In a one-tailed test, the alternative hypothesis tells you which tail to use. If the sample mean moves in the opposite direction, keep that tail. The p-value can then be greater than 0.5, because the observed statistic lies on the other side of the null mean.
The 2027 H2 Mathematics syllabus (9758) covers one-tailed and two-tailed tests for a population mean, p-values and interpreting results in context. Use the original example below to see why the smaller tail is sometimes the wrong one.
- Write the alternative hypothesis before looking at the test statistic.
- Find the probability in the direction stated by the alternative.
- Give the conclusion about the original claim in context.
Worked example | A test for shorter packing times
A team wants evidence that a new process reduces mean packing time below 40 seconds. Assume the packing times are normally distributed with known standard deviation 5 seconds and that the 25 observations are an independent random sample. The observed sample mean is 42 seconds.
Use H₀: μ = 40 and H₁: μ < 40. The standard error is 5 ÷ √25 = 1 second. The observed z-statistic is (42 − 40) ÷ 1 = +2.
Because H₁ points to lower values, the p-value is P(Z ≤ 2), about 0.9772 under the standard normal distribution. At the 5% significance level, fail to reject H₀. The data do not provide sufficient evidence that the new process reduces the population mean below 40 seconds.
Where the tempting 0.0228 comes from
The upper tail beyond +2 has probability about 0.0228. It would match an alternative of μ > 40. The original question asks for evidence of a decrease. Selecting the smaller tail after seeing the result changes the test.
A quick sketch puts +2 to the right of zero. Shade everything to the left of +2 for the lower-tail probability. The shaded region is clearly larger than half the curve. That catches the error before a calculator entry.
Change the sample mean and keep the same question
If the observed sample mean were 38 seconds, the z-statistic would be −2. The lower-tail probability would then be about 0.0228. At 5%, that result would support the stated reduction.
If the original alternative had instead been μ ≠ 40, the two-tail p-value at z = +2 would be approximately 0.0455. That test answers a different question. Choose the hypothesis from the problem before examining the result.
Ask a study partner to read your conclusion without seeing your calculations. They should be able to identify the direction of the original claim and whether the evidence met the significance threshold.
Frequently asked questions
Can a p-value exceed 0.5 in a one-tailed test?
Yes. The specified tail can include more than half of the null distribution.
Does failing to reject show that the population mean is exactly 40 seconds?
It means the data did not provide sufficient evidence for the stated alternative at the chosen significance level.
Official information
SEAB’s 2027 A-Level H2 Mathematics syllabus (9758), Section 6.5 Hypothesis Testing. Checked 9 November 2026.