Section 11.3
Inference about Two Means: Independent Samples
Compare means from two unrelated populations using the two-sample t-test.
1
Requirements
Independent Samples
Two separate, unrelated groups with no pairing or matching.
Independent random samples
Populations normal OR and
No outliers in either sample
2
Test Statistic
Under , the term
3
Degrees of Freedom
Welch's Formula (Complex)
Calculated by calculator/software using variances and sample sizes.
Conservative Approach
Use the smaller of the two.
4
Confidence Interval for
Estimates the difference between two independent population means.
5
Two-Sample t-Test Calculator
Two-Sample t-Test Calculator (Independent Samples)
Welch's t-test for comparing two population means
Sample 1
Sample 2
Null:
Alternative:
df:
49.30
Calculations
Difference
Standard Error
SE = 2.1365
Degrees of Freedom
df = 49.30
Test Statistic
t₀ = 2.0126
t-Distribution (df = 49.30)
Classical Approach
Critical Region: t < -2.012 or t > 2.012
t₀ = 2.0126 is in rejection region
P-Value Approach
P-value = 0.0496
P-value < α = 0.05
Reject H₀
At the α = 0.05 level, there is sufficient evidence to conclude that the two population means are different.
!
Common Pitfalls
Using Matched-Pairs Test
Independent samples require the two-sample t-test, not differences.
Using Pooled Variance When Inappropriate
Use Welch's t-test (unpooled) unless populations have equal variances.
Adaptive Engine
LogicLens Practice Suite
Log in to Access Adaptive Practice
Our AI engine generates unique practice problems.