 # Overview of Statistical Hypothesis Testing: The z-Test

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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 a One-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

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 and Nonsignificant 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 Rejection for a One-tailed Test of Whether Scores Increase

A Sampling Distribution Showing the Region of Rejection for 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 13 15 11 10 12 17

Example Choose a = 0.05 Reject H0 if zobt > or if zobt <

Example Since zobt lies within the rejection region, we reject H0 and accept Ha. Therefore, we conclude m does not equal 11.