Hypothesis Tests About μ (σ known): Critical-Value Approach
Enter μ0, x̄, σ (known), n, and α. The calculator finds the critical z value(s), shows the rejection region, and decides whether to reject H0 using the critical-value rule.
Hypothesis Tests About μ, σ Known, the Critical Value Approach
Statistics • Hypothesis Tests About the Mean and Proportion
Frequently Asked Questions
How do you do a hypothesis test for a mean when σ is known using the critical value approach?
Compute z = (x̄ - μ0) / (σ / sqrt(n)), find the critical z cutoff(s) from α and the tail type, then reject H0 if the observed z falls in the rejection region. This calculator performs all of these steps and shows the rejection region on a normal curve.
What critical z values are used for two-tailed, left-tailed, and right-tailed tests?
Two-tailed tests use ±z(α/2) and reject when |z| ≥ z(α/2). Left-tailed tests use z(α) and reject when z ≤ z(α), while right-tailed tests use z(1-α) and reject when z ≥ z(1-α).
What is the standard error in a one-sample z test for a mean?
When σ is known, the standard error of the sample mean is SE = σ / sqrt(n). It measures the typical sampling variation of x̄ around μ under the model.
When is it appropriate to use a z test for a mean with σ known?
It is used when the population standard deviation σ is known (or treated as known in an intro setting) and the sampling distribution of x̄ is approximately normal. This is reasonable when the population is roughly normal or when n is large enough for the central limit theorem to apply.
How does changing α affect the rejection region and the decision?
A smaller α makes the rejection region smaller and the critical cutoff(s) more extreme, so it becomes harder to reject H0. A larger α makes rejection easier because the rejection region is larger.