Difference Test Between Two Means

Two-Sample Mean Comparison

A statistical method used to decide whether two population means are equal by comparing the sample means from two independent groups.

Population Means

μₓ is the true average of population X; μᵧ is the true average of population Y. The hypothesis test examines the difference μₓ − μᵧ.

Known Population Standard Deviations

When the true standard deviations of both populations are known, the test uses a z-distribution instead of a t-distribution.

Independent Samples

Two samples are independent if the observations in one group do not affect or relate to observations in the other. This is required for two-sample z tests.

(X̄ - Ȳ) ~ N(μ_x - μ_γ, (((σ_x)^2)/n_X) +(((σ_Y)^2)/n_Y))

• The distribution of the difference between two sample means.

• When n_X and n_Y, are large

• Basically sampling distribution of (X̄ - Ȳ)

μ_x - μ_γ < 0

Statements comparing two populations means;

Hypothesis for 2-sample Z test for left-tailed

μ_x - μ_γ =/= 0

Statements comparing two populations means;

Hypothesis for 2-sample Z test for two-tailed

Null Hypothesis H_0: μ_x - μ_γ = 0

The assumption that the populations have the same mean; any observed difference is due to random chance.

Alternative Hypothesis

States how the means differ (greater, smaller, or simply not equal).

SE = ((((σ_x)^2)/n_X) +(((σ_Y)^2)/n_Y)))^1/2

Standard Error of (X̄ - Ȳ)

z = ((X̄ - Ȳ) - 0)/SE

• Z-statistic for Two Samples

• Measures how many standard errors the observed difference is from the value assumed under H_0.

P-Value (Two-Sample Z test)

The probability of observing a difference in sample means as extreme or more extreme than the one found, if H_0 is true.

One-Tailed Test (Left-tail or Right-tail)

A test that checks for a directional difference (e.g., μ_x < μ_γ). The p-value is the area in one tail of the z-curve.

Two-Tailed Test

A test for any difference (μ_x =/= μ_γ). The p-value equals the sum of both tail areas beyond ±|z|

Interpretation of Small P-Value

A small p-value (typically < 0.05) indicates that the observed difference is unlikely under H_0, providing evidence against the null hypothesis.

Practical Conclusion

The real-world statement about the populations based on the statistical evidence.

normal

With large n, the sampling distribution of X̄ and Ȳ is approximately ______, allowing the z-test even if the population itself isn’t perfectly that.

Yes, it is true

Is it true that if samples are not independent, the two-sample Z test is invalid because the variance formula for X̄ - Ȳ fails.

significantly different

The interval shows the range of plausible values for μ_x - μ_γ. If 0 is not inside the interval, the means are ____________________.