STA 291Fall 2009

Lecture 18Dustin Lueker

A way of statistically testing a hypothesis by comparing the data to values predicted by the hypothesis◦ Data that fall far from the predicted values

provide evidence against the hypothesis

Significance Test

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1. State a hypothesis that you would like to find evidence against

2. Get data and calculate a statistic1. Sample mean2. Sample proportion

3. Hypothesis determines the sampling distribution of our statistic

4. If the sample value is very unreasonable given our initial hypothesis, then we conclude that the hypothesis is wrong

Logical Procedure

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Assumptions◦ Type of data, population distribution, sample size

Hypotheses◦ Null hypothesis

H0

◦ Alternative hypothesis H1

Test Statistic◦ Compares point estimate to parameter value under the null hypothesis

P-value◦ Uses the sampling distribution to quantify evidence against null hypothesis◦ Small p-value is more contradictory

Conclusion◦ Report p-value◦ Make formal rejection decision (optional)

Useful for those that are not familiar with hypothesis testing

Elements of a Significance Test

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How unusual is the observed test statistic when the null hypothesis is assumed true?◦ The p-value is the probability, assuming that the

null hypothesis is true, that the test statistic takes values at least as contradictory to the null hypothesis as the value actually observed The smaller the p-value, the more strongly the data

contradicts the null hypothesis

P-value

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In addition to reporting the p-value, sometimes a formal decision is made about rejecting or not rejecting the null hypothesis◦ Most studies require small p-values like p<.05 or

p<.01 as significant evidence against the null hypothesis “The results are significant at the 5% level”

α=.05

Conclusion

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p-value<.01◦ Highly significant

“Overwhelming evidence” .01<p-value<.05

◦ Significant “Strong evidence”

.05<p-value<.1◦ Not Significant

“Weak evidence p-value>.1

◦ Not Significant “No evidence”

Whether or not a p-value is considered significant typically depends on the discipline that is conducting the study

P-values and Their Significance

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Significance level◦ Alpha level

α Number such that one rejects the null hypothesis if

the p-values is less than it Most common are .05 and .01

◦ Needs to be chosen before analyzing the data Why?

Terminology

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Type I and Type II Errors

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Decision

Reject H0

Do Not Reject H0

Condition of H0

TrueType I Error

Correct

False CorrectType II Error

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α=probability of Type I error β=probability of Type II error Power=1-β

◦ The smaller the probability of Type I error, the larger the probability of Type II error and the smaller the power If you ask for very strong evidence to reject the null

hypothesis (very small α), it is more likely that you fail to detect a real difference

In reality, α is specified, and the probability of Type II error could be calculated, but the calculations are often difficult

Type I and Type II Errors

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In a criminal trial someone is assumed innocent until proven guilty◦ What type of error (in terms of hypothesis testing)

would be made if an innocent person is found guilty?◦ What type of error would be made if a guilty person

is found not guilty?◦ What does the Power represent (1-β)?

Also, the reason we only do not reject H0 instead of saying that we accept H0 is because of the way our hypothesis tests are set up Just like in a criminal trial someone is found not guilty

instead of innocent

Example

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If the consequences of a Type I error are very serious, then α should be small◦ Criminal trial example

In exploratory research, often a larger probability of Type I error is acceptable

If the sample size increases, both error probabilities decrease

How to choose α?

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Which area of study would be most likely to use a very small level of significance?◦ Social Sciences◦ Medical◦ Physical Sciences

How to choose α?

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H0: p=p0

◦ p0 is the value we are testing against

H1: p≠p0

◦ Most common alternative hypothesis This is called a two-sided hypothesis since it includes

values falling on two sides of the null hypothesis (above and below)

Hypotheses

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The z-score has a standard normal distribution

◦ The z-score measures how many estimated standard errors the sample proportion falls from the hypothesized population proportion

The farther the sample proportion falls from p0 the larger the absolute value of the z test statistic, and the stronger the evidence against the null hypothesis ◦ Sample size restrictions

Test Statistic

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npp

ppz

)1(

ˆ

00

0

5

5

0

0

nq

np

Let p denote the proportion of Floridians who think that government environmental regulations are too strict

A telephone poll of 824 people conducted in June 1995 revealed that 26.6% said regulations were too strict◦ Test H0: p=.5 at α=.05◦ Calculate the test statistic◦ Find the p-value and interpret

Example

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npp

ppz

)1(

ˆ

00

0

Has the advantage that different test results from different tests can be compared◦ Always a number between 0 and 1, no matter why

type of data is being examined Probability that a standard normal

distribution takes values more extreme than the observed z-score

The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis

P-value

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The research hypothesis is usually the alternative hypothesis (H1 or HA)◦ The alternative is the hypothesis that we want to

prove by rejecting the null hypothesis Assume that we want to prove that μ is

larger than a particular number μ0 ◦ We need a one-sided test with hypotheses

Null hypothesis can also be written with an equals sign

One-Sided Significance Tests

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01

00

:

:

ppH

ppH

01

00

:

:

ppH

ppH

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For a large sample test of the hypothesis the z test statistic equals 1.04

◦ Find the p-value and interpret◦ Suppose z=2.5 rather than 1.04, find the p-value

Does this provide stronger or weaker evidence against the null hypothesis?

◦ Now consider the one-sided alternative Find the p-value and interpret

For one-sided tests, the calculation of the p-value is different “Everything at least as extreme as the observed value” is

everything above the observed value in this case Notice the alternative hypothesis

Example

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0:

0:

1

0

pH

pH

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0:

0:

1

0

pH

pH

Two sided tests are more common in practice

Look for formulations like◦ “test whether the mean has changed”◦ “test whether the mean has increased”◦ “test whether the mean is the same”◦ “test whether the mean has decreased”

One-Sided vs. Two-Sided Test

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If someone wanted to test to see if the average miles a social worker drives in a month was at least 2000 miles, what would H1 be? H0? 1. μ<20002. μ≤20003. μ≠20004. μ≥20005. μ>20006. μ=2000

Example

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