Presentation on theme: "Overview of Statistical Hypothesis Testing: The z-Test"— Presentation transcript:
1 Overview of Statistical Hypothesis Testing: The z-Test Chapter 7Overview 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 statisticsWhen experimental hypotheses lead to either one-tailed or a two-tailed testsHow to create the null and alternative hypothesesWhen 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
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 sampleTwo assumptions are common to all parametric procedures:The population of dependent scores should be at least approximately normally distributedThe scores should be interval or ratio
7 Nonparametric Procedures Nonparametric statistics are inferential procedures not requiring stringent assumptions about the populations being represented.
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 changeA 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 HypothesisThe 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 LogicWhen a relationship is indicated by the sample data, it may be becauseThe relationship operates in nature and it produced our dataORWe are misled by sampling error
18 The z-TestThe 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 sampleThe dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scaleWe know the mean of the population of raw scores under another condition of the independent variableWe 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 representsChoose the criterion, symbolized by a (alpha)Locate the region of rejection which, for a two-tailed test, involves defining an area in both tailsDetermine 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 zIn a two-tailed test, reject H0 and accept Ha if the z-score you computed isLess than the negative of the critical z-valueORGreater than the positive of the critical z-valueOtherwise, fail to reject H0
25 Interpreting Significant and Nonsignificant Results
26 Rejecting H0When 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 H0When 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 trueNonsignificant results provide no convincing evidence the independent variable does not work
29 Summary of the z-TestDetermine the experimental hypotheses and create the statistical hypothesisSelect a, locate the region of rejection, and determine the critical valueCompute and zobtCompare zobt to zcrit
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.
37 Type I ErrorsA Type I error is defined as rejecting H0 when H0 is trueIn a Type I error, there is so much sampling error we conclude the predicted relationship exists when it really does notThe theoretical probability of a Type I error equals a
38 Type II ErrorsA 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 errorThe probability that we will detect a relationship and correctly reject a false null hypothesis (H0)
40 ExampleUse 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 4.114131511101217
41 ExampleChoose a = 0.05Reject H0 if zobt > or if zobt <
42 ExampleSince zobt lies within the rejection region, we reject H0 and accept Ha. Therefore, we conclude m does not equal 11.