One-Sample t-Test Calculator
Test whether a population mean differs from a hypothesized value μ₀ when the population standard deviation is unknown. Enter raw data or just x̄, Sx, and n — the output matches the TI-84 T-Test screen: t, p, df, x̄, Sx, and n.
Comparing two groups instead? Use the two-sample and paired t-test calculator. Part of the TI-84 statistics functions guide.
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The One-Sample t-Test Formula
t = (x̄ − μ₀) / (s / √n)
df = n − 1
H₀: μ = μ₀ vs. Hₐ: μ ≠ μ₀, μ < μ₀, or μ > μ₀
Use it when you have one sample, the population standard deviation is unknown, and the data are roughly normal or n is large enough (about 30+) for the central limit theorem to apply. If σ is known, the z-test is the right tool.
How to Run T-Test on a TI-84
- Press STAT, arrow right to TESTS, and choose 2:T-Test.
- Choose Inpt: Data (list in L1) or Stats (type x̄, Sx, n).
- Enter μ₀, pick the alternative (≠, <, >), then highlight Calculate and press ENTER.
The screen shows t, p, x̄, Sx, and n — the same values this calculator reports, plus df = n − 1.
Worked Example: Is the Average Above 50?
A sample of n = 16 has x̄ = 52.3 and Sx = 4.8. Test H₀: μ = 50.
- SE = 4.8 / √16 = 1.2.
- t = (52.3 − 50) / 1.2 = 1.9167, with df = 15.
- Two-tailed p-value (μ ≠ 50): 0.074533 — not significant at 0.05.
- Right-tailed p-value (μ > 50): 0.037266 — significant at 0.05.
The same data lead to different conclusions depending on the alternative, which is why it must be chosen before looking at the results. For the matching confidence interval, use the TInterval calculator.
Frequently Asked Questions
When should I use a one-sample t-test?
When you compare one sample mean with a fixed, hypothesized value and do not know the population standard deviation. Examples: is the average fill volume 500 ml, is the mean score above 70, is the average wait time under 10 minutes.
What does the p-value mean in a t-test?
It is the probability of getting a t statistic at least as extreme as yours if the null hypothesis were true. A small p-value (typically below 0.05) means the observed mean would be unusual under H₀, so you reject it.
Can I run the test from summary statistics only?
Yes. Choose Stats input and enter x̄, Sx, and n, exactly like the TI-84's Stats option. You get the same t and p-value as from the raw data.
What are the assumptions of the one-sample t-test?
A random sample, independent observations, and a population that is roughly normal — or a sample large enough (commonly n ≥ 30) for the sampling distribution of the mean to be approximately normal. Strong skewness or outliers in small samples make the p-value unreliable.
What is the difference between a t-test and a z-test?
A z-test assumes the population standard deviation σ is known; a t-test estimates it from the sample, which adds uncertainty and gives the t-distribution heavier tails. In practice σ is rarely known, so the t-test is the usual choice.
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