Review for Exam 3: Psychological Statistics

Hypothesis Testing with t Statistic

• Application: Used when population parameters are unknown

• Standard Error: Substitute estimated standard error when population standard deviation ($ \sigma $) is unknown

• Basis: Relies on sample variance instead of population standard deviation

Formula for Single Sample t statistic

• Formula:
  t = \frac{M - \mu}{sM} where sM = \frac{s^2}{n}

• Variables:

  • $M$: Sample mean from obtained data

  • $\mu$: Population mean (hypothesized from $H_0$)

  • $s_M$: Estimated standard error computed from sample data

Characteristics of t statistic

• t statistic: Serves as an “estimated” z score

• Degrees of Freedom: Defined as $n - 1$, describing how well the t statistic represents a z-score

• t Distribution: Generally flatter and more variable than z distribution

  • The approximation of a t distribution to normal distribution improves as degrees of freedom increase

• Sample Size Effect: Larger sample sizes result in greater degrees of freedom ($ df $), improving the approximation to a normal distribution

Four Steps of Hypothesis Testing: Single Sample t-test

Step 1: State Hypotheses

• Null Hypothesis ($H_0$): Population mean will not change

• Alternative Hypothesis ($H_1$): Population mean will change

Step 2: Locate Critical Region

• Use t distribution table; requires knowledge of:

  • Alpha level

  • Whether the test is one-tailed or two-tailed

  • Degrees of freedom (df) for sample

Step 3: Calculate t Statistic

1. Calculate sample variance ($s^2$):
   s^2 = \frac{SS}{n-1}

2. Calculate estimated standard error ($sM$): sM = \frac{s^2}{n}

3. Calculate t statistic:
   t = \frac{M - \mu}{s_M}

Step 4: Make a Decision

• Compare the t statistic to critical t values to decide:

  • Reject $H_0$: If the t statistic is in the critical region

  • Fail to Reject $H_0$: If the t statistic is not in the critical region

Assumptions for Single Sample t-test

1. Independent observations in the sample

2. Population from which samples are selected must be normally distributed

   • More critical for small sample sizes

3. Variance and sample size affect the likelihood of rejecting $H_0$

4. Influences of estimated standard error, which influences t statistic

Effect Size and Confidence Intervals

• Effect Size: Cohen’s d measures the absolute size of the treatment effect

  • Calculated as:
    d = \frac{M - \mu}{s}

  • Where $s$: Sample standard deviation

• Confidence Intervals: Used as an alternative method to estimate population mean after treatment

  • Calculation:
    \mu = M \pm t (s_M)

  • Defines interval around the sample mean: $M + t (sM)$ and $M - t (sM)$

Independent Measures t-test

• Purpose: Evaluates mean difference between two separate samples (populations or treatment conditions)

• Design: Between-subjects design

• Formula:
  t = \frac{(M1 - M2) - (\mu1 - \mu2)}{s(M1 - M2)}

Statistical Hypotheses

• Null Hypothesis ($H_0$): No difference between the two population means

  • $H0: \mu1 - \mu_2 = 0$

• Alternative Hypothesis ($H_1$): There is a difference between the two population means

  • $H1: \mu1 - \mu_2 \neq 0$

Degrees of Freedom

• Formula: df = df1 + df2

  • Where $df1$ and $df2$ correspond to the two samples and can be expressed as:
    df = (n1 - 1) + (n2 - 1)

  • Total:
    df = n - 2

  • Used to find critical t values for hypothesis tests

Pooled Variance

• Pooled Variance ($s_p^2$): Weighted average of the variances from each sample

  • Formula:
    sp^2 = \frac{SS1 + SS2}{df1 + df_2}

  • Combines variances while giving more weight to the sample with the larger size

Estimated Standard Error

• Measures error expected between a sample mean difference

• Formula: s{M1 - M2} = sp^2 (\frac{1}{n1} + \frac{1}{n2})

  • Note: $s{M1 - M_2}$ denotes estimated standard error, used in the denominator of t statistic

Comparison: Single Sample vs. Independent Measures t-tests

• Single Sample t-test:
  t = \frac{M - \mu}{s_M}

• Independent Measures t-test:
  t = \frac{(M1 - M2) - (\mu1 - \mu2)}{s(M1 - M2)}

Directional Hypothesis in Independent Measures t-test

• One-tailed t-test: Used if a clear prediction based on prior research exists

• Alternative Hypothesis: States direction of prediction (e.g., $\muA > \muB$)

• Null Hypothesis: Not equivalent to alternative hypothesis (e.g., $\muA \leq \muB$)

Effect Size and Confidence Intervals in Independent Measures

• Effect Size: Estimated Cohen’s d

  • Formula:
    d = \frac{M1 - M2}{s_p^2}

• Confidence Intervals: \mu1 - \mu2 = (M1 - M2) \pm t(s{M1 - M_2})

  • Defines interval of confidence regarding where $\mu1 - \mu2$ is located

Assumptions of Independent Measures t-test

1. Independent observations within each sample

2. Normality of populations from which samples are selected (
   can be violated if $n > 30$)

3. Equal variances (homogeneity of variance) must be satisfied

   • Hartley’s F-max test to determine this

   • Formula:
     F{max} = \frac{s^2{largest}}{s^2_{smallest}}

   • Compare $F_{max}$ to critical F value (from F-max table)

Repeated Measures t-tests

• Design: One group of participants measured in two treatment conditions (“within subjects design”)

• Goal: Test population mean difference between two treatment conditions using sample data

  • Compute difference score (D score):
    D = X2 - X1

Hypotheses for Repeated Measures t-tests

• Null Hypothesis ($H_0$): Mean of difference scores is zero

  • $H0: \muD = 0$

• Alternative Hypothesis ($H_1$): Mean of difference scores is not zero

  • $H1: \muD \neq 0$

• Degrees of Freedom:
  df = n - 1

Repeated Measures t Statistic

• Formula: t = \frac{MD - \muD}{s{MD}}

  • Where $M_D$: Mean of difference scores

  • $\mu_D$: Population difference scores from the null hypothesis (always = 0)

  • $s{MD}$: Estimated standard error computed from variance of difference scores

Estimated Standard Error in Repeated Measures

• Formula:
  s{MD} = \frac{s^2}{n}

• Sample Variance:
  s^2 = \frac{SS}{n - 1}

• Calculations may require the sum of squares (SS) of difference scores

Effect Size Calculations for Repeated Measures

• Effect Size: Estimated Cohen’s d

  • Formula:
    d = \frac{mean \ difference}{s}

• Confidence Interval: \muD = MD \pm t(s{MD})

  • Sample variance and size influence t-test results through estimated standard error

Assumptions for Repeated Measures t-test

1. Observations within treatment conditions should be independent

2. Population distribution of difference scores (D scores) should be normal

3. Advantages: Fewer subjects required, well-suited for studying changes over time, controls for individual differences

4. Disadvantages: Time-related factors; order effects can influence data due to two time points per individual

Summary of t-statistic Formulas

• Single Sample t-test:
  t = \frac{M - \mu}{s_M}

• Independent Measures t-test:
  t = \frac{(M1 - M2) - (\mu1 - \mu2)}{s(M1 - M2)}

• Repeated Measures t-test:
  t = \frac{MD - \muD}{s{MD}}

Analysis of Variance: ANOVA

• Purpose: Evaluate mean differences between two or more treatments (populations)

  • Protect from excessive Type I error (alpha level) when comparing more than two treatments

• Factor: Independent variable designating groups

• Levels: Individual groups/treatment conditions in a factor

Hypotheses for ANOVA

• Null Hypothesis ($H_0$): States no differences between population means

  • $H0: \mu1 = \mu2 = \mu3$

• Alternative Hypothesis ($H_1$): At least one population mean differs (e.g., treatment conditions are not all the same)

ANOVA Variance Calculations

• Variance Measurement: Size of differences among sample means

• Between Subjects Variance: Variability across treatment conditions

• Within Subjects Variance: Variability within a treatment condition

• ANOVA F-ratio: Compares variances
  F = \frac{variance{between}}{variance{within}}

• When null hypothesis is true, any variance is due to random differences

• If alternative hypothesis is true, treatment effect has systematic differences

  • $F > 1$ indicates differences due to treatments

Sources of Unsystematic Variability in ANOVA

• Individual differences

• Differences in experiment administration

• Measurement errors

Questions?

• Encouragement for interaction and clarification of any material discussed.