Chapter Ten Introduction to Hypothesis Testing. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 2 New Statistical Notation The.

Chapter Ten

Introduction to Hypothesis

Testing

Copyright © Houghton Mifflin Company. All rights reserved. Chapter 10 - 2

New Statistical Notation

• The symbol for greater than is >.

• The symbol for less than is <.

• The symbol for greater than or equal tois ≥.

• The symbol for less than or equal tois ≤.

• The symbol for not equal to is ≠.


The Role of InferentialStatistics in Research


Sampling Error

Remember:

Sampling error results when random

chance produces a sample statistic that

does not equal the population parameter

it represents.


Parametric Statistics

• Parametric statistics are procedures that require certain assumptions about the characteristics of the populations being represented. Two assumptions are common to all parametric procedures:– The population of dependent scores forms a

normal distribution

and– The scores are interval or ratio.


Nonparametric Procedures

• Nonparametric statistics are

inferential procedures that do not

require stringent assumptions about the

populations being represented.


Robust Procedures

Parametric procedures are robust. If the

data don’t meet the assumptions of the

procedure perfectly, we will have only a

negligible amount of error in the

inferences we draw.


Setting up Inferential Procedures


Experimental Hypotheses

Experimental hypotheses describe the

predicted outcome we may or may not

find in an experiment.


Predicting a Relationship

• A two-tailed test is used when we predict that there is a relationship, but do not predict the direction in which scores will change.

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


Designing a One-Sample Experiment

To perform a one-sample experiment, we

must already know the population mean

under some other condition of the

independent variable.


Alternative Hypothesis

The alternative hypothesis describes

the population parameters that the

sample data represent if the predicted

relationship exists.


Null Hypothesis

The null hypothesis describes the

population parameters that the sample

data represent if the predicted relationship

does not exist.


A Graph Showing the Existence of a Relationship


A Graph Showing That 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.


)( X

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 some other condition of the independent variable

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


Setting up for a Two-Tailed Test

1.Choose alpha. Common values are 0.05 and 0.01.

2.Locate the region of rejection. For a two-tailed test, this will involve defining an area in both tails of the sampling distribution.

3.Determine the critical value. Using the chosen alpha, find the zcrit value that gives the appropriate region of rejection.


A Sampling Distribution for H0 Showing the Region of Rejection for = 0.05 in

a Two-tailed Test


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

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

Two-Tailed Hypotheses


X

Xz

obt

NX

X

Computing z

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

where


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 that the results are too unlikely to occur if the predicted relationship does not exist in the population.


Interpreting Significant Results

• When we reject H0 and accept Ha, we do not prove that H0 is false

• While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows that such means do occur once in a while


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 that the results are likely to occur if the predicted relationship does not exist in the population.


Interpreting Nonsignificant Results

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

• Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature


1.Determine the experimental hypotheses and create the statistical hypothesis

2.Compute and compute zobt

3.Set up the sampling distribution

4.Compare zobt to zcrit

Summary of the z-Test

X


The One-Tailed Test


One-Tailed Hypotheses

• In a one-tailed test, if it is hypothesized that the independent variable causes an increase in scores, then the null hypothesis is that the population mean is less than or equal to a given value and the alternative hypothesis is that the population mean is greater than the same value. For example:– H0: ≤ 50

– Ha: > 50


A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether

Scores Increase


One-Tailed Hypotheses

• In a one-tailed test, if it is hypothesized that the independent variable causes a decrease in scores, then the null hypothesis is that the population mean is greater than or equal to a given value and the alternative hypothesis is that the population mean is less than the same value. For example:– H0: ≥ 50

– Ha: < 50


A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores

Decrease

[Insert Figure 10.8 here]


Choosing One-Tailed Versus Two-Tailed Tests

Use a one-tailed test only when confident

of the direction in which the dependent

variable scores will change. When in

doubt, use a two-tailed test.


Errors in StatisticalDecision 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 that we conclude that 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 that we conclude that the predicted relationship does not exist when it really does

• The probability of a Type II error is


Power

• The goal of research is to reject H0 when H0 is false

• The probability of rejecting H0 when it is false is called power


Possible Results of Rejecting or Retaining H0


14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

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


764.2966.0

1167.13obt

X

Xz

966.018

1.4

NX

X

Example

1.H0: = 11; Ha: ≠ 11

2.Choose = 0.05

3.Reject H0 if zobt > +1.965 or if zobt < -1.965.


Example

Since zobt lies within the rejection region,

we reject H0 and accept Ha. Therefore, we

conclude that ≠ 11.

Chapter Ten Introduction to Hypothesis Testing. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 2 New Statistical Notation The.

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