Independent-Samples t Test: Statistical Methods for Comparing Two Means

Independent-Samples t Test

Introduction to the Independent-Samples t Test

The independent-samples t test is a statistical method used to compare the means of two groups in a between-groups design. Each participant is assigned to only one condition, making it suitable for experiments where groups are independent.

• Purpose: To determine if there is a statistically significant difference between the means of two independent groups.

• Application: Commonly used in behavioral sciences, psychology, and experimental research.

Distribution of Differences Between Means

When conducting an independent-samples t test, the focus is on the distribution of differences between the means of two samples. This distribution helps assess the likelihood that observed differences are due to chance.

• Key Concept: The comparison distribution is the distribution of differences between sample means.

• Example: Comparing happiness scores between two groups.

Steps for Calculating Independent-Samples t Tests

Step 1: Identify the Populations, Distribution, and Assumptions

Begin by clearly defining the populations, the type of distribution, and the assumptions underlying the test.

• Population 1: People told they are drinking wine from a $10 bottle.

• Population 2: People told they are drinking wine from a $90 bottle.

• Distribution: Distribution of differences between means.

• Assumptions:

  • Participants are not randomly selected; caution is needed when generalizing findings.

  • Normality of the population distribution is not guaranteed.

Step 2: State the Null and Research Hypotheses

Formulate the hypotheses to be tested.

• Null Hypothesis (H0): On average, people drinking wine they are told is from a $10 bottle will give it the same rating as people drinking wine they are told is from a $90 bottle.$H_0: \, ext{μ}_1 = ext{μ}_2$

• Research Hypothesis (H1): On average, people drinking wine they are told is from a $10 bottle will give it a different rating than people drinking wine they are told is from a $90 bottle.$H_1: \, ext{μ}_1 eq ext{μ}_2$

Step 3: Determine the Characteristics of the Comparison Distribution

Calculating the spread (variance) of the comparison distribution involves several steps:

1. Calculate corrected variance for each sample.

2. Pool the variances.

3. Convert the pooled variances from squared standard deviation by dividing by sample size.

4. Add the two variances.

5. Calculate the square root of this variance.

Formulas

• Corrected sample variance: $s^2_X = \frac{\sum (X - M)^2}{N - 1}$

• Pooled variance: $s^2_{pooled} = \frac{df_X}{df_{total}} s^2_X + \frac{df_Y}{df_{total}} s^2_Y$

• Variance of the difference: $s^2_{difference} = s^2_{M_X} + s^2_{M_Y}$

• Standard error of the difference: $s_{difference} = \sqrt{s^2_{difference}}$

Step 4: Determine Critical Values, or Cutoffs

Critical values are determined using the t distribution table, based on the chosen significance level (commonly 0.05 for a two-tailed test).

• Example: For a 95% confidence interval, critical values are typically ±2.365 for small sample sizes.

Step 5: Calculate the Test Statistic

The test statistic quantifies the difference between sample means relative to the variability of the difference.

• General formula: $t = \frac{(M_X - M_Y) - (\mu_X - \mu_Y)}{s_{difference}}$

• If population means are assumed equal: $t = \frac{M_X - M_Y}{s_{difference}}$

Step 6: Make a Decision

Compare the calculated t value to the critical value to decide whether to reject the null hypothesis.

• If $|t_{calc}| > t_{critical}$, reject the null hypothesis.

• If $|t_{calc}| t_{critical}$, fail to reject the null hypothesis.

Reporting the Statistics

• Report the t value, degrees of freedom, and p value: $t(df) = t_{calc}, \, p < 0.05$

• Use $p > 0.05$ if there is no significant difference; $p < 0.05$ if there is a significant difference.

• Example: $t(7) = -2.44, \, p < 0.05$

Beyond Hypothesis Testing

Confidence Intervals and Effect Size

In addition to hypothesis testing, confidence intervals and effect size provide further insight into the results of an independent-samples t test.

Steps for Calculating a Confidence Interval

1. Draw a normal curve with the sample difference between means in the center.

2. Indicate the bounds of the confidence interval (CI) on either end, with percentages.

3. Look up the t values for the lower and upper ends of the CI in the table.

4. Convert the t values to raw differences.

5. Check the answer; each end of the CI should be equidistant from the sample mean.

Confidence Interval Formulas

• General formula for CI bounds: $(M_X - M_Y)_{upper} = t \, (s_{difference}) + (M_X - M_Y)_{sample}$ $(M_X - M_Y)_{lower} = -t \, (s_{difference}) + (M_X - M_Y)_{sample}$

Effect Size (Cohen's d)

Effect size quantifies the magnitude of the difference between groups, supplementing hypothesis testing.

• Cohen's d formula: $d = \frac{(M_X - M_Y) - (\mu_X - \mu_Y)}{s_{pooled}}$

Effect Size	Convention	Overlap
Small	0.2	85%
Medium	0.5	67%
Large	0.8	53%

The Bayesian Approach to Data Analysis

Introduction to Bayesian Analysis

The Bayesian approach represents a shift in statistical thinking, offering a more intuitive and straightforward alternative to traditional hypothesis testing. It incorporates prior beliefs and probabilities into the analysis.

• Key Features:

  • Includes prior beliefs and probabilities.

  • Advocated by some statisticians as a replacement for hypothesis testing.

Additional info: These notes are based on behavioral science statistics and are not directly related to General Chemistry, but the statistical methods described (such as the independent-samples t test) are broadly applicable in scientific research, including chemistry experiments involving comparison of two groups.