Topic 13

PSYC 220 - Psychological Statistics

Topic 13: One Sample t Test and Confidence Intervals

Key Concepts

  • Population: Total group from which a sample is drawn.

    • Population Size ( extbf{N}): 10,000

  • Sample: A smaller group taken from the population for analysis.

    • Sample Size ( extbf{n}): 6

  • Sampling Parameters:

    • Population Mean: μ

    • Sample Mean: 

    • Sampling Error: The difference between the sample mean and the population mean.

Research Focus

  • Interest in prenatal alcohol exposure:

    • Study on the effect of prenatal alcohol exposure on the birth weight of rats.

    • Population Mean Birth Weight of Normal Rats: 18 grams

    • Population Standard Deviation: 4 grams

    • Sample Size of Rats with Prenatal Alcohol Exposure, extbf{n}: 16

    • Sample Mean Birth Weight: 15 grams

  • Questions: Does alcohol affect birth weight?

Approaches to Analyze Data

  • Approach #1: Conduct a hypothesis test using significance level (α) = 0.05.

  • Approach #2: Compute the 95% confidence interval for the mean birth weight of rats exposed to prenatal alcohol.

Review of Hypothesis Testing and Confidence Intervals under Z Statistic

  • Hypothesis Testing:

    • Null Hypothesis (H0): The sample mean is equal to the population mean (e.g., μ = 18 grams).

    • Alternative Hypothesis (H1): The sample mean is not equal to the population mean (e.g., μ ≠ 18 grams).

  • Confidence Interval Formula:

    • For a sample mean:  ± (Z * (σ/√n))

    • Where:  = sample mean, Z = z-score based on confidence level, σ = population standard deviation, n = sample size.

  • Effect Size Calculation:

    • Effect Size (Cohen's d):
      ext{Effect Size} = rac{( - ext{Population Mean})}{ ext{Standard Deviation}}

    • Determine if extEffectSizeext{Effect Size} is significant using confidence intervals based on established limits.

  • Example Limits:

    • Lower Limit: 13.04 grams

    • Upper Limit: 16.96 grams

Limitations of Z Statistic

  • Z-Score Requirements: Requires knowledge of population standard deviation (σ), which may not be available.

  • More commonly, researchers have sample data available, which leads to the use of sample standard deviation (s).

Transitioning to t Tests

  • Use a t-test when σ is unknown:

    • Switch from Z to t distribution.

    • New Formula for Confidence Interval:
      ext{CI} =  ext{ ± } (t ext{ * } (s/ ext{√n}))

    • Where: s = sample standard deviation.

  • Checking t distribution using t tables rather than Z tables for hypothesis tests.

Application of t Distribution

  • Determining which test (Z or t):

    • Population Mean Birth Weight: 18 grams, Sample Mean Birth Weight (n=16): 15 grams.

    • Conducting hypothesis tests and calculating confidence intervals, α = 0.05.

  • Additional Example of Hypothesis Testing:

    • Mean Absences in Local School District: 8.45 days/year.

    • Sample Mean (Vocational Training Program): 6.79 days/year; Sample Standard Deviation: 2.56 days/year.

    • Analyze if absence rate is significantly different from the local average.

General Steps for t Test Hypothesis Testing

  1. State the Hypotheses: Formulate H0 and H1.

  2. Criterion for Decision: Identify the critical t value based on the chosen significance level and degrees of freedom (df).

  3. Collect Data: Calculate sample statistics using t formulas:
    t = rac{( - ext{Population Mean})}{s/ ext{√n}}

  4. Statistical Decision: Compare calculated t-statistic with critical values to reject or fail to reject H0.

  5. Report Effect Size: Calculate effect size based on measure of interest.

Confidence Intervals for t Distribution

  1. Decide the Level of Confidence (e.g., 95%).

  2. Find Critical Values based on df.

  3. Compute Standard Error:
    SE=s/ext√nSE = s/ ext{√n}

  4. Margin of Error Calculation:
    extMarginofError=text∗SEext{Margin of Error} = t ext{ * } SE

  5. Compute Limits: Add and subtract the margin of error from the sample mean.

  6. Construct and Interpret CI.

Historical Context: Student's t Distribution

  • Introduced by William Sealy Gosset (pseudonym: "Student") employed at Guinness Brewery.

  • Originated to enable hypothesis testing with small sample sizes and unknown population parameters.

Properties of the t Distribution

  • T-distribution approximates shape of the normal distribution but is flatter and more spread out, thereby accommodating more variability (