t-test Calculator
Perform one-sample, two-sample, and paired t-tests with comprehensive statistical analysis
t-test Configuration
Test if the population mean equals a hypothesized value
One-Sample t-test Parameters
Example: New Drug Treatment
One-Sample t-test
Scenario: Test if new drug reduces average recovery time from 10 days
Sample mean: 8.5 days
Hypothesized mean: 10 days
Sample std dev: 2.1 days
Sample size: 25 patients
Expected Result
t = (8.5 - 10) / (2.1 / √25) = -3.57
df = 24, p-value ≈ 0.001
Conclusion: Significant evidence that the drug reduces recovery time
When to Use Each Test
One-Sample
Compare sample mean to known population value
Example: Is average height different from 170cm?
Two-Sample
Compare means of two independent groups
Example: Men vs women average salaries
Paired
Compare before/after or matched pairs
Example: Weight before vs after diet
t-test Assumptions
Data should be approximately normally distributed
Observations should be independent
For two-sample: consider equal vs unequal variances
Sample size should be adequate (typically n ≥ 30)
Understanding t-tests
What is a t-test?
A t-test is a statistical hypothesis test that uses the t-distribution to determine if there is a significant difference between group means or if a sample mean differs significantly from a population mean. It's particularly useful when the population standard deviation is unknown.
Types of t-tests
- •One-sample: Tests if sample mean equals hypothesized value
- •Two-sample: Compares means of two independent groups
- •Paired: Tests differences in matched pairs or repeated measures
Key Formulas
One-sample t-test
t = (x̄ - μ₀) / (s / √n)
df = n - 1
Two-sample (equal variances)
t = (x̄₁ - x̄₂ - Δ) / (sp√(1/n₁ + 1/n₂))
df = n₁ + n₂ - 2
Paired t-test
t = (d̄ - Δ) / (sd / √n)
df = n - 1
Historical Context
The t-test was developed by William Sealy Gosset in 1908, who published under the pseudonym "Student" while working at the Guinness Brewery in Dublin. He developed this test as an economical way to monitor the quality of beer with small sample sizes.
Fun Fact: The "Student's t-distribution" is named after Gosset's pseudonym, not because it was designed for students! 🍺
Interpretation Guidelines
- •p-value < α: Reject null hypothesis (significant result)
- •p-value ≥ α: Fail to reject null hypothesis
- •|t| > critical value: Significant at chosen α level
- •Effect size: Consider practical significance beyond statistical significance