STEM Calculators

Inferences About the Difference Between Two Population Means for Independent Samples, σ₁ and σ₂ Known

Statistics • Estimation and Hypothesis Testing, Two Populations

Written by STEM Calculators Team Published Updated
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Assumes two independent simple random samples with σ₁ and σ₂ known. Use the normal model when sample sizes are large, or when populations are approximately normal.

Hypotheses

Decision settings

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Normal model visualization (standard normal)
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-4 -2 0 2 4
The shaded area updates after calculation.

Independent vs dependent samples

Independent samples come from different populations (or unrelated selections). The choice of one sample does not affect the other.
Dependent samples involve related observations (paired/matched). This calculator is for independent samples only.

Sampling distribution facts
If conditions hold, then x̄₁ − x̄₂ is (approximately) normal with:
\[ \begin{aligned} \mu_{\bar{x}_1-\bar{x}_2} &= \mu_1-\mu_2 \\ \sigma_{\bar{x}_1-\bar{x}_2} &= \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \end{aligned} \]
Enter values and click Calculate.
Batch mode: paste CSV data (compute many rows at once)

Paste rows as CSV (comma-separated). Header is optional. Supported columns: x1, x2, sigma1, sigma2, n1, n2, and (optional) conf, alpha, delta0, alt (two/gt/lt), task (ci/ht).

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Frequently Asked Questions

What does it mean that sigma1 and sigma2 are known in a two-sample mean problem?

It means the population standard deviations for both groups are treated as known constants rather than estimated from the samples. Under this assumption, inference about mu1 - mu2 uses the standard normal (z) model.

How is the standard error for mu1 - mu2 computed in this calculator?

The standard error is computed from the known population standard deviations and sample sizes: SE = sqrt(sigma1^2/n1 + sigma2^2/n2). This SE is used for both confidence intervals and hypothesis tests.

How do I interpret a confidence interval for mu1 - mu2?

A confidence interval gives a plausible range of values for the true difference mu1 - mu2 based on the sample information and the chosen confidence level. If the interval includes 0, it indicates the data are consistent with no difference at that confidence level.

What is delta0 in the hypothesis test about mu1 - mu2?

Delta0 is the hypothesized difference under the null hypothesis H0, often set to 0. The test compares the observed difference (xbar1 - xbar2) to delta0 after standardizing by the standard error.

When should I use a two-tailed vs one-tailed alternative for mu1 - mu2?

Use a two-tailed alternative when you want to detect any difference (mu1 - mu2 not equal delta0). Use a right-tailed or left-tailed alternative only when the research question is directional and you care about differences greater than or less than delta0.

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