1. Overview of Statistical Hypothesis Testing: The z-Test
Chapter 7
Overview of Statistical Hypothesis Testing: The z-Test

2. 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

3. Going Forward How to interpret significant and nonsignificant results
What Type I errors, Type II errors, and power are

4. The Role of Inferential Statistics in Research

5. Inferential Statistics
Inferential statistics are used to decide whether sample data represent a particular relationship in the population.

6. 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

7. Nonparametric Procedures
Nonparametric statistics are inferential procedures not requiring stringent assumptions about the populations being represented.

8. Setting up Inferential Procedures

9. Experimental Hypotheses
Experimental hypotheses describe the possible outcomes of a study.

10. 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

11. 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.

12. Alternative Hypothesis
The alternative hypothesis (Ha) describes the population parameters the sample data represent if the predicted relationship exists in nature.

13. Null Hypothesis
The null hypothesis (H0) describes the population parameters the sample data represent if the predicted relationship does not exist in nature.

14. A Graph Showing the Existence of a Relationship

15. 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

16. A Graph Showing a Relationship Does Not Exist

17. Performing the z-Test

18. The z-Test
The z-test is the procedure for computing a z-score for a sample mean on the sampling distribution of means.

19. 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

20. 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

21. 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

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

23. Computing z The z-score is computed using the same formula as before
where

24. 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

25. Interpreting Significant and Nonsignificant Results

26. 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.

27. 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.

28. 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

29. 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

30. The One-Tailed Test

31. 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:

32. 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:

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

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

35. 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.

36. Errors in Statistical Decision Making

37. 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

38. 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

39. 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)

40. 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

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

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