Perform a one-sample hypothesis test and interpret the $p$-value

Primary Tool

Hypothesis Test Calculator

The hypothesis test calculator shows the test statistic, $p$ -value, critical value, and decision from the same inputs. That keeps attention on interpretation instead of arithmetic while still exposing every quantity used in the test.

Problem

A tutoring program claims that its students average $70$ points on a placement exam. A random sample of $40$ students from the program has mean $\overset{x}{ˉ} = 74.2$ and sample standard deviation $s = 8.4$ .

At significance level $α = 0.05$ , test whether the true program mean differs from $70$ .

Set up the hypotheses

The null hypothesis is the model being tested: $H_{0} : μ = 70$ . The alternative is two-sided because the question asks whether the mean differs from $70$ , not only whether it is higher or lower.

• $H_{0} : μ = 70$
• $H_{a} : μ \neq = 70$
•Use a one-sample $t$ test because the population standard deviation is unknown and the sample standard deviation $s$ is used.

Step-by-step walkthrough

The calculation turns the sample mean into a standardized distance from the null value.

1Compute the standard error: $SE = s / n = 8.4/ 40 \approx 1.328$ .
2Compute the test statistic: $t = (\overset{x}{ˉ} - μ_{0}) / SE = (74.2 - 70) /1.328 \approx 3.16$ .
3Use $df = n - 1 = 39$ degrees of freedom for the $t$ distribution.
4For a two-sided test, the $p$ -value is the probability, assuming $H_{0}$ is true, of seeing a $t$ statistic at least this far from $0$ in either direction.
5The calculator reports a small $p$ -value, below $0.05$ , so the decision rule says to reject $H_{0}$ at the 5% level.

Interpret the result

Rejecting $H_{0}$ means the sample provides statistically significant evidence that the program mean is not $70$ . The direction of the sample, $74.2 > 70$ , suggests the mean is higher, but the two-sided test itself was built to detect any difference.

The $p$ -value is not the probability that the tutoring program's claim is true. It is the probability of getting a sample result this extreme, or more extreme, if the true mean were exactly $70$ and the model assumptions held.

Statistical significance is not the same as practical importance. A difference of about $4.2$ points might matter a lot or very little depending on the exam scale, cost of the program, and how the sample was collected.

Common Pitfall

Rejecting $H_{0}$ does not prove the alternative hypothesis, and failing to reject $H_{0}$ would not prove the null. The test only measures how compatible the sample is with the null model under the chosen assumptions.

Try a Variation

Change the alternative hypothesis to 'greater than' while keeping the same sample summary. How does the $p$ -value change, and why is choosing the alternative after seeing the sample mean a bad statistical habit?

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A $p$ -value is a measure of how surprising the observed test statistic would be if the null hypothesis were true. It is not the probability that the null hypothesis itself is true.

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