Chapter 11 notes

Chapter 11: Independent Samples t-Test

Learning Objectives

• Understand how to compare two different populations in a research study.

• Calculate estimated standard error with two samples.

• Compute a t-score for two samples.

• Define pooled variance and its utility in handling unequal sample sizes for hypothesis testing.

Example Research Question

• Investigate whether self-proclaimed psychics are more accurate than non-psychics.

Key Concepts

• Independent Samples:

  • Defined as two separate, independent samples.

  • Each sample provides a mean and a standard deviation.

  • The primary question is whether these means significantly differ from each other.

• Types of t-tests:

  • One sample t-test.

  • Independent samples t-test.

  • Paired samples t-test.

Between-Subjects Design

• Involves comparing two distinct and independent samples.

• Samples may originate from different populations.

• Alternatively, they can be drawn from the same population but subjected to different treatment conditions.

Independent Samples t-Test Overview

• The mean and variability of scores for each sample can be calculated and compared using the independent samples t-test.

• t-score Calculation:

  • Numerator: Mean difference observed between the two samples.

  • Denominator: Mean difference expected by chance due to sampling error.

  • The calculated t-score is then compared with the critical t-value.

Review of Statistical Concepts

• Sample Standard Deviation:
  $s = rac{SS}{n - 1}$

  • Where $SS$ is the sum of squares and $n$ is the sample size.

• Sample Variance:
  $s^2 = rac{SS}{n - 1}$

Estimated Standard Error

• Estimated standard error is calculated by taking the standard deviation divided by the square root of the sample size ($n$).

• In variance form for t-tests:

  • For One-Sample t-test:
    $s_X = rac{s}{ ext{n}}$

  • For Paired-Samples t-test:
    $s_D = rac{s^2_D}{n}$

Importance of Variance Form

• Variance ($s^2$) is the square of the standard deviation, allowing for calculations that estimate sampling error across two samples via pooled variance.

Pooled Variance

• Definition:

  • Pooled variance ($s_p^2$) combines variance terms from two samples weighted by their respective sample sizes.

  • It estimates the sampling error for the independent samples t-test and is derived from variability between the two samples.

SPSS Functionality

• SPSS can automatically calculate the pooled variance for conducting an independent samples t-test.

Calculating the Independent Samples t-Test

• The formula for our t-test value is derived from understanding the steps below: ### Step 1: State Hypotheses

  • Alternative Hypothesis (H1): The difference in population means is greater than 0 (e.g., psychics are more accurate).

  • Null Hypothesis (H0): The population means will not differ (the difference is less than or equal to 0).

Example Hypotheses - Psychic Study

• H1: $ext{Mean}<em>{ ext{psychics}} > ext{Mean}</em>{ ext{non-psychics}}$

• H0: $ext{Mean}<em>{ ext{psychics}} ext{ } ext{is} ext{ } ext{less than or equal to} ext{ } ext{Mean}</em>{ ext{non-psychics}}$

Step 2: Set Decision Criterion

• Typical significance level (alpha) is typically set at 0.05.

Step 3: Collect Sample Data

• Example collection resulted in:

  • n (psychics) = 45

  • n (non-psychics) = 50

• Important note on sample sizes being different.

Step 4: Calculate t-test Value

• Sum of Squares (SS): Measures how much individual data points deviate from the mean.

  • Higher SS indicates greater variability; lower SS indicates less variability.

  • Calculation: Subtract mean from each data point, square differences, and sum them.

Calculating Necessary Values:

• Pooled Variance Formula:
  $s_p^2 = rac{SS_{ ext{psychics}} + SS_{ ext{non-psychics}}}{df_{ ext{psychics}} + df_{ ext{non-psychics}}}$

  • Example:
    $s_p^2 = rac{550 + 350}{44 + 49} = 20.93$

• Estimating Standard Error:
  $S_{X_1 - X_2} = rac{s_p^2}{n_{ ext{psychics}}} + rac{s_p^2}{n_{ ext{non-psychics}}} = rac{20.93}{45} + rac{20.93}{50} = 0.94$

• Sample t-value Calculation:
  $t_{ ext{sample}} = rac{(X_1 - X_2) - (m_1 - m_2)}{S_{X_1 - X_2}}$

  • Example:
    $t_{ ext{sample}} = rac{(26 - 25) - 0}{0.94} = 1.06$

Step 5: Make a Decision

• Degrees of Freedom Calculation:
  $df = df_{ ext{psychics}} + df_{ ext{non-psychics}} = 44 + 49 = 93$

• Comparison with Critical Value:

  • From the t-table, critical t-value ($t_{crit}$) = 1.658.

  • Since sample t-value (1.06) does not fall in the critical region, fail to reject the Null Hypothesis.

Using SPSS for the Independent Samples t-Test

1. Enter data in a column of the Data Window.

2. Code groups in another column.

3. Define codes in the Values column (Variable View tab).

4. Use the Analyze menu to select Compare Means.

5. Choose Independent Samples t-Test from the tests list.

6. Add the respective columns as Variable 1 and Variable 2.

7. Indicate the paired variable and click OK to generate output.

8. Compare p-value from the output with your alpha level.

T-Test Example Output in SPSS

Group Statistics

• Gender

  • Men:

  • N = 10, Mean = 53.5000, Std. Deviation = 10.40566, Std. Error Mean = 3.29056

  • Women:

  • N = 10, Mean = 52.9000, Std. Deviation = 14.44107, Std. Error Mean = 4.56667

Independent Samples Test Results

• Levene's Test for Equality of Variances

  • t-test for Equality of Means:

    • F = 2.217, Sig. = 0.154, df = 18, t = 0.916, Mean Difference = 60.000,

    • Std. Error Difference = 5.62870, Confidence Interval = -11.22545 to 12.42545

Test Assumptions

1. Normal Distribution: The population of difference scores must follow a normal distribution to use the t-distribution for finding the critical value.

2. Independence: Participants within each sample must be independent; one participant's score cannot correlate with another's.

3. Homogeneity of Variances: This assumes that the scores' variances in the populations are equal across groups.

   • Levene’s test indicates whether this assumption holds. If significant, use the row labeled “Equal variances not assumed” for analysis. If not significant, use “Equal variances assumed" row values.