A one-sample t-test has n = 25, x = 12, s = 5, testing H_0: μ = 10. Find the test statistic and its degrees of freedom. [2 marks]

ScotlandStatisticsSyllabus dot point

How do you test claims about one or two population means when the standard deviation is unknown?

Carry out the one-sample, two-sample (independent) and paired t-tests for population means, stating the hypotheses, computing the test statistic, using degrees of freedom, and interpreting the result, while checking the normality assumption.

A focused answer to the SQA Advanced Higher Statistics t-test content: the one-sample t-test, the two-sample (independent) t-test and the paired t-test, with the test statistics, the degrees of freedom, the normality assumption and how to interpret the outcome.

Generated by Claude Opus 4.813 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The one-sample t-test
The two-sample (independent) t-test
The paired t-test
Checking the assumption
Try this

What this dot point is asking

The t-tests are the standard tools for claims about means when the population standard deviation is unknown, which is almost always. The SQA wants you to run three versions: a one-sample test (does a single mean differ from a stated value?), a two-sample test (do two independent groups have different means?), and a paired test (does a before-and-after change differ from zero?), each time stating hypotheses, computing the statistic, using the right degrees of freedom, and checking the normality assumption.

The one-sample t-test

This tests whether a single population mean equals a claimed value.

Run a one-sample t-test

We want to decide whether the evidence from a small sample is strong enough to reject the claim $\mu = 100$ . Because the population standard deviation $\sigma$ is unknown, we use the t-statistic with $n - 1$ degrees of freedom rather than a $z$ -statistic.

Step 1: State the hypotheses, significance level and normality assumption

We test $H_0: \mu = 100$ against $H_1: \mu \neq 100$ (two-tailed) at $\alpha = 0.05$ . The t-test requires the data to be approximately normally distributed; we assume this holds here. Sample values: $n = 9$ , $\bar{x} = 105$ , $s = 6$ .

Step 2: Compute the test statistic

Subtract the hypothesised mean from the sample mean, then divide by the standard error $s/\sqrt{n}$ . The degrees of freedom are $n - 1 = 8$ .

t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}} = \dfrac{105 - 100}{6/\sqrt{9}} = \dfrac{5}{6/3} = \dfrac{5}{2} = 2.5

Step 3: Compare with the critical value and state the conclusion

For a two-tailed test at $\alpha = 0.05$ on $8$ degrees of freedom, the critical value from the t-table is $t_{0.025, 8} = 2.306$ .

2.5 > 2.306 \implies \text{reject } H_0

Final answer: there is significant evidence at the 5% level that the population mean differs from $100$ .

The two-sample (independent) t-test

This compares the means of two independent groups.

The key requirement is independence of the two samples: the groups must be separate, with no natural pairing between an observation in one and an observation in the other. If there is a pairing, the paired test is correct instead.

The paired t-test

When each observation in one group is naturally matched with one in the other (the same individual before and after, or matched pairs), the paired test exploits that link.

Checking the assumption

All t-tests assume the underlying data (or, for the paired test, the differences) come from an approximately normal population. For small samples this matters; for larger samples the central limit theorem makes the test robust. If normality is clearly violated and the sample is small, a non-parametric test is the safer choice.

Try this

Q1. A one-sample t-test has $n = 25$ , $\bar{x} = 12$ , $s = 5$ , testing $H_0: \mu = 10$ . Find the test statistic and its degrees of freedom. [2 marks]

Cue. $t = \dfrac{12 - 10}{5/\sqrt{25}} = \dfrac{2}{1} = 2$ on $24$ degrees of freedom.

Q2. State the assumption all t-tests share and what to do for a small, skewed sample. [1 mark]

Cue. They assume the data (or differences) are approximately normal; for a small, clearly skewed sample use a non-parametric test instead.

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AH style: one-sample t4 marksA sample of

n = 16

items has mean

\bar{x} = 52

and sample standard deviation

s = 8

. Test at the

5\%

level whether the population mean exceeds

50

. (Use

t_{0.05, 15} = 1.753

Show worked answer →

Hypotheses: $H_0: \mu = 50$ against $H_1: \mu > 50$ (one-tailed) (1 mark).

Test statistic: $t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}} = \dfrac{52 - 50}{8/\sqrt{16}} = \dfrac{2}{2} = 1$ on $n - 1 = 15$ degrees of freedom (2 marks).

Compare with the critical value: $1 < 1.753$ , so the statistic is not in the critical region; do not reject $H_0$ . There is insufficient evidence at the $5\%$ level that the mean exceeds $50$ (1 mark). Markers reward the hypotheses, the test statistic with degrees of freedom, and the contextual conclusion.

AH style: paired t3 marksEight people have their blood pressure measured before and after a programme. Explain why a paired t-test is appropriate and state its hypotheses in terms of the mean difference.

Show worked answer →

The two measurements come from the same individuals, so the observations are paired, not independent; pairing removes the large person-to-person variation and tests the within-pair change (1 mark).

Work with the differences $d = \text{before} - \text{after}$ for each person, treating them as a single sample (1 mark).

Hypotheses on the mean difference $\mu_d$ : $H_0: \mu_d = 0$ (no change) against $H_1: \mu_d \neq 0$ (or one-sided if a direction is expected), tested with $t = \dfrac{\bar{d}}{s_d/\sqrt{n}}$ on $n - 1$ degrees of freedom (1 mark). Markers reward the justification for pairing, the use of differences and the hypotheses on $\mu_d$ .

Related dot points

Sources & how we know this

SQA Advanced Higher Statistics Course Specification (C803 77) — SQA (2023)
SQA Advanced Higher Statistics Data Booklet — SQA (2019)