t-Test — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	10 of 27 (37.0%)
Priority	🟠 HIGH
Section Distribution	Sec A: ~8
Most Repeated	"Compute t test for the following data"
Note	More common in older papers (2011–2020); ANOVA dominates Section A post-2018

🎯 Must-Know Concepts

1. t-test for independent samples (two groups)
2. t-test for correlated/paired samples
3. Formula, df, interpretation
4. Relationship to SE (standard error)
5. Assumptions of t-test

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🔥 Core Computation: Independent Samples t-Test

Step 1: Compute means and SDs for each group

Step 2: Compute standard error of the difference between means: $SE_{diff} = \sqrt{\frac{SD_1^2}{n_1} + \frac{SD_2^2}{n_2}}$

For small samples with equal variances (pooled SD): $t = \frac{\bar{X}_1 - \bar{X}_2}{SD_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Where pooled SD: $SD_p = \sqrt{\frac{(n_1-1)SD_1^2 + (n_2-1)SD_2^2}{n_1+n_2-2}}$

Step 3: Degrees of freedom: df = n₁ + n₂ − 2

Step 4: Compare t obtained with t critical at df and α = 0.05

Decision: If |t obtained| ≥ t critical → Reject H₀ → Significant difference

Paired Samples t-Test (Correlated):

$t = \frac{\bar{D}}{SE_D}$

Where D = difference scores for each pair, D̄ = mean of differences, SE_D = SD of differences / √n

df = n − 1 (n = number of pairs)

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📝 Assumptions of t-Test (Section B — ~200 words)

1. Data measured on interval or ratio scale
2. Normal distribution in the population
3. Homogeneity of variance (for independent samples)
4. Random sampling
5. Independence of observations (for independent samples t-test)

When to use t-test vs ANOVA: t-test compares two groups; ANOVA compares three or more groups. Running multiple t-tests inflates Type I error; ANOVA avoids this.

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🧠 Memory Aid

t-test = two groups: t = difference between means / standard error of that difference.

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📚 Related Content

• t-Test: Independent and Correlated Samples
• Significance of Difference Between Two Means
• Standard Error and Sample Size

✅ Self-Assessment

1. Write the formula for independent samples t-test.
2. What are the degrees of freedom for a t-test with n₁ = 10, n₂ = 12?
3. What is the difference between independent and paired t-tests?
4. Why use ANOVA instead of multiple t-tests?
5. What assumptions must be met for a t-test?

💡 Exam Tips

• Always show the SE calculation step explicitly.
• State H₀ and H₁ before computing.
• Questions often pair: "Describe SE. Compute t-test for the following data."
• df must be correct — losing marks here is avoidable.