Chapter 5: Introduction to Hypothesis Testing and Inferential Statistics

The Two Branches of Statistical Methods

Descriptive Statistics

Descriptive statistics are used to summarize and organize data collected from a research study. They provide a way to present large amounts of information in a manageable form.

• Definition: Descriptive statistics involve methods for organizing, displaying, and describing data using tables, graphs, and summary measures such as mean, median, and mode.

• Example: Calculating the average score of students taught by Method A and Method B.

• Application: Used to present the central tendency, variability, and distribution of data.

Inferential Statistics

Inferential statistics allow researchers to draw conclusions and make inferences that go beyond the data collected in a study. They help determine whether observed differences or relationships are likely to be genuine or due to chance.

• Definition: Inferential statistics use sample data to make generalizations about a population.

• Example: Determining if the difference in average scores between Method A and Method B is statistically significant.

• Application: Used in hypothesis testing, estimation, and prediction.

Descriptive Statistics: Organize and Simplify

Sample Data Comparison

Descriptive statistics help organize and simplify data from different groups. For example, scores from students taught by two different methods can be summarized using averages and visualized with histograms.

• Method A: Average = 76

• Method B: Average = 71

• Visualization: Histograms can show the distribution of scores for each method.

Inferential Statistics: Drawing Conclusions

Interpreting Sample Differences

When a difference is observed between groups, inferential statistics help determine whether this difference is real or due to sampling error.

• Key Point: There are two possible interpretations for sample differences:

  1. No real difference exists; the observed difference is due to chance (sampling error).

  2. A real difference exists; the sample data accurately reflect this difference.

• Goal: Inferential statistics help researchers decide between these interpretations.

Hypothesis Testing

Definition and Purpose

Hypothesis testing is a systematic procedure for deciding whether the results of a research study support a hypothesis that applies to a population.

• Hypothesis: A prediction intended to be tested in a research study.

• Theory: A set of principles that explains facts, relationships, or events, and gives rise to specific hypotheses.

• Application: Used to test the effect of an experimental procedure or intervention.

Core Logic of Hypothesis Testing

The core logic involves considering the probability that the result of a study could have occurred if the experimental procedure had no effect.

• If this probability is low, the scenario of no effect is rejected, and the theory behind the experimental procedure is supported.

• In behavioral and social sciences, support for the research hypothesis is determined by how unlikely it is that there is no effect.

The Hypothesis-Testing Process

Five Steps of Hypothesis Testing

Hypothesis testing follows a structured process:

1. Restate the question as a research hypothesis and a null hypothesis about the populations.

2. Determine the characteristics of the comparison distribution.

3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.

4. Determine your sample's score on the comparison distribution.

5. Decide whether to reject the null hypothesis.

Example: Hypothesis Testing in Practice

A psychologist tests a training program to reduce stress in childless men who marry women with teenage children. Previous research shows a mean stress level of 85 (SD = 15). After the program, one man's stress level is 60. Using a significance level of 0.05, the researcher must decide if the program was effective.

• Independent variable: The training program

• Dependent variable: Stress level

Step 1: State Hypotheses

• Null hypothesis ($H_0$): There is no difference in stress level between the population and the individual who underwent the training program.

• Research hypothesis ($H_1$): There is a difference in stress level between the population and the individual who underwent the training program.

Step 2: Determine Characteristics of the Comparison Distribution

• The comparison distribution is the normal curve.

• Greek letters are used: $\mu$ (mean), $\sigma$ (standard deviation).

• For this example: $\mu = 85$, $\sigma = 15$

Step 3: Determine Cutoff Sample Score (Critical Value)

• The cutoff sample score is the critical value against which the study results are compared.

• Represents how extreme a sample score must be to reject the null hypothesis.

• Conventional levels of significance: $p < .05$, $p < .01$

One-Tailed and Two-Tailed Hypothesis Tests

Directional vs. Nondirectional Hypotheses

• Directional hypothesis: Predicts a specific direction of effect (e.g., scores will be higher).

• One-tailed test: Used for directional hypotheses; rejection region is in one tail of the distribution.

• Nondirectional hypothesis: Predicts an effect but not the direction.

• Two-tailed test: Used for nondirectional hypotheses; rejection regions are in both tails of the distribution.

Note: In psychology research, hypothesis testing is typically two-tailed (nondirectional).

Determining Cutoff Points with Two-Tailed Tests

• Divide the significance level between the two tails.

• For $p = .05$, the null hypothesis is rejected if the sample score is in the top 2.5% or bottom 2.5% of the distribution.

• Critical value for $p = .05$ is $z = \pm 1.96$

• Critical value for $p = .01$ is $z = \pm 2.58$

Decision Errors in Hypothesis Testing

Type I and Type II Errors

Even with correct calculations, hypothesis testing can lead to incorrect conclusions due to the probabilistic nature of inference.

Error Type	Description	Probability
Type I Error	Rejecting the null hypothesis when it is actually true (false positive)	Equal to the significance level ($\alpha$), e.g., $p < .05$
Type II Error	Failing to reject the null hypothesis when the research hypothesis is true (false negative)	Can be reduced by increasing the significance level (e.g., $p < .10$), but this increases Type I error

• Type I Error: Finding an effect when none exists; serious because it can lead to false theories or ineffective programs.

• Type II Error: Missing a real effect; can be reduced by using a more lenient significance level, but this increases the risk of Type I error.

• Compromise: Standard significance levels ($p < .05$, $p < .01$) are used to balance the risks.

Reporting Hypothesis Tests in Research Articles

Standard Practices

• Researchers report whether results are statistically significant.

• Symbols for statistical methods (e.g., t, F, z) are provided.

• Significance level is indicated (e.g., $p < .05$ or $p < .01$).

• Exact p-values may be reported.

• Researchers specify whether a one-tailed or two-tailed test was used.

• Null hypothesis is explicitly stated.