Chapter 7 OVERVIEW OF STATISTICAL HYPOTHESIS TESTING: THE z-TEST.

Chapter 7

OVERVIEW OF STATISTICAL HYPOTHESIS TESTING: THE z-TEST

Going Forward

Your goals in this chapter are to learn:• Why the possibility of sampling error causes

researchers to perform inferential statistics• When experimental hypotheses lead to

either one-tailed or a two-tailed tests• How to create the null and alternative

hypotheses• When and how to perform the z-test

Going Forward

• How to interpret significant and nonsignificant results

• What Type I errors, Type II errors, and power are

The Role of Inferential Statistics in Research

Inferential Statistics

Inferential statistics are used to decide whether

sample data represent a particular relationship

in the population.

Parametric Statistics

• Parametric statistics are inferential procedures requiring certain assumptions about the raw score population being represented by the sample

• Two assumptions are common to all parametric procedures:– The population of dependent scores should be at

least approximately normally distributed– The scores should be interval or ratio

Nonparametric Procedures

Nonparametric statistics are inferential

procedures not requiring stringent assumptions

about the populations being represented.

Setting up Inferential Procedures

Experimental Hypotheses

Experimental hypotheses describe the possible

outcomes of a study.

Predicting a Relationship

• A two-tailed test is used when we do not predict the direction in which dependent scores will change

• A one-tailed test is used when we do predict the direction in which dependent scores will change

Designing aOne-Sample Experiment

To perform a one-sample experiment, we must already know the population mean for participants tested under another condition of the independent variable.

Alternative Hypothesis

The alternative hypothesis (Ha) describes the

population parameters the sample data

represent if the predicted relationship exists in

nature.

Null Hypothesis

The null hypothesis (H0) describes the

population parameters the sample data

represent if the predicted relationship does not

exist in nature.

A Graph Showing the Existence of a Relationship

The Logic

When a relationship is indicated by the sample data, it may be because•The relationship operates in nature and it produced our data

OR•We are misled by sampling error

A Graph Showing a Relationship Does Not Exist

Performing the z-Test

The z-Test

The z-test is the procedure for computing a z-

score for a sample mean on the sampling

distribution of means.

Assumptions of the z-Test

1. We have randomly selected one sample

2. The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale

3. We know the mean of the population of raw scores under another condition of the independent variable

4. We know the true standard deviation of the population described by the null hypothesis)( X

Setting up for a Two-Tailed Test

Create the sampling distribution of means from the underlying raw score population that H0 says our sample represents

Choose the criterion, symbolized by (alpha)

Locate the region of rejection which, for a two-tailed test, involves defining an area in both tails

Determine the critical value by using the chosen to find the zcrit value resulting in the appropriate region of rejection

Two-Tailed Hypotheses

• In a two-tailed test, the null hypothesis states the population mean equals a given value. For example, H0: m = 100.

• In a two-tailed test, the alternative hypothesis states the population mean does not equal the same given value as in the null hypothesis. For example, Ha: 100.

A Sampling Distribution for H0 Showing the Region of Rejection for = 0.05 in a Two-tailed Test

Computing z

• The z-score is computed using the same formula as before

whereX

Xz

obt

NX

X

Comparing Obtained z

• In a two-tailed test, reject H0 and accept Ha if the z-score you computed is– Less than the negative of the critical z-value

OR– Greater than the positive of the critical z-value

• Otherwise, fail to reject H0

Interpreting Significant andNonsignificant Results

Rejecting H0

• When the zobt falls beyond the critical value, the statistic lies in the region of rejection, so we reject H0 and accept Ha.

• When we reject H0 and accept Ha we say the results are significant. Significant indicates the results are unlikely to occur if the predicted relationship does not exist in the population.

Failing to Reject H0

• When the zobt does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H0.

• When we fail to reject H0 we say the results are nonsignificant. Nonsignificant indicates the results are likely to occur if the predicted relationship does not exist in the population.

Nonsignificant Results

• When we fail to reject H0, we do not prove H0 is true

• Nonsignificant results provide no convincing evidence the independent variable does not work

Summary of the z-Test

1. Determine the experimental hypotheses and create the statistical hypothesis

2. Select , locate the region of rejection, and determine the critical value

3. Compute and zobt

4. Compare zobt to zcrit

X

The One-Tailed Test

One-Tailed Hypotheses

In a one-tailed test, if it is hypothesized the independent variable causes an increase in scores, then the null hypothesis states the population mean is less than or equal to a given value and the alternative hypothesis states the population mean is greater than the same value. For example:

50:

50:0

aH

H

One-Tailed Hypotheses

In a one-tailed test, if it is hypothesized the independent variable causes a decrease in scores, then the null hypothesis states the population mean is greater than or equal to a given value and the alternative hypothesis states the population mean is less than the same value. For example:

50:

50:0

aH

H

A Sampling Distribution Showing the Region of Rejectionfor a One-tailed Test of Whether Scores Increase

A Sampling Distribution Showing the Region of Rejectionfor a One-tailed Test of Whether Scores Decrease

Choosing One-Tailed Versus Two-Tailed Tests

Use a one-tailed test only when it is the

appropriate test for the independent variable.

That is, when the independent variable can

“work” only if scores go in one direction.

Errors in Statistical Decision Making

Type I Errors

• A Type I error is defined as rejecting H0 when H0 is true

• In a Type I error, there is so much sampling error we conclude the predicted relationship exists when it really does not

• The theoretical probability of a Type I error equals

Type II Errors

• A Type II error is defined as retaining H0 when H0 is false (and Ha is true)

• In a Type II error, the sample mean is so close to the described by H0 we conclude the predicted relationship does not exist when it really does

Power

Power is

• The probability of rejecting H0 when it is false

• The probability of not making a Type II error

• The probability that we will detect a relationship and correctly reject a false null hypothesis (H0)

Example

Use the following data set and conduct a two-tailed z-test to determine if = 11 and the population standard deviation is known to be 4.1

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

1. 2. Choose = 0.053. Reject H0 if zobt > +1.965 or if zobt < -1.965.

11:;11:0 aHH

764.2966.0

1167.13obt

X

Xz

966.018

1.4

NX

X

Example

Since zobt lies within the rejection region, we

reject H0 and accept Ha. Therefore, we conclude

does not equal 11.