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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

- We have randomly selected one sample
- The dependent variable is at least approximately

normally distributed in the population and

involves an interval or ratio scale - We know the mean of the population of raw scores

under another condition of the independent

variable - We know the true standard deviation of the

population described by the null

hypothesis

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 a (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

a 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 m 100.

A Sampling Distribution for H0 Showing the Region

of Rejection for a 0.05 in a Two-tailed Test

Computing z

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

before - where

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

- Determine the experimental hypotheses and create

the statistical hypothesis - Select a, locate the region of rejection, and

determine the critical value - Compute and zobt
- Compare zobt to zcrit

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

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

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 a

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 m 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 m 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

- Choose a 0.05
- Reject H0 if zobt gt 1.965 or if zobt lt -1.965.

Example

- Since zobt lies within the rejection region, we

reject H0 and accept Ha. Therefore, we conclude m

does not equal 11.