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CHAPTER 8

Hypothesis Testing

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Objectives

Understand the definitions used in hypothesis testing.

State the null and alternative hypotheses.

Find critical values for the z test.

State the five steps used in hypothesis testing.

Test means for large samples using the z test.

Test means for small samples using the t test.

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Objectives (cont’d.)

Test proportions using the z test.

Test variances or standard deviations using the chi square test.

Test hypotheses using confidence intervals.

Explain the relationship between type I and type II errors and the power of a test.

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Introduction

Statistical hypothesis testing is a decision-making process for evaluating claims about a population.

In hypothesis testing, the researcher must define the population under study, state the particular hypotheses that will be investigated, give the significance level, select a sample from the population, collect the data, perform the calculations required for the statistical test, and reach a conclusion.

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Hypothesis Testing

Hypotheses concerning parameters such as means and proportions can be investigated.

The z test and the t test are used for hypothesis testing concerning means.

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Methods to Test Hypotheses

The three methods used to test hypotheses are:

1. The traditional method.

2. The P-value method.

3. The confidence interval method.

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Statement of a Hypothesis

A statistical hypothesis is a conjecture about a population parameter which may or may not be true.

There are two types of statistical hypotheses for each situation: the null hypothesis and the alternative hypothesis.

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Hypotheses

The null hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

The alternative hypothesis, symbolized by H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters.

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Hypothesis-Testing Common Phrases

Is greater than Is less than

Is increased Is decreased or reduced from

Is greater than or equal to

Is less than or equal to

Is at least Is at most

Is equal to Is not equal to

Has not changed from

Has changed from

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Design of the Study

After stating the hypotheses, the researcher’s next step is to design the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study.

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Statistical Test

A statistical test uses the data obtained from a sample to make a decision about whether or not the null hypothesis should be rejected.

The numerical value obtained from a statistical test is called the test value.

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Possible Outcomes of a Hypothesis Test

Error Type I

Error Type II

Correct Decision

Correct Decision

Do not reject H0

Reject H0

H0 True H0 False

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Summary of Possible Outcomes

A type I error occurs if one rejects the null hypothesis when it is true.

A type II error occurs if one does not reject the null hypothesis when it is false.

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Error Probabilities

The level of significance is the maximum probability of committing a type I error. This probability is symbolized by ; that is , .

The probability of a type II error is symbolized by . That is, .

P( )type I error

P( )type II error

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and Probabilities

In most hypothesis testing situations, cannot easily be computed; however, and are related in that decreasing one increases the other.

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Hypothesis Testing

In a hypothesis testing situation, the researcher decides what level of significance to use.

After a significance level is chosen, a critical value is selected from a table for the appropriate test.

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Critical Values

The critical value(s) separates the critical region from the noncritical region. The symbol for critical value is C.V.

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Critical Values (cont’d)

The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected.

The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.

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One-Tailed Test

A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean.

A one-tailed test is either right-tailed or left-tailed, depending on the direction of the inequality of the alternative hypothesis.

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Left-Tailed Test

0-z

Criticalregion

Noncriticalregion

H k

H k0

1

010 128

0 05 165

0 01 2 33

: . , .

: . , .

. , .

C. V.

C. V.

C. V.

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Right-Tailed Test

0 +z

Criticalregion

Noncriticalregion

H k

H k0

1

010 128

0 05 165

0 01 2 33

: . , .

: . , .

. , .

C. V.

C. V.

C. V.

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Two-Tailed Test

In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions.

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Two-Tailed Test (cont’d.)

0 +z

Criticalregion

Noncriticalregion

0

1

: 0.10,C.V. 1.65: 0.05,C.V. 1.96

0.01,C.V. 2.58

H kH k

0 +z-z

Criticalregion

Noncriticalregion Critical

region

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Hypothesis-Testing (Traditional Method)

Step 1 State the hypothesis, and identify the claim.

Step 2 Find the critical value from the appropriate table.

Step 3 Compute the test value.

Step 4 Make the decision to reject or not reject the null hypothesis.

Step 5 Summarize the results.

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The z Test

The z test is a statistical test for the mean of a population. It can be used when , or when the population is normally distributed and is known.

30n

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The z Test Formula

The formula for the z test is:

where = sample mean = hypothesized population mean = population standard deviation = sample size

Xz

n

X

n

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The z Test When is Unknown

The central limit theorem states that when the population standard deviation is unknown, the sample standard deviation s can be used in the formula as long as the sample size is 30 or more.

Xz

s n

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The P-value

The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.

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The P-value (cont’d.)

The P-value is the actual area under the standard normal distribution curve (or other curve depending on what statistical test is being used) representing the probability of a particular sample statistic or a more extreme sample statistic occurring if the null hypothesis is true.

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Decision Rule When Using a P-Value

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Statistical vs. Practical Significance

The researcher should distinguish between statistical significance and practical significance.

When the null hypothesis is rejected at a specific significance level, it can be concluded that the difference is probably not due to chance and thus is statistically significant. However, the results may or may not have any practical significance.

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The t Test

The t test is a statistical test of the mean of a population and is used when the population is normally or approximately normally distributed, is unknown, and the sample size is less than 30.

The formula for the t test is:

The degrees of freedom are d.f. = n–1.

Xt

s n

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z Test for a Proportion

A hypothesis test involving a population proportion can be considered as a binomial experiment when there are only two outcomes and the probability of a success does not change from trial to trial.

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Formula for the z Test for Proportions

/

population proportion

=sample size

p pz

pq n

where

Xp

np

n

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22

2

2

2

( 1)

with d.f.= -1 where sample size

= sample variance

=population variance

n s

nn

s

Chi-Square Test for Single Variance

The formula is:

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Assumptions for Chi-Square Test for Single Variance

The sample must be randomly selected from the population.

The population must be normally distributed for the variable under study.

The observations must be independent of each other.

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Confidence Intervals & Hypothesis Testing

There is a relationship between confidence intervals and hypothesis testing.

When the null hypothesis is rejected in a hypothesis testing situation, the confidence interval for the mean using the same level of significance will not contain the hypothesized mean.

Likewise, when the null hypothesis is not rejected, the confidence interval computed will contain the hypothesized mean.

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Power of a Statistical Test

The power of a test measures the sensitivity of the test to detect a real difference in parameters if one actually exists.

The higher the power, the more sensitive the test is to detecting a real difference between parameters if there is a difference.

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Summary

A statistical hypothesis is a conjecture about a population.

There are two types of statistical hypotheses: the null hypothesis states that there is no difference, and the alternative hypothesis specifies a difference.

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Summary (cont’d.)

The z test is used when the population standard deviation is known and the variable is normally distributed or when is not known and the sample size is greater than or equal to 30.

When the population standard deviation is not known and the variable is normally distributed, the sample standard deviation is used, but a t test should be conducted if the sample size is less than 30.

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Summary (cont’d.)

Researchers compute a test value from the sample data in order to decide whether the null hypothesis should or should not be rejected.

Statistical tests can be one-tailed or two-tailed, depending on the hypotheses.

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Summary (cont’d.)

The null hypothesis is rejected when the difference between the population parameter and the sample statistic is said to be significant.

The difference is significant when the test value falls in the critical region of the distribution.

The critical region is determined by , the level of significance of the test.

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Summary (cont’d.)

The significance level of a test is the probability of committing a type I error.

A type I error occurs when the null hypothesis is rejected when it is true.

The type II error can occur when the null hypothesis is not rejected when it is false.

One can test a single variance by using a chi-square test.

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Conclusions

Researchers are interested in answering many types of questions. For example:

“Will a new drug lower blood pressure?”

“Will seat belts reduce the severity of injuries caused by accidents?”

These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population.

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Homework

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