Hypothesis Testing

HYPOTHESIS TESTINGStatistics

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 when we are interested in deviations in either direction away from the hypothesized parameter value. €

HA : p ≠ p0( )

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

referring the statistic to a Standard Normal model.

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z =ˆ p − p0

SD ˆ p ( )

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

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χ 2( )

REASONING OF HYPOTHESIS TESTING The test about proportions is called a

one-proportion 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

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z =ˆ p − p0( )SD ˆ p ( )

REASONING OF HYPOTHESIS TESTING We use the hypothesized proportion to

find the standard deviation,

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 P-value.

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SD ˆ p ( ) = p0q0

n.

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

two-sided 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 two-sided 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 two-sided, 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.