Lecture 5.1-7 significance tests for mean

Lecture Overview

• Course: Sociology/Anthropology 10B Probability and Statistics

• Professor: David Schaefer

• Institution: University of California, Irvine

• Copyright: David Schaefer 2022

Week's Focus

• Similar to last week: Calculate t-score for a sample

• Different goal: Steps for calculations vary

• Emphasis on understanding logic behind each step

Introduction to Significance Tests

• Reading Assignment: Chapter 6 (Pages 139 – 142)

Overview of Topics

• Inference: Estimation and Testing

• Logic of Significance Tests

• Hypotheses:

  • Null Hypothesis (H0)

  • Alternative Hypothesis (Ha)

Estimation vs. Significance Testing

• Estimation: Identifying plausible population parameter values based on sample statistic

• Significance Testing: Evaluating if sample evidence supports or refutes a given hypothesis about a population parameter.

Role in Research

• Model/Theory: Hypotheses derived from theories and tested with empirical data.

Understanding Hypotheses

• Definition: Testable assertions about population parameters.

• Example of a Non-testable Hypothesis: "Invisible unicorns cause the Earth's rotation."

• Valid Hypotheses:

  • H0: Median household income in the US is $72,000.

  • Ha: Median household income is not $72,000.

Characteristics of Hypotheses in Social Sciences

• Typically about relationships between variables or category differences.

• Initial focus on hypotheses using one variable at a time.

Logic of Significance Tests

• Investigate if evidence supports or refutes a claim about a population parameter.

Steps for Hypotheses Testing

• Null Hypotheses (H0): Specific assertion (e.g., H0: µ = $72,000)

• Alternative Hypothesis (Ha): The parameter differs from H0 (e.g., Ha: µ ≠ $72,000)

Evaluating Hypotheses

• Sample data determines if it aligns with:

  • H0 (e.g., mean income = $72,000)

  • Ha (e.g., mean income ≠ $72,000)

Summary of Hypothesis Testing

• A hypothesis is:

  • Testable assertion about a population

  • Null: Specific value for population parameter

  • Alternative: Range of values for population parameter.

• Next step: Conduct significance test to analyze evidence supporting H0.

Steps in a Significance Test

• Five essential steps:

  1. Assumptions

  2. Hypotheses

  3. Test Statistic

  4. P-Value

  5. Conclusion and Interpretation

Assumptions Overview

• Type of data: nominal, ordinal, interval

• Sampling type: random/probability sample

• Shape of population distribution

• Sample size adequacy

Hypothesis Pairs

• Only one hypothesis survives the significance test.

• Example Null Hypothesis (H0): Mean weeks worked = 48

  • Alternative Hypothesis (Ha): Mean weeks worked ≠ 48

Test Statistics

• Calculated from sample data to evaluate null hypothesis.

• The t-score indicates distance of sample mean from hypothesized population mean.

P-Value

• Definition: Probability corresponding to the test statistic under the null hypothesis.

• Small P-value suggests evidence against H0; large P-value suggests consistency with H0.

Decision Making

• If P-value < α: Reject H0 in favor of Ha.

• If P-value ≥ α: Fail to reject H0 (never accept H0).

• Common α-levels: .05, .01, .001.

Conclusion Interpretation

• Does sample evidence support or refute H0?

• Implications of results for the research question.

Summary of the Five Steps in Significance Testing

1. Verify Assumptions

2. Set Hypotheses

3. Calculate Test Statistic

4. Determine P-Value

5. Draw Conclusions

Tables Overview

Table 6.1: Parts of a Statistical Significance Test

• Assumptions: Data type, randomization, distribution shape, sample size

• Hypotheses: Null and Alternative

• Test Statistic: Comparison against H0

• P-Value: Weight of evidence

• Conclusion: Interpretation of findings

Table 6.4: Significance Tests for Population Means

• Components including assumptions, hypotheses, test statistics, P-values, and conclusions.