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.