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