agresti/franklin statistics, 1 of 122 chapter 8 statistical inference: significance tests about...

Agresti/Franklin Statistics, 1 of 122

Chapter 8Statistical inference: Significance Tests

About Hypotheses

Learn ….

To use an inferential method called

a Significance Test

To analyze evidence that data provide

To make decisions based on data


Two Major Methods for Making Statistical Inferences about a Population

Confidence Interval

Significance Test


Questions that Significance Tests Attempt to Answer

Does a proposed diet truly result in weight loss, on the average?

Is there evidence of discrimination against women in promotion decisions?

Does one advertising method result in better sales, on the average, than another advertising method?


Section 8.1

What Are the Steps For Performing a Significance

Test?


Hypothesis A hypothesis is a statement about a

population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values

The main goal in many research studies is to check whether the data support certain hypotheses


Significance Test

A significance test is a method of using data to summarize the evidence about a hypothesis

A significance test about a hypothesis has five steps


Step 1: Assumptions

A (significance) test assumes that the data production used randomization

Other assumptions may include:• Assumptions about the sample size

• Assumptions about the shape of the population distribution


Step 2: Hypotheses

Each significance test has two hypotheses:

• The null hypothesis is a statement that the parameter takes a particular value

• The alternative hypothesis states that the parameter falls in some alternative range of values


Null and Alternative Hypotheses

The value in the null hypothesis usually represents no effect

• The symbol Ho denotes null hypothesis

The value in the alternative hypothesis usually represents an effect of some type

• The symbol Ha denotes alternative hypothesis


Null and Alternative Hypotheses A null hypothesis has a single

parameter value, such as Ho: p = 1/3

An alternative hypothesis has a range of values that are alternatives to the one in Ho such as

• Ha: p ≠ 1/3 or

• Ha: p > 1/3 or

• Ha: p < 1/3


Step 3: Test Statistic

The parameter to which the hypotheses refer has a point estimate: the sample statistic

A test statistic describes how far that estimate (the sample statistic) falls from the parameter value given in the null hypothesis


Step 4: P-value To interpret a test statistic value, we use a

probability summary of the evidence against the null hypothesis, Ho

• First, we presume that Ho is true

• Next, we consider the sampling distribution from which the test statistic comes

• We summarize how far out in the tail of this sampling distribution the test statistic falls


Step 4: P-value

We summarize how far out in the tail the test statistic falls by the tail probability of that value and values even more extreme

• This probability is called a P-value

• The smaller the P-value, the stronger the evidence is against Ho


Step 4: P-value


Step 4: P-value

The P-value is the probability that the test statistic equals the observed value or a value even more extreme

It is calculated by presuming that the null hypothesis H is true


Step 5: Conclusion

The conclusion of a significance test reports the P-value and interprets what it says about the question that motivated the test


Summary: The Five Steps of a Significance Test

1. Assumptions

2. Hypotheses

3. Test Statistic

4. P-value

5. Conclusion


Is the Statement a Null Hypothesis or an Alternative Hypothesis?

In Canada, the proportion of adults who favor legalize gambling is 0.50.

a. Null Hypothesis

b. Alternative Hypothesis


Is the Statement a Null Hypothesis or an Alternative Hypothesis?

The proportion of all Canadian college students who are regular smokers is less than 0.24, the value it was ten years ago.

a. Null Hypothesis

b. Alternative Hypothesis


Section 8.4

Decisions and Types of Errors in Significance Tests


Type I and Type II Errors

When H0 is true, a Type I Error occurs when H0 is rejected

When H0 is false, a Type II Error occurs when H0 is not rejected


Significance Test Results


An Analogy: Decision Errors in a Legal Trial


P(Type I Error) = Significance Level α

Suppose H0 is true. The probability of rejecting H0, thereby committing a Type I error, equals the significance level, α, for the test.


P(Type I Error)

We can control the probability of a Type I error by our choice of the significance level

The more serious the consequences of a Type I error, the smaller α should be


Type I and Type II Errors

As P(Type I Error) goes Down, P(Type II Error) goes Up

• The two probabilities are inversely related


A significance test about a proportion is conducted using a significance level of 0.05.

The test statistic is 2.58. The P-value is 0.01. If Ho is true, for what probability of a Type I error was the test designed?

a. .01

b. .05

c. 2.58

d. .02


A significance test about a proportion is conducted using a significance level of 0.05.

The test statistic is 2.58. The P-value is 0.01. If this test resulted in a decision error, what type of error was it?

a. Type I

b. Type II

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