t Confidence Interval Calculator
Build a confidence interval for a population mean — or for the difference between two means — when the population standard deviation is unknown. The output matches the TI-84 TInterval and 2-SampTInt screens.
Know σ? Use the z confidence interval calculator. Part of the TI-84 statistics functions guide.
Related Calculators
One-Sample t-Test Calculator
Test a mean against a hypothesized value from data or x̄, Sx, and n — the TI-84 T-Test output.
Confidence Interval Calculator
Estimate confidence intervals for means and proportions with visible assumptions.
invT Calculator
Find t critical values with invT(area, df) for left, right, or two-tailed areas.
Formulas
One mean: x̄ ± t* · s / √n, df = n − 1
Two means (Welch): (x̄₁ − x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂)
Two means (pooled): (x̄₁ − x̄₂) ± t* · sp · √(1/n₁ + 1/n₂), df = n₁ + n₂ − 2
t* = invT((1 + C) / 2, df)
How to Use TInterval and 2-SampTInt on a TI-84
- Press STAT → TESTS.
- Choose 8:TInterval for one mean or 0:2-SampTInt for two means.
- Pick Data or Stats, enter the values and C-Level (e.g. 0.95).
- For 2-SampTInt, set Pooled: No unless you have a reason to assume equal variances.
- Highlight Calculate and press ENTER.
Worked Example: One Mean
A sample of n = 16 has x̄ = 52.3 and Sx = 4.8. Find a 95% confidence interval for μ.
- df = 15 and t* = invT(0.975, 15) = 2.1314.
- SE = 4.8 / √16 = 1.2, so the margin of error is 2.1314 × 1.2 = 2.5577.
- Interval: 52.3 ± 2.5577 = (49.7423, 54.8577).
The interval contains 50, which matches the two-tailed one-sample t-test of H₀: μ = 50 not rejecting at α = 0.05.
Worked Example: Two Means
Group 1: x̄₁ = 78.2, Sx₁ = 6.1, n₁ = 25. Group 2: x̄₂ = 74.5, Sx₂ = 8.3, n₂ = 22. Find a 95% interval for μ₁ − μ₂ (Pooled: No).
- Difference: 78.2 − 74.5 = 3.7.
- SE = √(6.1²/25 + 8.3²/22) = 2.1494 and Welch df = 38.1635.
- Interval: (−0.6505, 8.0505).
Because the interval contains 0, the data do not show a significant difference between the group means at the 5% level.
Frequently Asked Questions
When do I use a t interval instead of a z interval?
Use a t interval whenever the population standard deviation is unknown and estimated by the sample standard deviation — which is almost always. A z interval is only correct when σ is known.
Should I choose Pooled Yes or No for 2-SampTInt?
No, in almost all cases. The unpooled (Welch) interval is valid whether or not the variances are equal; pooling is only justified when you have strong reason to believe the two population variances are the same. AP Statistics uses the unpooled version.
Why is the degrees of freedom not a whole number?
With Pooled: No, the TI-84 and this calculator use the Welch–Satterthwaite formula, which usually produces a decimal df. It is more accurate than the conservative rule of using the smaller of n₁ − 1 and n₂ − 1.
How do I interpret a 95% confidence interval?
If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true mean. It does not mean there is a 95% probability that this particular interval contains it.
How does sample size affect the interval?
The margin of error shrinks roughly with 1/√n, and t* also gets smaller as df grows. Quadrupling the sample size roughly halves the width of the interval.
Embed This Calculator
Add this free calculator to your course page or LMS.
Adjust the height value to fit your page.