Hypothesis Testing and Inferential Statistics

Part 2: Hypothesis Testing & Inferential Statistics Psychology 272 Spring 2025

Dedication

• Dedicated to the memory of Dr. David O. Belcher (1957 - 2018)

Objectives

• Understand various hypotheses: null hypothesis and research hypothesis

• Distinguish between one-tailed and two-tailed tests, knowing when to use each

• Recognize potential errors in hypothesis testing

• Comprehend what is meant by statistical significance

• Understand and calculate sampling distributions and standard error of the mean

• Perform and interpret one-tailed and two-tailed z-tests and t-tests

• Calculate confidence intervals

• Explore assumptions behind z-tests and t-tests

Introduction

• Emphasis on methodical thinking in hypothesis testing

• Overview of logical progression in inferential statistics

Conceptual Framework

• Hypothesis Testing: A systematic method used in inferential statistics to test assumptions (hypotheses) about a population based on sample data.

The Null Hypothesis and Research Hypothesis

• The null hypothesis (H₀) posits no difference or effect - typically formulated as “There is no difference between…”

  • Statistical Notation: H₀: μ1 = μ2 (mean of group 1 equals mean of group 2)

• The research hypothesis (H₁) is contrary to the null and reflects the researcher's prediction - posits there is a difference.

  • Statistical Notation: H₁: μ1 ≠ μ2 (mean of group 1 is not equal to mean of group 2)

Examples of Hypotheses

1. NCCPA Score Example:

   • Null hypothesis: H₀: "No difference between my NCCPA score (59.5) and the national average (55.6)."

   • Research hypothesis: H₁: "There is a difference between my NCCPA score and the national average."

2. IQ Test Example:

   • Null hypothesis: H₀: "Average IQ of my Psych 272 students is not significantly higher than WCU average (100)."

   • Research hypothesis: H₁: "Average IQ of my Psych 272 students is significantly higher than WCU average."

3. After-School Program Example:

   • Null hypothesis: H₀: "Children attending programs have the same IQ as those who do not."

   • Research hypothesis: H₁: "Children attending programs have higher IQs than those who do not."

Karl Popper's Principle of Falsification

• Emphasizes that one cannot prove a hypothesis true, only falsifiable.

• Propose an opposite hypothesis to what one believes to be true to disprove it.

Testing Hypotheses: The Logic of Hypothesis Testing

Steps in Hypothesis Testing

1. State the null hypothesis (H₀)

2. Test the null hypothesis: Collect data and conduct statistical tests.

3. Make a conclusion about H₀: Determine if H₀ is rejected or not.

4. Decision-making: Based on the conclusion, decide on the validity of the hypothesis.

Errors in Hypothesis Testing

• Errors that can occur during testing of hypotheses include:

  • Type I Error (α): Rejecting a true null hypothesis (false positive).

  • Example: Incorrectly concluding treatment effects exist when they do not.

  • Type II Error (β): Failing to reject a false null hypothesis (false negative).

  • Example: Failing to find a significant treatment effect that is present.

Statistical Significance

• Defined in terms of the probability of observing the results if the null hypothesis is true.

• Common significance levels:
  1.
  α = .05 - 5% risk of Type I error - threshold to reject null hypothesis should be met or exceeded.

• A significant p-value (p < α) indicates a statistically significant difference.

Single-Sample Research

1. Single-group design compared to the population.

   • Minimum sample size suggested: 30 participants.

2. Types of single-sample inferential statistics:

   • z-test

   • t-test

   • chi-square goodness-of-fit test

z-Test

• Used when population standard deviation (σ) is known to determine if a sample differs significantly from the population.

• Formula for z-score: $z = \frac{\bar{X} - \mu}{\frac{σ}{\sqrt{N}}}$

  • Where\bar{X} is the sample mean; µ is the population mean.

Central Limit Theorem

• States that the distribution of sample means approaches normality as sample size increases (N).

• The standard error of the mean (SEM) is calculated as:
  $SEM = \frac{σ}{\sqrt{N}}$

Conducting a One-Tailed z-Test

1. Construct hypotheses:

   • H₀: No difference in IQ.

   • H₁: Students in programs have higher IQ.

2. Critical values and areas of rejection:

   • Using p-value to determine if observed z > critical value

3. Decision based on results:

   • If observed z exceeds critical value, reject H₀.

Example of One-Tailed z-Test

1. Sample of 75 students; mean IQ = 103.5; population mean = 100; σ = 15.

2. Determine critical value for α = 0.05: critical z-value determined from z-table is 1.645.

3. Observed z value is 2.02:

   • Since 2.02 > 1.645, we reject H₀.

   • Conclude significant difference exists.

Conducting a Two-Tailed z-Test

• No directionality specified, just determining if there is a difference.

• Critical values determined for both lower and upper tails, e.g., ±1.96 for α = 0.05 when split (2.5% each tail).

Understanding Statistical Power

• Power: Probability of correctly rejecting a false H₀.

• Increased sample size can enhance power without raising Type I error risk.

• Balancing α and β is essential in testing: lowering α increases β.

Confidence Intervals

• A range that estimates the population mean (μ) within a specified confidence level (e.g., 95% or 99%).

• Confidence Interval (CI) formula: $CI = \bar{X} \pm Z_{α/2} \cdot SE$

  • Where Z is the critical z-value for the CI level.

t-Test

• Used when population standard deviation is unknown; t-distributions depend on sample size.

• In calculations, t is not perfectly normal but approximates z as N increases.

• The t-test formula: $t = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{N}}}$

  • Where s = sample standard deviation.

Degrees of Freedom (df)

• Represent the number of independent values that can vary in a statistical calculation; for sample size N, df = N - 1.

Running a One-Sample t-Test

1. Set hypotheses:

   • H₀: μ ≤ μ₀ (population mean)

   • H₁: μ > μ₀ (expected to be higher)

2. Collect sample data (N = 10) and calculate t-value.

3. Determine critical t from t-table based on df and α.

4. Conclusion based on comparison of observed t with critical t-value.

Conclusion of the t-Test

• Results indicates whether to reject or accept H₀.

SPSS Statistical Analysis

• Instructions for conducting one-sample t-tests using SPSS.

• Procedure includes selecting variables and specifying a population mean to test against.

References

• Jackson, S.L. (2016). Research methods and statistics: A critical thinking approach (5th ed.). Belmont, CA: Cengage Learning.

• Popper, K. R. (1963). Conjectures and refutations. London: Routledge and Kegan Paul.