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Chapter 8 Introduction to Hypothesis Testing. PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau. Chapter 8 Learning Outcomes. Concepts to review. z- Scores (Chapter 5) - PowerPoint PPT Presentation
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Chapter 8 Introduction to Hypothesis Testing
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences Seventh Edition
by Frederick J. Gravetter and Larry B. Wallnau
Chapter 8 Learning Outcomes
Concepts to review
• z-Scores (Chapter 5)
• Distribution of sample means (Chapter 7)– Expected value– Standard error– Probability and sample means
8.1 Logic of Hypothesis Testing
• Hypothesis testing is one of the most commonly used inferential procedures
• Definition: a statistical method that uses sample data to evaluate a hypothesis about a population
Logic of hypothesis test
1. State hypothesis about a population
2. Predict the characteristics of the sample based on the hypothesis
3. Obtain a random sample from the population
4. Compare the obtained sample data with the prediction made from the hypothesis
– If consistent, hypothesis is reasonable– If discrepant, hypothesis is rejected
Figure 8.1 Basic experimental situation
Figure 8.2 Unknown population in experimental situation
Four steps of Hypothesis Testing
• State the hypotheses
• Set the criteria for a decision
• Collect data and compute sample statistics
• Make a decision
Step 1: State hypotheses
• Null hypothesis (H0) states that, in the general population, there is no change, no difference, or not relationship
• Alternative hypothesis (H1) states that there is a change, a difference, or a relationship in the general population
Step 2: Set the criteria for decision
• Distribution of sample means is divided– Those likely if H0 is true
– Those very unlikely if H0 is true
• Alpha level, or level of significance, is a probability value used to define “very unlikely”
• Critical region is composed of the extreme sample values that are very unlikely
• Boundaries of critical region are determined by alpha level.
Figure 8.3 Division of distribution of sample means
Figure 8.4 Critical regions for α = .05
Learning Check
• A sports coach is investigating the impact of a new training method. In words, what would the null hypothesis say?
Learning Check - Answer
• A sports coach is investigating the impact of a new training method. In words, what would the null hypothesis say?
Learning Check
• Decide if each of the following statements is True or False.
Answer
Step 3: Collect data and Compute sample statistics
• Data collected after hypotheses stated
• Data collected after criteria for decision set
• This sequence assures objectivity
• Compute a sample statistic (z-score) to show the exact position of the sample.
Step 4: Make a decision
• If sample data are in the critical region, the null hypothesis is rejected
• If the sample data are not in the critical region, the researcher fails to reject the null hypothesis
Box 8.1: Proving the alternative hypothesis
• Seems odd to focus on null hypothesis, which we do not believe to be true.
• In logic, it is easier to demonstrate that a universal hypothesis is false than true.
Jury trial: an analogy
• Trial begins with null hypothesis (innocent until proven guilty).
• Police and prosecutor gather evidence (data) to show reject innocent plea and conclude guilt.
• If there is sufficient evidence, jury rejects innocence claim and concludes guilt.
• If there is not enough evidence, jury fails to convict (but does not conclude defendant is innocent).
Learning Check
• Decide if each of the following statements is True or False.
Answer FF
8.2 Uncertainty and Errors in Hypothesis Testing
• Hypothesis testing is an inferential process– Uses limited information to reach general
conclusion– Sample data used to draw conclusion about a
population
• Errors are possible
Type I Errors
• Researcher rejects a null hypothesis that is actually true
• Researcher concludes that a treatment has an effect when it has none
• Alpha level is the probability that a test will lead to a Type I error.
Type II Errors
• Researcher fails to reject a null hypothesis that is really false.
• Researcher has failed to detect a real treatment effect.
Table 8.1
Figure 8.5 Location of critical region boundaries
Learning Check TF
• Decide if each of the following statements is True or False.
Answer TF
8.3 Example of a Hypothesis Test
• Step 1: H0: alcohol exposure =18 (Even with alcohol exposure, the rats still average 18 grams at birth.)
• Step 2: α = .05Critical region: z beyond ±1.96
• Step 3: z = 3.00
• Step 4: Reject H0
Figure 8.6 Structure of study in example
Figure 8.7 Critical region for Example 8.2
In the Literature
• A result is significant or statistically significant if it is very unlikely to occur when the null hypothesis is true.
• In APA format– State significance– Report value of test statistic– Report alpha level or p-value
Factors influencing hypothesis test
• Size of difference between sample mean and original population mean– Appears in numerator of the z-score
• Variability of the scores – Influences size of the standard error
• Number of scores in the sample– Influences size of the standard error
Assumptions for hypothesis tests
• Random sampling
• Independent observations
• Value of standard deviation is unchanged by the treatment
• Normal sampling distribution
Figure 8.8 Rejection / critical region
Learning Check
• A researcher uses a hypothesis test to evaluate H0 µ = 80. Which combination of factors is most likely to result in rejecting the null hypothesis?
Learning Check - Answer
• A researcher uses a hypothesis test to evaluate H0 µ = 80. Which combination of factors is most likely to result in rejecting the null hypothesis?
Learning Check
• Decide if each of the following statements is True or False.
Answer
8.4 Directional (One-Tailed) Hypothesis Tests
• The procedure in Section 8.3 is two-tailed because critical region is divided on both tails of the distribution.
• Researchers usually have a specific prediction about the direction of a treatment effect before they begin.
• In a directional hypothesis or one-tailed test, the hypotheses specify an increase or decrease in the population mean
Figure 8.9 Critical region for Example 8.3
Learning Check
• A researcher is predicting that a treatment will decrease scores. If this treatment is evaluated using a directional hypothesis test, then the critical region for the test
Learning Check
• A researcher is predicting that a treatment will decrease scores. If this treatment is evaluated using a directional hypothesis test, then the critical region for the test
8.5 Concerns about Hypothesis Testing: Measuring Effect Size
• Although commonly used, some researchers are concerned about hypothesis testing– Focus of test is data, not hypothesis– Significant effects are not always substantial
• Effect size measures the absolute magnitude of a treatment effect, independent of sample size
Cohen’s d : measure of effect size
σμμ treatment notreatment
deviation standard
difference mean d sCohen'
−==
Figure 8.10 A 15-point difference in two situations
Learning Check
• Decide if each of the following statements is True or False.
Answer
8.6 Statistical Power
• The power of a test is the probability that the test will correctly reject a false null hypothesis– It will detect a treatment effect if one exists– Power = 1 – β or 1 – p (Type II error)
• Power usually computed before starting study– Requires assumptions about factors that
influence power
Figure 8.11 Demonstration of Measuring Power
Influences on Power
• As effect size increases, power also increases
• Larger sample sizes produce greater power
• Reducing the alpha level (making the test more stringent) reduces power
• Using a one-tailed test increases power
Figure 8.12 Demonstration of Power
Learning Check
• The power of a statistical test is the probability of _____.
Learning Check
• The power of a statistical test is the probability of _____.
Learning Check
• Decide if each of the following statements is True or False.
Answer