Hypothesis Test and Confidence Interval Calculator

This calculator performs hypothesis tests and calculates confidence intervals for population means. It helps determine whether sample data provides enough evidence to reject a null hypothesis and estimates the range where the true population mean likely falls.

What is a Hypothesis Test and Confidence Interval?

A hypothesis test evaluates whether sample data provides enough evidence to reject a null hypothesis. A confidence interval estimates the range where the true population parameter likely falls with a specified probability.

Common hypothesis tests include z-tests and t-tests, while confidence intervals are calculated using sample statistics and standard errors.

Key Formulas

Z-test statistic: \( z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \)

T-test statistic: \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \)

Confidence interval: \( \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \) (for z-test) or \( \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \) (for t-test)

Note: This calculator assumes normally distributed data. For small samples (n < 30), use the t-distribution. For larger samples, use the z-distribution.

How to Use This Calculator

Enter your sample mean (x̄)
Enter the population standard deviation (σ) or sample standard deviation (s)
Enter the sample size (n)
Enter the hypothesized population mean (μ₀)
Select the significance level (α)
Choose between z-test or t-test
Click "Calculate" to get results

Formulas and Assumptions

The calculator uses these statistical formulas:

Z-test for large samples (n ≥ 30)
T-test for small samples (n < 30)
Confidence intervals based on standard error

Assumptions:

Data is normally distributed
Samples are independent and random
Population standard deviation is known for z-tests

Worked Example

Suppose we want to test if the average height of a population is 170 cm, with a sample mean of 172 cm, standard deviation of 5 cm, and sample size of 50.

Calculate standard error: 5/√50 ≈ 0.707
Calculate z-score: (172-170)/0.707 ≈ 2.83
Determine p-value from z-table: ≈ 0.0023
Compare to α = 0.05: p < α → Reject null hypothesis
Calculate 95% confidence interval: 172 ± 1.96*0.707 → [170.6, 173.4]

Interpreting Results

For hypothesis tests:

If p-value < α, reject the null hypothesis
If p-value ≥ α, fail to reject the null hypothesis

For confidence intervals:

95% CI means there's 95% probability the true mean falls within the interval
Wider intervals indicate more uncertainty

FAQ

What's the difference between a hypothesis test and confidence interval?

A hypothesis test evaluates whether sample data provides enough evidence to reject a null hypothesis, while a confidence interval estimates the range where the true population parameter likely falls.

When should I use a z-test vs. t-test?

Use a z-test when the population standard deviation is known and the sample size is large (n ≥ 30). Use a t-test when the population standard deviation is unknown and the sample size is small (n < 30).

What does a p-value of 0.03 mean?

A p-value of 0.03 means there's a 3% probability of observing the sample data (or more extreme) if the null hypothesis were true. With a common α = 0.05, you would reject the null hypothesis.