A researcher tests H_0: μ = 20 against H_1: μ > 20 at α = 0.05 and obtains a p-value of 0.08. State the conclusion. [1 mark]

ScotlandStatisticsSyllabus dot point

How do you set up a hypothesis test and decide whether to reject the null hypothesis?

Set up null and alternative hypotheses, choose a significance level, compute and use a test statistic and p-value, decide between one- and two-tailed tests, identify the critical region, and distinguish Type I and Type II errors.

A focused answer to the SQA Advanced Higher Statistics hypothesis testing framework: forming null and alternative hypotheses, the significance level, the test statistic, the p-value and critical region, one- and two-tailed tests, and Type I and Type II errors.

Generated by Claude Opus 4.812 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
Setting up the hypotheses
The significance level, test statistic and p-value
One- and two-tailed tests and the critical region
Type I and Type II errors
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What this dot point is asking

Hypothesis testing is the formal procedure for deciding whether sample data give convincing evidence against a claim. The SQA wants you to state hypotheses correctly, fix a significance level, compute a test statistic and p-value, choose between a one- and two-tailed test, identify the critical region, and reach a conclusion while understanding the two kinds of error you might make. This framework is shared by every named test in the area.

Setting up the hypotheses

Every test begins with a pair of hypotheses about a population parameter.

You never "prove" $H_0$ ; you either reject it (the evidence is strong enough) or fail to reject it (the evidence is insufficient). This asymmetry is deliberate: the test protects $H_0$ unless the data are convincing.

The significance level, test statistic and p-value

The decision rests on comparing a p-value with a pre-chosen significance level.

Apply the p-value decision rule

The p-value measures how surprising the data would be if $H_0$ were true. A small p-value (below $\alpha$ ) means the data are too extreme to be easily explained by $H_0$ alone, so we reject it. A large p-value means the data are consistent with $H_0$ , so we do not reject.

Step 1: State the hypotheses and significance level

We are testing $H_0: \mu = 50$ against $H_1: \mu \neq 50$ (two-tailed) at $\alpha = 0.05$ . The significance level $\alpha$ was fixed before seeing the data; it is the maximum probability of wrongly rejecting a true $H_0$ that we are prepared to tolerate.

Step 2: Compare the p-value with $\alpha$

The analysis produces a p-value of $0.012$ . We apply the decision rule: reject $H_0$ if p-value $< \alpha$ .

0.012 < 0.05 \implies \text{reject } H_0

The result is more extreme than the 5% threshold would allow if $H_0$ were true.

Step 3: State the conclusion in context

Reject $H_0$ : there is significant evidence at the 5% level that the population mean differs from 50.

Final answer: p-value $= 0.012 < 0.05$ , so we reject $H_0$ and conclude the mean differs significantly from $50$ .

One- and two-tailed tests and the critical region

Whether you look in one tail or both depends entirely on the alternative hypothesis.

Find a critical value for a two-tailed z-test

For a two-tailed test, we reject $H_0$ if the test statistic is extreme in either direction. The significance level $\alpha$ is therefore shared equally between the two tails, giving each tail an area of $\alpha/2$ .

Step 1: Identify the area in each tail

The alternative hypothesis $H_1: \mu \neq \mu_0$ is two-sided, so at $\alpha = 0.05$ we place half the rejection area in each tail.

\alpha/2 = 0.05/2 = 0.025

Step 2: Find the critical $z$ -values from the table

We need the $z$ -value such that $P(Z > z) = 0.025$ , equivalently $P(Z < z) = 0.975$ . From the standard normal table:

z = 1.96 \quad \text{(and by symmetry } {-1.96} \text{ for the lower tail)}

Step 3: State the critical region

The critical region is the set of test-statistic values for which $H_0$ is rejected. The statistic must fall beyond the critical values in either direction.

Final answer: Reject $H_0$ if $z < -1.96$ or $z > 1.96$ ; otherwise do not reject.

Type I and Type II errors

Because the decision is based on a sample, two mistakes are possible.

Try this

Q1. A researcher tests $H_0: \mu = 20$ against $H_1: \mu > 20$ at $\alpha = 0.05$ and obtains a p-value of $0.08$ . State the conclusion. [1 mark]

Cue. Since $0.08 > 0.05$ , do not reject $H_0$ : there is insufficient evidence at the $5\%$ level that the mean exceeds $20$ .

Q2. State which error becomes more likely if the significance level is lowered from $5\%$ to $1\%$ , keeping the sample size fixed. [1 mark]

Cue. A Type II error becomes more likely, because a stricter rejection threshold makes a true effect harder to detect.

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: set up test3 marksA manufacturer claims the mean lifetime of a bulb is

1000

hours. A consumer group suspects it is less. State the null and alternative hypotheses, say whether the test is one- or two-tailed, and explain how the significance level is used.

Show worked answer →

Hypotheses: $H_0: \mu = 1000$ against $H_1: \mu < 1000$ , where $\mu$ is the true mean lifetime (1 mark).

The test is one-tailed (lower tail) because the suspicion is specifically that the mean is less than $1000$ , not merely different (1 mark).

The significance level $\alpha$ (for example $0.05$ ) is the probability of rejecting $H_0$ when it is in fact true; if the p-value is less than $\alpha$ , or the test statistic falls in the critical region, $H_0$ is rejected (1 mark). Markers reward the correctly directed hypotheses, the one-tailed identification and the role of $\alpha$ .

AH style: errors3 marksDefine a Type I and a Type II error in hypothesis testing, and state what reducing the significance level does to each.

Show worked answer →

A Type I error is rejecting $H_0$ when it is actually true (a false positive); its probability is the significance level $\alpha$ (1 mark).

A Type II error is failing to reject $H_0$ when it is actually false (a false negative); its probability is denoted $\beta$ (1 mark).

Reducing $\alpha$ lowers the chance of a Type I error but, for a fixed sample size, raises the chance of a Type II error, because a stricter rejection rule makes it harder to detect a real effect (1 mark). Markers reward both definitions and the trade-off between the two error types.

Related dot points

Sources & how we know this

SQA Advanced Higher Statistics Course Specification (C803 77) — SQA (2023)
SQA Advanced Higher Statistics Course Overview and Resources — SQA (2023)