Example Data Table
| Case | Calculator Type | Inputs | Claim | Suggested Method |
|---|---|---|---|---|
| Product weight | One Mean | n = 40, mean = 52, s = 9 | Mean equals 50 | T interval and test |
| Conversion rate | One Proportion | n = 120, successes = 74 | Proportion equals 0.50 | Z interval and test |
| Two class scores | Two Means | n1 = 45, mean1 = 84, n2 = 42, mean2 = 79 | Difference equals 0 | Welch t interval and test |
| Two campaign rates | Two Proportions | x1 = 32 of 45, x2 = 25 of 42 | Difference equals 0 | Z interval and test |
Formula Used
One Mean
Confidence interval: x̄ ± critical × standard error. Standard error is σ / √n for known σ, or s / √n for unknown σ.
Test statistic: z = (x̄ − μ0) / standard error, or t = (x̄ − μ0) / standard error.
One Proportion
Confidence interval: p̂ ± z × √[p̂(1 − p̂) / n]. Test statistic: z = (p̂ − p0) / √[p0(1 − p0) / n].
Two Means
Difference: x̄1 − x̄2. Welch standard error: √[(s1² / n1) + (s2² / n2)]. Pooled testing uses a shared variance estimate.
Two Proportions
Difference: p̂1 − p̂2. Standard error: √[p̂1(1 − p̂1) / n1 + p̂2(1 − p̂2) / n2].
Decision Rule
Reject the null hypothesis when the p value is less than alpha. Otherwise, do not reject the null hypothesis.
How to Use This Calculator
- Select the calculator type that matches your sample design.
- Enter the confidence level and alpha level.
- Choose the alternative hypothesis direction.
- Enter sample sizes, means, deviations, or successes.
- Enter the null value for the claim being tested.
- Press Calculate to view the interval and test result.
- Use CSV or PDF to save the current result.
Confidence Intervals and Hypothesis Tests
Why these tools matter
Statistical decisions need more than one raw number. A sample mean or sample proportion can move because of natural sampling error. A confidence interval shows a reasonable range for the unknown population value. A hypothesis test checks whether a claimed value fits the observed data. Used together, they give a clearer view of evidence.
What the calculator compares
This calculator covers common research situations. You can estimate one mean, one proportion, two independent means, or two independent proportions. You can also test a null claim. The claim may concern an average, a rate, or a difference between groups. The tool reports the estimate, standard error, critical value, interval limits, test statistic, p value, and decision.
How to read the interval
A wider interval means more uncertainty. Small samples, high variation, and high confidence levels increase width. A narrow interval means the estimate is more precise. Precision improves when sample size grows or variation falls. The confidence level describes the long run success of the interval method, not the chance that one fixed parameter moves.
How to read the test
The p value measures how unusual the sample result is, assuming the null claim is true. A small p value gives evidence against that claim. The alpha level is your chosen cutoff. When the p value is less than alpha, reject the null claim. When it is not less than alpha, do not reject it.
Good practice
Choose the test before viewing results. Check sample independence. Use t methods when the population standard deviation is unknown. Use z methods when a population standard deviation is known or when proportions are being tested with adequate counts. For two means, Welch testing is often safer because it does not assume equal variances. Always report the method, assumptions, sample size, interval, p value, and practical meaning.
Limitations to remember
Statistics cannot prove a claim with perfect certainty. It only weighs evidence under stated assumptions. Extreme outliers, biased sampling, paired observations, or small expected proportion counts can require another method. Treat the output as a guide. Use subject knowledge to decide whether a statistically significant result is also useful in real work. It supports future decisions.
FAQs
What is a confidence interval?
A confidence interval is a range of likely values for a population parameter. It uses sample data, standard error, and a critical value. Wider intervals show more uncertainty. Narrower intervals show more precision.
What is hypothesis testing?
Hypothesis testing checks whether sample evidence disagrees with a stated claim. The claim is called the null hypothesis. The p value helps decide whether the evidence is strong enough to reject that claim.
When should I use a t test?
Use a t test for means when the population standard deviation is unknown. This is common in real sample studies. The calculator uses the sample standard deviation and degrees of freedom.
When should I use a z test?
Use a z test for proportions or for a mean when the population standard deviation is known. For proportions, the sample should have enough expected successes and failures for a normal approximation.
What does alpha mean?
Alpha is the rejection cutoff. A common value is 0.05. If the p value is smaller than alpha, the result is considered statistically significant under the chosen test setup.
What does a p value show?
A p value shows how unusual the observed result is under the null hypothesis. Smaller p values provide stronger evidence against the null claim. They do not measure practical importance.
Can I compare two groups?
Yes. Choose two means for average differences. Choose two proportions for rate differences. Enter both sample sizes and the matching statistics for each group.
Why do results change with confidence level?
A higher confidence level uses a larger critical value. This makes the interval wider. A lower confidence level uses a smaller critical value and gives a narrower interval.