handouts-of-psy516-from-lesson-19-45

Page 1: Confidence Interval for Population Mean

Confidence Interval for Population Means

  • Estimated from a sample of measurements. Requires:

    • Sample mean (X̄)

    • Standard deviation (σ)

    • Number of observations (n)

  • Sample mean does not affect width of confidence interval.

  • Normal distribution is bell-shaped.

Confidence Interval Formula

  • For standard normal distribution:

    • CI = X̄ ± Z * (σ/√n)

    • Where:

      • Z* = critical value (z-distribution)

      • σ = population standard deviation

      • n = sample size

  • For t-distribution, replace Z* with t*.

Page 2: Confidence Interval for Sample Means

The Rule for Sample Means

  • If numerous samples of the same size are taken:

    • Frequency curve of means will approximate bell-shaped.

    • Mean of sample means equals population mean.

  • Formula for standard error of mean (SEM):

    • SEM = σ/√n

95% Confidence Interval Formula

  • Uses 2 standard errors:

    • CI = Sample mean ± 2 * SEM

    • Valid for samples of n ≥ 30.

  • For smaller samples, use t-distribution multiplier greater than 2.

Page 3: Comparing Two Means

Confidence Interval for Two Means

  • Collect independent samples.

  • Calculate mean and standard deviation for each sample.

Steps

  1. Compute standard error (SEM) for each sample.

  2. Combine the SEMs:

    • Measure of variability = √[(SEM1)^2 + (SEM2)^2]

  3. CI for difference:

    • CI = (Mean1 - Mean2) ± 2 * Measure of variability

  • Valid only with independent measurements.

Page 4: Reporting Confidence Intervals

Reporting

  • Basics of reporting:

    • Confidence intervals include upper & lower bounds.

  • Example of educational levels:

    • Average for nonsmokers was higher than for smokers by 0.67 years.

Page 5: Hypothesis Testing - Introduction

Hypothesis Testing Overview

  • Common statistical procedure in research.

  • General process:

    1. State a hypothesis about a population.

    2. Predict characteristics for sample.

    3. Collect random sample.

    4. Compare sample data to prediction.

Hypothesis Explained

  • Null Hypothesis (H0): Treatment has no effect.

  • Sample should approximate population.

Page 6: Steps in Hypothesis Testing

Steps in Hypothesis Testing

  1. State the hypothesis:

    • Two opposing hypotheses: H0 (no effect) and H1 (effect).

  2. Set criteria for decision.

  3. Collect and compute sample statistics.

Decision Criteria

  • Alpha level: probability of Type I error.

  • Select alpha typically at 0.05.

Page 7: Null and Alternative Hypotheses

Null and Alternative Hypotheses

  • Null Hypothesis (H0): Treatment has no effect.

  • Alternative Hypothesis (H1): Treatment has an effect.

Representing Hypotheses

  • H0: μ = 80 (no change)

  • H1: μ ≠ 80 (change)

  • Directional tests indicate expected result.

Page 8: Collecting Data

Collecting and Analyzing Data

  • Sample chosen (e.g., 200 adults).

  • Collect data on treatment impact.

  • Compute sample mean and z-score:

    • z = (M - μ) / SEM

Decision Outcome

  • Data in critical region -> Reject H0.

  • Else, fail to reject H0.

Page 9: z-scores and Decision Making

Calculating z-scores

  • Measure of location relative to H0.

  • Use obtained sample mean to assess.

Standard Error and Decision

  • If z-score is in critical region:

    • Conclude treatment had an effect.

  • If not, conclude treatment had no significant effect.

Page 10: Decision Outcomes

Possible Outcomes

  1. Sample data in critical region -> Reject H0.

  2. Sample data not in critical region -> Fail to reject H0.

Types of Hypothesis

  • Simple Hypothesis: specifies exact parameter.

  • Composite Hypothesis: specifies range.

  • Directional and Non-directional hypothesis tests.

Page 11: Differences Between Hypotheses

One-tailed vs Two-tailed Hypothesis

  • One-tailed: specific direction specified.

  • Two-tailed: detects any difference.

Examples of Hypotheses

  • H0: Test scores not increased (treatment does not work).

  • H1: Test scores increased (treatment works).

Page 12: Determining Significance of Outcomes

Critical Regions and Outcomes

  • High-probability sample means identified.

  • Boundary lines defined by alpha level.

Understanding Alpha Level and Critical Region

  • Alpha levels (0.05, 0.01, 0.001) influence critical regions.

  • If data falls within critical region -> reject hypothesis.

Page 13: Conclusion Approaches

Summation of Hypothesis Testing

  • Decisions depend on similarities between sample data and hypotheses.

  • Outcome decisions frame the conclusions about treatments.

Page 14: Relating Data to Hypotheses

Relating Data to Hypotheses

  • Overall, make accurate predictions about population.

  • Use previous decisions for future evaluations.

  • Rely on well-structured data analysis to inform results.

Page 15: Checking Assumptions of Tests

Key Points to Remember

  • Each hypothesis test assumes certain conditions are met.

  • Failure to meet these conditions can lead to incorrect results.

Page 16: T-Tests and Other Measures

Importance of Checking Assumptions

  • Significant differences assessed based on statistical tests.

  • Include clear definitions for hypothesis tests and structures.

Page 17: Evaluating Data Outcomes

  • Evaluate frequencies along groups based on assumptions.

  • Make distinctions between matched pairs and independent samples.

Page 18: Introduction to ANOVA

Overview of ANOVA Usage

  • ANOVA is used for analyzing variance among more than two groups.

  • Helps in understanding treatment effects.

Page 19: Handling ANOVA properly

Calculating ANOVA Terms

  • Calculate terms for different effective variables to control outcomes.

  • Handle data variances and effects properly.