Left Tailed Test Calculator
Run a left-tailed hypothesis test (H₀: μ ≥ μ₀ vs Hₚ: μ < μ₀). Enter your test statistic directly, or use the optional sample-data mode to compute it from your sample mean, hypothesized mean, standard deviation and sample size: the tool returns the critical value, the exact p-value, the reject/fail decision, a plain-English conclusion, and Cohen’s d effect size when sample data is given.
Assumes a random sample, independent observations, and approximate normality (or n ≥ 30). A p-value is not the probability that H₀ is true.
Worked example, checked against this exact tool
Sample mean 98, hypothesized mean 100, standard deviation 5, sample size 36, α = 0.05. Open the sample-data section and enter those, then "Fill test statistic": t = (98 − 100) ÷ (5 ÷ √36) = −2.4000, df = 35. Calculate: critical value = −1.6896, p-value = 0.0109, decision = reject the null hypothesis, Cohen's d = −0.40 (a small effect). These exact figures are what the calculator above produces.
What "left-tailed" means
The alternative hypothesis points left: Hₚ says the population mean is below the hypothesized value. So the rejection region is the left tail, and you reject H₀ when the test statistic falls below the critical value (equivalently, when the p-value is below α). A right-tailed test would mirror this; a two-tailed test splits α across both tails.
Z-test or t-test?
Use the z-test when the population standard deviation σ is known (rare outside textbooks); use the t-test with sample standard deviation s and n−1 degrees of freedom in every real analysis. With large samples the t distribution converges to the normal, which is why the two answers nearly agree past n = 100 or so.
Significant is not the same as important
A p-value of 0.0109 says the result is unlikely under H₀, not that it matters. Cohen's d measures the standardized size of the difference: 0.2 is small, 0.5 medium, 0.8 large. Here d = −0.40 sits between small and medium: statistically significant, modest in magnitude. Report both, always.
Check the assumptions first
The test assumes a random sample, independent observations, and approximate normality of the sampling distribution (satisfied by n ≥ 30 via the central limit theorem, or by roughly normal data for small n). Violated assumptions break the p-value before the arithmetic even starts.
What competing calculators miss
StatTrek teaches the steps without a dedicated tool; SocSciStatistics splits the workflow across separate calculators; GraphPad and DanielSoper each cover one piece. This page runs the whole left-tailed test in one place with exact p-values, adds the sample-data mode students actually need, and reports effect size alongside significance.
Frequently asked questions
When do I reject the null hypothesis?
In a left-tailed test, when the test statistic is below the critical value, equivalently when the p-value is below α.
What is the p-value?
The probability of a test statistic at least this extreme if H₀ were true. Small p = evidence against H₀.
Z-test vs t-test?
Z when σ is known; t with sample s otherwise. They nearly agree for large n.
What is Cohen's d?
The standardized mean difference: (x̄ − μ₀) / s. 0.2 small, 0.5 medium, 0.8 large.
What alpha should I use?
0.05 is the convention; 0.01 for stronger evidence requirements, 0.10 for exploratory work.
Can I enter the test statistic directly?
Yes. That is the primary input; the sample-data section is optional.
What are the degrees of freedom?
For a one-sample t-test, n − 1.
Is a small p-value proof H₀ is false?
No. It is evidence against H₀ under the model's assumptions, not a probability that H₀ is true.