Porisma

Hypothesis tests for a mean

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By the end, you can

  • Formulate hypotheses before examining a result
  • Choose one-sided or two-sided evidence
  • Interpret p-values and conclusions in context

about 30 minutes for the lesson · practise at your own pace.

The central idea

A test asks how unusual the observed statistic would be if the null model were true. State H₀, H₁ and significance level first. For a normal population with known variance, the standardised sample mean uses z=(xˉ−μ0)/(σ/n)z=(\bar x-\mu_0)/(\sigma/\sqrt n). A p-value measures a tail probability under H₀, not the probability that H₀ is true.

Organise a z-test around the claim

The source’s eight steps are: define X and population mean μ; state the distributions of X and the sample mean under H0H_0 (including CLT justification if needed); state hypotheses; state the test statistic; state α; specify rejection using a p-value or critical value; calculate; conclude in context.

Its scenario assumes normal masses with variance 80 kg280\ \mathrm{kg}^2, samples 100 men and observes mean 77.2 kg. The claim “more than 75 kg” at 5% gives H0:μ=75H_0:\mu=75, H1:μ>75H_1:\mu>75 and Xˉ∼N(75,80/100)\bar X\sim N(75,80/100) under H0H_0, with an upper-tail test.

Here μ0=75\mu_0=75 is the hypothesised population mean, not the observed sample mean 77.2; the source paragraph calling μ₀ a sample mean is inaccurate. A test assesses how unusual the observation is under the null model, not the probability that H0H_0 is true. The full mass scenario is worked below; the later examples below apply the large-sample approximation and reverse test decisions to find unknowns.

Why take a sample?

The source contrasts testing a whole population with drawing a practical sample. Its introductory contexts are mean filtration efficiency in product-quality checks and a school mean-height claim of 1.6 metres assessed with a sample of 50 students.

A sample can supply evidence about a population parameter while leaving uncertainty; it cannot prove or disprove a claim with certainty.

The source also lists one-sample, two-sample and paired t-tests, ANOVA and chi-squared tests as examples of other test families. This chapter concentrates on a population mean using a z-test, with the exact or approximate reference model justified for the case.

Critical regions: allocate the significance level

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Three normal curves show 10% on the left below −1.28155, 10% on the right above 1.28155, and 5% in each tail beyond ±1.64485 for a two-sided 10% test.
Three normal curves show 10% on the left below −1.28155, 10% on the right above 1.28155, and 5% in each tail beyond ±1.64485 for a two-sided 10% test.
Three normal curves show 10% on the left below −1.28155, 10% on the right above 1.28155, and 5% in each tail beyond ±1.64485 for a two-sided 10% test.
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Observed tail probabilities

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Three normal curves show observed tails beyond −2, beyond 2, and beyond both −2 and 2. One-sided probability is 0.0228; the two-sided total is 0.0455.
Three normal curves show observed tails beyond −2, beyond 2, and beyond both −2 and 2. One-sided probability is 0.0228; the two-sided total is 0.0455.
Three normal curves show observed tails beyond −2, beyond 2, and beyond both −2 and 2. One-sided probability is 0.0228; the two-sided total is 0.0455.
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Compare p-values and critical values at 5%

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For observations −1.5 and 1.5, the one-sided p-value 0.0668 exceeds 0.05, so do not reject. For two-sided z=2, p=0.0455 and |z|>1.960, so reject. Darker shading marks the probability beyond the observation.
For observations −1.5 and 1.5, the one-sided p-value 0.0668 exceeds 0.05, so do not reject. For two-sided z=2, p=0.0455 and |z|>1.960, so reject. Darker shading marks the probability beyond the observation.
For observations −1.5 and 1.5, the one-sided p-value 0.0668 exceeds 0.05, so do not reject. For two-sided z=2, p=0.0455 and |z|>1.960, so reject. Darker shading marks the probability beyond the observation.
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Explore

Use the three 5% decisions above. Before calculating again, predict which decisions could change at 1%. For a two-sided test, sketch the rejection tails at both levels on the same standard normal curve. Explain why the observed statistic and its p-value stay fixed while the critical values move.

Worked example

A machine is intended to fill bottles to a population mean of 500 ml. Independent fills are normally distributed with known standard deviation 8 ml. A random sample of 16 bottles has mean 504 ml. Before taking the sample, the inspector chose a 5% test for an increase.

Define μ\mu as the mean fill volume and state H0:μ=500H_0:\mu=500, H1:μ>500H_1:\mu>500. Under H0H_0, Xˉ∼N(500,82/16)=N(500,4)\bar X\sim N(500,8^2/16)=N(500,4) exactly.

The standard error is 2 ml, so the observed z=(504−500)/2=2z=(504-500)/2=2.

The upper-tail probability is approximately 0.0228. Reject at 5%: there is evidence that the population mean fill exceeds 500 ml.

The test concerns the mean, not whether every bottle exceeds 500 ml.

A p-value is calculated under the null model. It is neither the probability that the claim is true nor a measure of the practical size of the difference.

Check your understanding

A supplier claims that the population mean lifetime of its lamps is 1200 hours. An inspector suspects a decrease and chooses a 5% test before collecting data. Lifetimes are normally distributed with known standard deviation 80 hours; 25 independent randomly selected lamps have mean lifetime 1176 hours. Set up and carry out the test, explain the p-value in context, and state what the conclusion can and cannot establish.

Mastery learning objectives

At the end of this chapter, I should be able to:

Direct hypothesis tests

Understand the procedure for carrying out a zz-test.

Make the necessary adjustments to a hypothesis-test presentation when the situation changes.

For a non-normal population, justify the large-sample normal approximation using the CLT and its sampling conditions.

When the population variance σ2\sigma^2 is unknown, use the unbiased estimate s2s^2 with the appropriate large-sample qualification.

Find unknowns from a test conclusion

Use the p-value method to find an unknown significance percentage α\alpha.

Use critical-value inequalities to find unknown μ0\mu_0, xˉ\bar x, nn or s2s^2, preserving strictness, domains and precision.

Choose a practice question

The following nine rows retain the printed source map and its question references. A dash means the source cell is blank. The qualifications below identify misplaced references; the linked Discussion and Review sets retain their own reviewed wording.

Common question typeLecture examplesDiscussionReview follow-up
Explain significance or p-value in context—1(ii), 5, 7(iv)6(ii), 9(b)(iii)
Explain whether a sample is random——9(a)
Compare one-tailed and two-tailed conclusions—4(a)(b)7(iii)
Test a normal-population mean using z1, 2, 5(i)1(i)—
Test a non-normal-population mean using a large sample and CLT32, 4(1st part), 5, 7(i)(ii), 8(i)(ii)1(i), 4, 6, 8, 9(b)(i)(ii), 10(iv)
Find unknown significance using p47(iii)3
Find unknown reference mean using critical values5(ii)62, 5
Find unknown observed mean using critical values6(i)(ii)3, 8(iii)7(i)(ii), 9(b)(iv), 10(v)
Find unknown n using critical values6(iii)1(iii)1(ii)

Source-map qualifications: Example 5(i) is printed in the normal-population row, but its cereal population is not stated normal; its large n=80 working uses CLT.

The printed Review 9(a)/9(b) labels do not occur on the current Review 9, which is the virologist question with parts (i)–(v).

For the OUT sampling-frame and score questions use Review 8(a), 8(b)(i)–(iv).

For a one-sided deduction from two-sided rejection use Review 6(iii); printed Review 7(iii) instead pools bearing samples.

Review 6(ii), printed in the p-value-meaning row, asks for waiting-mean bounds; Review 5’s final ask explicitly requests a contextual p-value explanation. These source references are retained rather than silently renumbered.

Practice questions

Practice section 1 · Questions 1–6

Assess Alice’s claim about mean mass · Turn three claims into hypotheses · +4 more

Practice section 2 · Questions 7–12

Test cereal mass and invert a two-sided rejection · Reverse a known-variance test for mean time and sample size · +4 more

Practice section 3 · Questions 13–18

Pool battery samples before calculating a test · Find the reference means rejected by an upper-tail test · +4 more

Practice section 4 · Questions 19–24

Find significance levels that reject a wage claim · Test a filling machine using a sample standard deviation · +4 more

Practice section 5 · Questions 25–26

Separate a large-sample test from a known-variance test · Distinguish the target population and invert a non-rejection condition

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