Left-Tailed Test: Hypotheses, Critical Value, p-Value and Worked Example

A left-tailed test is a hypothesis test whose alternative hypothesis says the true value is smaller than the value in the null hypothesis. You reject the null hypothesis only when the test statistic lands far enough in the left (negative) tail of its sampling distribution. This guide gives the hypotheses, the critical-value rule, the left-tailed p-value, a fully worked t-test with the arithmetic shown, and the Excel, R and Python calls that reproduce it.

What is a left-tailed test?

A left-tailed test (also called a lower-tailed or lower one-sided test) puts the entire significance level α in the lower tail of the test statistic’s distribution. The sign in the alternative hypothesis decides the tail: Ha with “<” is a left-tailed test. The NIST/SEMATECH e-Handbook of Statistical Methods writes the lower one-sided two-sample t-test as Ha: μ1 < μ2 and rejects H0 when the test statistic T is less than t(α, ν).

Left-tailed hypotheses (Ha: μ < μ0)

For a one-sample test of a mean against a stated value μ0:

• H0: μ = μ0 (some textbooks write μ ≥ μ0; the decision rule is identical)
• Ha: μ < μ0

Choose the direction before you collect data, from the research question, not from which way the sample mean happened to fall. Wording such as “reduces”, “is lower than”, “is below a limit” or “decreases” usually signals a left-tailed alternative. For the general pairing of the two hypotheses, see null hypothesis vs alternative hypothesis and the null hypothesis explained.

Critical value and rejection region

The critical value is the point that cuts off area α in the lower tail. Reject H0 when the test statistic is at or below it. NIST’s table of Student’s t critical values gives the lower-tail rule directly: for a lower one-sided test, find the column corresponding to 1 − α, and reject the null hypothesis if the test statistic is less than the negative of the table value. This works because the t distribution is symmetric, so t(1−α, ν) = −t(α, ν).

Test statistic	α = 0.05 lower critical value	Reject H0 when
z (standard normal)	−1.645	z ≤ −1.645
t, ν = 10	−1.812	t < −1.812

The t value for ν = 10 comes from the row of the NIST table (1.812 in the 0.95 column). The z value is the same −1.645 discussed on the z-alpha page, and the general idea is covered in rejection region.

Left-tailed p-value

The p-value is the probability, assuming H0 is true, of a test statistic as small as or smaller than the one observed. For a left-tailed test it is a lower-tail area:

• z test: p = Φ(z), the standard normal cumulative distribution function at the observed z.
• t test: p = F(t; ν), the cumulative distribution function of the t distribution with ν degrees of freedom at the observed t.

Decision rule: reject H0 when p ≤ α. This always agrees with the critical-value method. For converting z to a probability by hand see how to find a p-value from a z-score; for what the number does and does not mean, see what is a p-value.

Worked example with real numbers

A lab claims a process yields a mean of μ0 = 50 units. You suspect it yields less. A random sample of n = 11 gives a mean x̄ = 47.2 and a standard deviation s = 4.8. Test at α = 0.05.

1. Hypotheses. H0: μ = 50; Ha: μ < 50 (left-tailed).
2. Standard error. SE = s / √n = 4.8 / √11 = 4.8 / 3.3166 = 1.4473.
3. Test statistic. t = (x̄ − μ0) / SE = (47.2 − 50) / 1.4473 = −2.8 / 1.4473 = −1.935.
4. Degrees of freedom. ν = n − 1 = 10.
5. Critical value. The NIST table gives 1.812 for ν = 10 in the 0.95 column, so the lower critical value is −1.812. Because −1.935 < −1.812, the statistic is in the rejection region.
6. p-value. p = P(T ≤ −1.935) with 10 degrees of freedom = 0.0409. Because 0.0409 ≤ 0.05, reject H0.

Conclusion: at the 5% level there is evidence that the mean is below 50. The two methods agree, as they must.

Two checks show why direction matters. The same data tested two-tailed give p = 2 × 0.0409 = 0.082, which does not reject at 0.05. Tested right-tailed (the wrong direction here), p = 1 − 0.0409 = 0.959, which is nowhere near significant. Choosing the tail after seeing the data inflates the false-positive rate, which is why it has to be fixed in advance. See one-tailed vs two-tailed tests and two-tailed hypothesis testing for the trade-off.

Interpretation: reading the result

What you observe	Reading
Test statistic below the negative critical value	In the rejection region: reject H0 in favour of Ha: μ < μ0
p ≤ α	Reject H0 (same decision as the critical-value method)
p > α	Fail to reject H0; this is not evidence that H0 is true
Positive test statistic	The sample mean is above μ0; the lower-tail p-value will be above 0.5 and H0 cannot be rejected

Assumptions and common mistakes

• Assumptions of the t version: a random sample, independent observations, and a roughly normal population (or a sample large enough that the sampling distribution of the mean is approximately normal). The z version additionally assumes σ is known or n is large.
• Picking the tail after the data. The direction must come from the hypothesis, not the sign of x̄.
• Forgetting the sign. The critical value is negative. Comparing t = −1.935 with +1.812 would wrongly fail to reject.
• Looking up the wrong column. A one-sided α = 0.05 uses the 0.95 column of a cumulative table, not the 0.975 column used for a two-sided test at the same α.
• Treating “fail to reject” as proof of H0.

How to report it

Give the test statistic, degrees of freedom, p-value, the direction of the test and the sample estimates. For the worked example: “A one-sample, left-tailed t-test showed the mean (M = 47.2, SD = 4.8, n = 11) was lower than 50, t(10) = −1.93, p = .041 (one-tailed).” State that the test was one-tailed so readers can compare it correctly. For conventions on formatting p-values, see how to report p-values.

How to calculate it in Excel, R or Python

Excel

=T.DIST((47.2-50)/(4.8/SQRT(11)), 10, TRUE)   // p-value, about 0.0409
=T.INV(0.05, 10)                               // lower critical value, about -1.812
=NORM.S.DIST(-1.645, TRUE)                     // z version: about 0.050

R

t <- (47.2 - 50) / (4.8 / sqrt(11))
pt(t, df = 10)            # p-value, about 0.0409
qt(0.05, df = 10)         # critical value, about -1.812
# with raw data x:  t.test(x, mu = 50, alternative = "less")

Python

import numpy as np
from scipy import stats
t = (47.2 - 50) / (4.8 / np.sqrt(11))
p = stats.t.cdf(t, df=10)      # about 0.0409
crit = stats.t.ppf(0.05, df=10)  # about -1.812
# with raw data x:  stats.ttest_1samp(x, 50, alternative="less")

The same t statistic drives the broader family of mean comparisons; see the t-test guide for the two-sample and paired versions.

FAQ

When do I use a left-tailed test?

Use it when your research hypothesis predicts a decrease or a value below a reference, and a result in the opposite direction would be of no interest or would not change your decision. Decide this before seeing the data.

Is the critical value negative?

Yes. For a lower-tail test at α = 0.05 the z critical value is −1.645 and, with 10 degrees of freedom, the t critical value is −1.812. You reject when the statistic is less than that value.

How is the left-tailed p-value different from the two-tailed p-value?

For a symmetric distribution and a statistic in the left tail, the two-tailed p-value is twice the left-tailed one (0.082 versus 0.041 in the example above).

What if my t statistic is positive?

Then the sample mean is above μ0, the left-tailed p-value exceeds 0.5, and you cannot reject H0.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.3, Two-Sample t-Test for Equal Means: itl.nist.gov/div898/handbook/eda/section3/eda353.htm (checked 2026-10-29).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.2, Critical Values of the Student’s t Distribution: itl.nist.gov/div898/handbook/eda/section3/eda3672.htm (checked 2026-10-29).