Decision Rule Hypothesis Testing Calculator

Calculator

Test model

Alternative hypothesis

Significance level

Hypothesis label

Null value

Sample mean

Sample size

Population standard deviation

Sample standard deviation

Successes

Sample variance

Manual distribution

Manual statistic

Manual degrees of freedom

Claim note

Example Data Table

Scenario	Test	Alpha	Tail	Statistic	Decision
Mean delivery time	t test	0.05	Right	2.11	Reject if t is beyond the cutoff.
Defect rate	Proportion z test	0.01	Left	-2.45	Compare statistic with lower rejection region.
Process variance	Chi square test	0.05	Two	41.62	Check both chi square critical limits.

Formula Used

One mean z test: z = (x̄ - μ0) / (σ / √n)

One mean t test: t = (x̄ - μ0) / (s / √n), with df = n - 1

One proportion z test: z = (p̂ - p0) / √(p0(1 - p0) / n)

One variance test: χ² = (n - 1)s² / σ0², with df = n - 1

Decision rule: reject H0 when the statistic enters the rejection region, or when p value ≤ alpha.

How to Use This Calculator

Select the correct test model. Choose the alternative hypothesis direction. Enter alpha before reviewing the result. Add the sample values required by that model. Use the manual mode when you already have a statistic. Submit the form. Read the rejection rule, p value, and final decision together.

Understanding Decision Rules

A decision rule turns a hypothesis test into a clear action. It states when evidence is strong enough to reject the null hypothesis. The rule uses the significance level, the direction of the alternative hypothesis, and the selected test distribution. This calculator helps you compare a test statistic with the correct rejection region.

Why the Rule Matters

A test can feel confusing without a rule. The same statistic may lead to different actions when alpha, tail type, or degrees of freedom change. A right tailed test checks unusually large values. A left tailed test checks unusually small values. A two tailed test checks both extremes. The calculator shows these regions before you interpret the result.

Critical Value Method

The critical value method builds a cutoff from alpha. If the statistic falls inside the rejection region, reject the null hypothesis. For a z test, the cutoff comes from the standard normal curve. For a t test, it also depends on degrees of freedom. For a chi square variance test, the cutoff is not symmetric.

P Value Method

The p value method measures how extreme the observed statistic is under the null hypothesis. A small p value means the data would be unlikely if the null were true. When the p value is less than or equal to alpha, reject the null hypothesis. Otherwise, fail to reject it.

Practical Interpretation

Statistical decisions should be written with care. Rejecting the null does not prove the alternative with certainty. Failing to reject the null does not prove the null is true. It only means the sample did not provide enough evidence at the chosen alpha level. You should also review sample size, assumptions, and study design.

Using This Tool

Choose the test model first. Enter sample values, alpha, and tail direction. The calculator can compute z, t, proportion z, and chi square variance statistics. You may also enter a manual statistic. After submission, read the decision rule, critical value, p value, and conclusion together. This gives a stronger and more transparent testing summary. Good inputs matter. Use a planned alpha level. Avoid changing tails after seeing data. Record assumptions. Share both methods when reporting important results. Keep raw calculations for review.

FAQs

What is a decision rule in hypothesis testing?

It is a preset rule that tells when to reject the null hypothesis. It usually uses alpha, tail direction, and a critical value or p value.

Should I use the p value method or critical value method?

Both methods should agree when calculated correctly. Use the critical value method for rejection regions. Use the p value method for reporting strength of evidence.

What does alpha mean?

Alpha is the chosen significance level. It is the risk of rejecting a true null hypothesis under the test assumptions.

When do I choose a two tailed test?

Choose a two tailed test when the alternative hypothesis checks for any difference. It tests both unusually low and unusually high results.

What is degrees of freedom?

Degrees of freedom describe available independent information. They affect t and chi square cutoffs. A one sample t test usually uses n minus one.

Can I enter my own test statistic?

Yes. Select manual statistic mode. Then choose the distribution, tail direction, alpha, statistic value, and degrees of freedom when required.

What does fail to reject mean?

It means the sample did not provide enough evidence against the null hypothesis at the chosen alpha level. It does not prove the null is true.

Is this calculator suitable for reports?

It can support reports by showing rules and calculations. Always verify assumptions, sampling design, and context before making final conclusions.