Calculator Form
Formula Used
Standard error equals sigma divided by the square root of sample size.
SE = σ / √n
The shifted z value uses the assumed true mean.
δ = (μ1 - μ0) / SE
For a right-tailed test, beta equals Φ(zα - δ).
For a left-tailed test, beta equals 1 - Φ(zα - δ).
For a two-tailed test, beta equals Φ(zu - δ) - Φ(zl - δ).
Power equals one minus beta.
How to Use This Calculator
- Select the test direction used in your hypothesis test.
- Enter alpha, sample size, null mean, true mean, and sigma.
- Press the calculate button.
- Review beta, power, effect size, and critical rules.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Case | Tail | Alpha | n | μ0 | μ1 | σ | Expected Meaning |
|---|---|---|---|---|---|---|---|
| A | Two | 0.05 | 36 | 100 | 104 | 12 | Moderate shift from the null mean |
| B | Right | 0.01 | 64 | 50 | 54 | 10 | Strict right-tail decision rule |
| C | Left | 0.10 | 25 | 80 | 76 | 8 | Higher alpha lowers beta risk |
Beta Hypothesis Testing Guide
What Beta Means
Beta is the probability of a Type II error. It happens when a test fails to reject a false null hypothesis. This calculator estimates beta for a one sample mean test. It also reports power, which is one minus beta.
Why Power Matters
Power shows the chance that a test detects a real effect. A larger power value is usually better. Researchers often want power near 0.80 or higher. Low power means a useful effect may be missed, even when the sample data looks reasonable.
Inputs Explained
The null mean is the value stated in the null hypothesis. The assumed true mean is the real value you want to test against. Standard deviation measures spread. Sample size controls the standard error. Alpha controls how hard it is to reject the null hypothesis.
Tail Selection
A right-tailed test checks for an increase. A left-tailed test checks for a decrease. A two-tailed test checks for any meaningful change. The tail choice changes the critical region. It also changes beta and power.
Interpreting Results
A smaller beta means less Type II error risk. A larger power means stronger detection ability. If power is weak, increase sample size, raise alpha when suitable, reduce variation, or target a larger practical effect. Always keep the research design in mind.
Advanced Use
This tool is useful during planning. It can compare alternate sample sizes and effect assumptions. It can also explain why a completed test missed an effect. The calculation assumes normal sampling behavior and a known standard deviation. Use a dedicated t power method when sigma is estimated from small samples.
FAQs
What is beta in hypothesis testing?
Beta is the probability of not rejecting the null hypothesis when the null hypothesis is false. It measures Type II error risk.
What is test power?
Power is one minus beta. It estimates the chance of detecting a real effect when the assumed true mean is correct.
Does a lower beta always help?
Yes, lower beta means less Type II error risk. But reducing beta may need a larger sample, higher alpha, or lower variation.
What alpha should I enter?
Common alpha values are 0.05, 0.01, and 0.10. Choose the value that matches your study rule or risk tolerance.
Can I use this for two-tailed tests?
Yes. Select the two-tailed option. The calculator splits alpha across both tails and estimates beta between both critical limits.
What does assumed true mean mean?
It is the alternative mean used for planning. It represents the real effect size you want the test to detect.
Why does sample size affect beta?
Larger samples reduce standard error. That makes real effects easier to detect and usually lowers beta.
Is this calculator for proportions?
No. This version is designed for one sample mean testing with known standard deviation. Proportion tests need different formulas.