To plan a statistical hypothesis test, specify the
model you will use to test the null hypothesis and
the parameter of interest.
 Models require an assumption so you will need to
state them and check any corresponding
conditions.
 The model should end with a statement like:
Because the conditions are satisfied, I can model
the sampling distribution with a Normal model.

REASONING OF HYPOTHESIS TESTING
Models—
 This is the place you run your tests
 There are many tests that can be run:

 One-proportion
z-test
 One-proportion t-test
 Two-proportion z-test
 Two-proportion t-test
 Chi-squared  test
2
REASONING OF HYPOTHESIS TESTING
The test about proportions is called a oneproportion z-test.
 The conditions for the one-proportion z-test are
the same for the one-proportion z-interval.
 We test the hypothesis H0: p = p0 using the
statistic
 pˆ  p 0 

z

SD  pˆ 
REASONING OF HYPOTHESIS TESTING

We use the hypothesized proportion to find the
standard deviation,
p 0q 0
ˆ
SD  p  

.
n
When the conditions are met and the null
hypothesis is true, the static follows the Normal

model, so we can use the model to obtain a Pvalue.
REASONING OF HYPOTHESIS TESTING
Mechanics—
 This is the section we use our calculations of
our test statistic from the data.
 Different tests require different formulas.
 The ultimate calculation is to obtain a P-value–
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 will reject the
null hypothesis.

REASONING OF HYPOTHESIS TESTING
Conclusion—
 This is a statement about the null hypothesis.
 The conclusion must state either that we reject
or that we fail to reject the null hypothesis.
 The conclusion is always stated in context.

ALTERNATIVES
An alternative hypothesis is known as a twosided alternative.
 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 on either direction from
the null hypothesis value.

ALTERNATIVES
How do you determine if you need a two-sided
test?
 Look at the W’s specifically the Why of the
study.
 The way the null hypothesis is stated is will
determine if you are doing a one-sided or twosided test.

WHAT CAN GO WRONG?
Don’t base your null hypothesis on what you
see in the data.
 Don’t base your alternative hypothesis on the
data, either.
 Don’t make your null hypothesis what you want
to show to be true.
 Don’t forget to check your conditions.

WHAT WE HAVE LEARNED
We start with a null hypothesis specifying the
parameter of a model we will test using our
data.
 Our alternative hypothesis can be one- or twosided, depending on what we want to learn.
 We must check the appropriate assumptions
and conditions before proceeding with our test.
 If the data are out of line with the null
hypothesis model, the P-value will be small and
we will reject the null hypothesis.

WHAT WE HAVE LEARNED
If the data are consistent with the null
hypothesis model, the P-value will be small and
we will reject the null hypothesis.
 We must always state our conclusion in the
context of the original question.


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