Independent Measures t-Test- Practice Problems

What Is an Independent Measures t-Test?

It's a statistical test that compares two group means from separate, unrelated groups. Think: different people in each group, no overlap, no pairing.

You use this when you want to know if the difference between two group averages is real or just random noise.

When Should You Use It?

Choose this test when:

• You're comparing exactly two separate groups
• Participants appear in only one group
• Your data is continuous (height, score, time, etc.)
• You want to test if one condition produces different results than another

The Formula

Here's what you're working with:

t = (X̄₁ - X̄₂) / SE

Where:

• X̄₁ and X̄₂ = the two group means
• SE = standard error of the difference

The standard error calculation:

SE = √(s₁²/n₁ + s₂²/n₂)

For equal variances assumed, pool the variance:

s² = ((n₁-1)s₁² + (n₂-1)s₂²) / (n₁ + n₂ - 2)

Degrees of Freedom

df = n₁ + n₂ - 2

That's it. Simple arithmetic.

Step-by-Step Calculation

1. Gather Your Numbers

You need: both means, both standard deviations, both sample sizes.

2. Calculate the Difference

Subtract one mean from the other. The sign doesn't matter for a two-tailed test—you're just checking if a difference exists.

3. Find the Standard Error

Plug your values into the SE formula above. Work through the squares, divisions, and additions first. Then take the square root.

4. Divide and Compare

Divide your mean difference by the standard error. This gives you your t-value.

5. Check Your Critical Value

Match your calculated t against the critical value from a t-table using your df and chosen alpha level (usually .05).

If |t| > critical value → significant difference.

If |t| < critical value → no significant difference.

Practice Problems

Problem 1: Test Scores 📊

A researcher wants to know if students perform differently when taught with Method A versus Method B.

Method A: n = 25, X̄ = 78, s = 12

Method B: n = 25, X̄ = 84, s = 10

Step 1: State your hypotheses

H₀: μ₁ = μ₂ (no difference)

H₁: μ₁ ≠ μ₂ (difference exists)

Step 2: Calculate pooled variance

s² = [((25-1)(144) + (25-1)(100)] / (25+25-2)

s² = [(24×144 + 24×100)] / 48

s² = (3456 + 2400) / 48

s² = 5856 / 48

s² = 122

Step 3: Calculate standard error

SE = √(122/25 + 122/25)

SE = √(4.88 + 4.88)

SE = √(9.76)

SE = 3.12

Step 4: Calculate t

t = (78 - 84) / 3.12

t = -6 / 3.12

t = -1.92

Step 5: Decision

df = 48, α = .05 (two-tailed)

Critical value = ±2.01

Since |-1.92| < 2.01, you fail to reject H₀.

No significant difference between methods at α = .05.

Problem 2: Reaction Times ⚡

Does caffeine affect reaction time?

Caffeine group: n = 16, X̄ = 245ms, s = 22

Placebo group: n = 16, X̄ = 268ms, s = 18

Step 1: Hypotheses

H₀: μ_caffeine = μ_placebo

H₁: μ_caffeine ≠ μ_placebo

Step 2: Pooled variance

s² = [((15)(484) + (15)(324)] / 30

s² = (7260 + 4860) / 30

s² = 12120 / 30

s² = 404

Step 3: Standard error

SE = √(404/16 + 404/16)

SE = √(25.25 + 25.25)

SE = √50.5

SE = 7.11

Step 4: t-value

t = (245 - 268) / 7.11

t = -23 / 7.11

t = -3.24

Step 5: Decision

df = 30, α = .05

Critical value = ±2.04

|−3.24| > 2.04 → Reject H₀

Significant difference exists. Caffeine improves reaction time.

Problem 3: Salary Comparison 💰

Compare salaries between two companies.

Company X: n = 30, X̄ = $52,000, s = $8,000

Company Y: n = 35, X̄ = $48,500, s = $6,500

Step 1: Hypotheses

H₀: μx = μy

H₁: μx ≠ μy

Step 2: Pooled variance

s² = [((29)(64,000,000) + (34)(42,250,000)] / 63

s² = [1,856,000,000 + 1,436,500,000] / 63

s² = 3,292,500,000 / 63

s² = 52,261,905

Step 3: Standard error

SE = √(52,261,905/30 + 52,261,905/35)

SE = √(1,742,063.5 + 1,493,197.3)

SE = √3,235,260.8

SE = $1,798.68

Step 4: t-value

t = (52,000 - 48,500) / 1798.68

t = 3,500 / 1798.68

t = 1.95

Step 5: Decision

df = 63, α = .05

Critical value ≈ ±2.00

1.95 < 2.00 → Fail to reject H₀

No significant salary difference at α = .05.

Equal vs. Unequal Variances

Use Welch's t-test when variances differ substantially between groups.

Check with an F-test or Levene's test if you're unsure.

Common Mistakes to Avoid

• Wrong test for your design. If you measured the same people twice, use a paired t-test instead.
• Forgetting df. Your critical value depends on this number.
• Misreading the table. Two-tailed vs. one-tailed changes the critical value.
• Rounding too early. Keep full precision until your final answer.
• Ignoring assumptions. Normality and independence matter for validity.

Quick Reference: Independent t-Test Checklist

• Two separate groups ✓
• Continuous dependent variable ✓
• Sample means and standard deviations ✓
• Calculate t-statistic ✓
• Find df = n₁ + n₂ - 2 ✓
• Compare to critical value ✓
• Report results with t, df, and p-value ✓

Reporting Your Results

Use this format:

t(df) = [value], p [less/greater] than [.05], d = [effect size]

Example:

t(48) = -1.92, p > .05, d = 0.55

Always include effect size (Cohen's d) when possible. Statistical significance alone doesn't tell the whole story.

When to Use What: Test Selection Guide