Further Hypothesis Testing

This lesson extends the hypothesis testing framework to include tests for the mean of a normal distribution and the correlation coefficient. These are key topics at A-Level and bring together the concepts of the normal distribution, sampling, and statistical inference.

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Hypothesis Test for the Mean of a Normal Distribution

If $X \sim N(\mu, \sigma^2)$X∼N(μ,σ2) and $\sigma^2$σ2 is known, we can test hypotheses about $\mu$μ using the sample mean $\bar{X}$Xˉ.

Distribution of the Sample Mean

If $X \sim N(\mu, \sigma^2)$X∼N(μ,σ2), then the sample mean of a random sample of size $n$n follows:

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$Xˉ∼N(μ,nσ2​)

The standard error of the mean is:

$\text{SE} = \frac{\sigma}{\sqrt{n}}$SE=n​σ​

Test Procedure

1. State hypotheses:

   • $H_0: \mu = \mu_0$H0​:μ=μ0​
   • $H_1: \mu > \mu_0$H1​:μ>μ0​ or $\mu < \mu_0$μ<μ0​ or $\mu \neq \mu_0$μ=μ0​

2. Assume $H_0$H0​ is true: $\bar{X} \sim N\left(\mu_0, \frac{\sigma^2}{n}\right)$Xˉ∼N(μ0​,nσ2​)

3. Calculate the test statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$z=σ/n​xˉ−μ0​​

4. Compare with the critical value or find the p-value.

5. Make a decision and state the conclusion in context.

Example: A machine fills bottles with a mean of 500 ml ($\sigma = 5$σ=5 ml). A sample of 25 bottles has a mean of 498.2 ml. Test at 5% whether the mean has decreased.

$H_0: \mu = 500$H0​:μ=500, $H_1: \mu < 500$H1​:μ<500 (one-tailed)

$z = \frac{498.2 - 500}{5 / \sqrt{25}} = \frac{-1.8}{1} = -1.8$z=5/25​498.2−500​=1−1.8​=−1.8

Critical value at 5% (one-tailed): $z = -1.6449$z=−1.6449

Since $-1.8 < -1.6449$−1.8<−1.6449, reject $H_0$H0​. There is sufficient evidence at the 5% level to suggest the mean volume has decreased.

Exam Tip: Always state the distribution of $\bar{X}$Xˉ under $H_0$H0​ before calculating the test statistic. This shows the examiner you understand the sampling distribution, which is worth method marks.

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Large Samples and the Central Limit Theorem

The Central Limit Theorem states that for a large sample (typically $n \geq 30$n≥30), the distribution of $\bar{X}$Xˉ is approximately normal regardless of the underlying distribution:

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{(approximately)}$Xˉ∼N(μ,nσ2​)(approximately)

This allows us to apply normal-based hypothesis tests even when the population is not normally distributed, provided the sample is large enough.

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Hypothesis Test for the Correlation Coefficient

To test whether a population correlation coefficient $\rho$ρ is significantly different from zero:

1. State hypotheses:

   • $H_0: \rho = 0$H0​:ρ=0 (no linear correlation)
   • $H_1: \rho > 0$H1​:ρ>0 or $\rho < 0$ρ<0 or $\rho \neq 0$ρ=0

2. Calculate the sample PMCC $r$r.

3. Compare $|r|$∣r∣ with the critical value from the PMCC table for the appropriate $n$n and $\alpha$α.

4. State conclusion in context.

Example: A sample of 12 pairs of data gives $r = 0.65$r=0.65. Test at 5% whether there is positive correlation.

$H_0: \rho = 0$H0​:ρ=0, $H_1: \rho > 0$H1​:ρ>0 (one-tailed)

Critical value for $n = 12$n=12 at 5% (one-tailed): 0.4973

Since $0.65 > 0.4973$0.65>0.4973, reject $H_0$H0​. There is sufficient evidence at the 5% level of positive correlation.

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Confidence and Significance

Significance Level	Confidence Level	Meaning
10%	90%	Weak evidence required to reject $H_0$H0​
5%	95%	Moderate evidence required (most common)
1%	99%	Strong evidence required

Lower significance levels require stronger evidence to reject $H_0$H0​, reducing the risk of a Type I error but increasing the risk of a Type II error.

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Common Mistakes in Hypothesis Testing

1. Not stating hypotheses in terms of the population parameter (use $p$p, $\mu$μ, or $\rho$ρ, not $\hat{p}$p^​ or $\bar{x}$xˉ).
2. Confusing one-tailed and two-tailed tests — read the question carefully.
3. Failing to state the conclusion in context — always refer back to the original question.
4. Using the wrong distribution — check whether binomial or normal is appropriate.
5. Incorrect inequality direction when calculating probabilities.

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Summary

• Test for the mean $\mu$μ using $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$z=σ/n​xˉ−μ0​​ when $\sigma$σ is known.
• $\bar{X} \sim N(\mu, \sigma^2/n)$Xˉ∼N(μ,σ2/n) — the standard error decreases as $n$n increases.
• The Central Limit Theorem justifies normal approximation for large samples.
• Test for correlation using the PMCC table.
• Always state hypotheses, significance level, and conclusion in context.

Exam Tip: The most common reason students lose marks in hypothesis testing questions is failing to give a contextual conclusion. After making your statistical decision, translate it into plain English: "There is sufficient evidence at the 5% significance level to suggest that the average weight of apples has decreased."

A-Level Deep Dive: Further Hypothesis Testing

Spec mapping

AQA 7357 specification, Paper 3 — Statistics, Section S: Statistical Hypothesis Testing (Year 2 content) covers conduct a statistical hypothesis test for the mean of a normal distribution with known, given or assumed variance and interpret the results in context (refer to the official specification document for exact wording).; and "Understand and apply the language of statistical hypothesis testing developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value; extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value (calculation of correlation coefficients is excluded)." Section S sits alongside Section O (probability), Section P (statistical distributions, including the normal distribution), and Section R (sampling), and is examined principally on Paper 3 alongside Mechanics. Critical values for the standard normal distribution are provided in the AQA formulae and statistical tables booklet; critical values of the product-moment correlation coefficient are also tabulated.

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Worked example with full mark scheme

Question (8 marks):

A machine fills bottles with mineral water. The volume $X$X ml dispensed per bottle is modelled as $X \sim N(\mu, 4^2)$X∼N(μ,42), where the standard deviation $\sigma = 4$σ=4 ml is known from the manufacturer's calibration data. The machine is set so that $\mu = 500$μ=500 ml. After a maintenance visit, a quality engineer suspects the mean volume has changed. A random sample of $n = 25$n=25 bottles gives a sample mean of $\bar{x} = 498.4$xˉ=498.4 ml.

Test, at the 5% significance level, whether the mean volume has changed.

Solution with mark scheme:

Step 1 — state hypotheses.

Let $\mu$μ denote the population mean volume after maintenance. Then:

$H_0: \mu = 500, \qquad H_1: \mu \neq 500$H0​:μ=500,H1​:μ=500

B1 — both hypotheses correct, with $\mu$μ defined as a population parameter (not sample mean) and $H_1$H1​ two-tailed (the engineer suspects a change in either direction).

Step 2 — state distribution of the sample mean under $H_0$H0​.

Under $H_0$H0​, since $X \sim N(500, 16)$X∼N(500,16) with $\sigma$σ known:

$\bar{X} \sim N\!\left(500,\ \dfrac{16}{25}\right) = N(500, 0.64)$Xˉ∼N(500, 2516​)=N(500,0.64)

so the standard error is $\sigma/\sqrt{n} = 4/5 = 0.8$σ/n​=4/5=0.8.

M1 — quoting the sampling distribution of $\bar{X}$Xˉ with the correct variance $\sigma^2/n$σ2/n. The single most common error is using $\sigma^2$σ2 instead of $\sigma^2/n$σ2/n — that costs the M1 and propagates a wrong z-value through the rest of the question.

Step 3 — compute the test statistic.

$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \dfrac{498.4 - 500}{0.8} = \dfrac{-1.6}{0.8} = -2.00$z=σ/n​xˉ−μ0​​=0.8498.4−500​=0.8−1.6​=−2.00

M1 — correct standardisation formula applied.

A1 — $z = -2.00$z=−2.00 to at least 2 d.p. Sign matters: an answer of $+2.00$+2.00 (forgetting which way round subtraction goes) loses this A1 and may invalidate the conclusion.

Step 4 — compare with critical value.

For a two-tailed test at the 5% level, the critical values are $z = \pm 1.96$z=±1.96. The critical region is $|z| > 1.96$∣z∣>1.96.

B1 — correct critical value $\pm 1.96$±1.96 for a two-tailed 5% test. Candidates who quote $\pm 1.645$±1.645 (the one-tailed 5% value) lose this mark.

Step 5 — make a decision.

Since $|z| = 2.00 > 1.96$∣z∣=2.00>1.96, the test statistic falls inside the critical region. Reject $H_0$H0​.

M1 — explicit comparison of test statistic with critical value, stating decision in symbolic form.

Step 6 — conclude in context.

There is sufficient evidence at the 5% significance level to suggest that the population mean volume has changed from 500 ml after the maintenance visit. The sample evidence (mean 498.4 ml) is consistent with a slight under-fill.

A1 — conclusion stated in context of the original problem (mineral-water filling), referring to the population mean, with appropriate tentative language ("evidence to suggest", not "proves"). This is the most frequently lost mark on hypothesis-test questions.

Total: 8 marks (B1 M1 M1 A1 B1 M1 A1, with the second A1 carrying the contextual conclusion).

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Specimen question modelled on the AQA 7357 Paper 3 format

Question (6 marks): A psychologist measures the product-moment correlation coefficient between hours of sleep and reaction-time score for a random sample of $n = 20$n=20 adults. The sample value is $r = -0.524$r=−0.524. The critical value of the PMCC at the 1% one-tailed significance level for $n = 20$n=20 is given in the table as $0.5155$0.5155.

Test, at the 1% significance level, whether there is evidence of negative correlation between hours of sleep and reaction-time score in the underlying population. (6)

Mark scheme decomposition by AO:

• B1 (AO2.5) — defining $\rho$ρ as the population PMCC and stating $H_0: \rho = 0$H0​:ρ=0, $H_1: \rho < 0$H1​:ρ<0 (one-tailed, negative direction matching the question stem).
• B1 (AO1.1a) — identifying the correct critical value as $-0.5155$−0.5155 for a one-tailed test in the negative tail, recognising that the tabulated value $0.5155$0.5155 is positive and must have its sign flipped for a lower-tail test.
• M1 (AO1.1b) — comparing the sample value $r = -0.524$r=−0.524 with the critical value $-0.5155$−0.5155, e.g. "since $-0.524 < -0.5155$−0.524<−0.5155" or equivalently "since $|r| = 0.524 > 0.5155$∣r∣=0.524>0.5155".
• A1 (AO2.2a) — correct decision: reject $H_0$H0​.
• A1 (AO3.5b) — contextual conclusion referring to population correlation, sleep, and reaction time, with appropriate hedging.
• B1 (AO2.4) — critical commentary: noting that the test indicates evidence of correlation but not causation, or that the result is sample-dependent.

Total: 6 marks split AO1 = 2, AO2 = 3, AO3 = 1. PMCC tests are unusual in that calculation of $r$r is excluded from AQA — the entire question rests on interpretation, comparison, and contextual reasoning, which pushes AO2/AO3 weight unusually high.

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Synoptic links

Connects to:

1. Section P — The normal distribution: the Z-test for a mean is an application of the normal distribution to the sampling distribution of $\bar{X}$Xˉ. Confidence in standardising $Z = (X - \mu)/\sigma$Z=(X−μ)/σ extends directly to $Z = (\bar{X} - \mu)/(\sigma/\sqrt{n})$Z=(Xˉ−μ)/(σ/n​) — same formula, different denominator.

2. Section R — Statistical sampling and the sampling distribution of $\bar{X}$Xˉ: the result $\bar{X} \sim N(\mu, \sigma^2/n)$Xˉ∼N(μ,σ2/n) when $X$X is normal (or by the Central Limit Theorem when $n$n is large) is the engine of the Z-test. Without the CLT, the Z-test would only apply to genuinely normal populations.

3. Section S — Hypothesis test for a binomial proportion (Year 1): the Year 1 binomial test introduces the language of $H_0$H0​, $H_1$H1​, significance level, critical region, p-value. The Z-test re-uses every word, swapping the discrete binomial test statistic for the continuous $Z$Z.

4. Section O — Probability: the significance level $\alpha$α is $P(\text{reject } H_0 \mid H_0 \text{ true})$P(reject H0​∣H0​ true) — a conditional probability. Misinterpreting $\alpha$α as $P(H_0 \text{ true})$P(H0​ true) is the most common conceptual error and lies in elementary probability, not statistics.

5. PMCC / regression (Section T): the PMCC test connects hypothesis testing to bivariate data. Although calculation of $r$r is excluded at A-level, its interpretation synthesises Section P (normality assumptions on each variable), Section S (hypothesis-testing language), and the broader framing of inference about populations.

6. Mechanics (Paper 2): quality-control problems in a manufacturing context (filling machines, packaging weights, component lengths) are ubiquitous in Paper 3 statistics questions but routinely reference physical contexts students meet in mechanics — units, tolerances, and engineering-meaningful conclusions matter.

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Mark-scheme literacy

Hypothesis-test questions on AQA 7357 Paper 3 split AO marks more evenly than typical pure-mathematics questions:

AO	Typical share	Earned by
AO1 (knowledge / procedure)	35–45%	Stating hypotheses, computing test statistic, looking up critical value, making the formal comparison
AO2 (reasoning / interpretation)	35–45%	Choosing one- vs two-tailed correctly, interpreting significance level, writing the conclusion in context, defending the decision
AO3 (problem-solving / modelling)	10–25%	Critiquing modelling assumptions (was $\sigma$σ really known? is the population normal? was sampling random?), commenting on sample size adequacy

Examiner-rewarded phrasing: "Let $\mu$μ denote the population mean ..."; "Under $H_0$H0​, $\bar{X} \sim N(\mu_0, \sigma^2/n)$Xˉ∼N(μ0​,σ2/n)"; "Since $|z| = 2.00 > 1.96$∣z∣=2.00>1.96, the test statistic lies in the critical region"; "There is sufficient evidence at the 5% significance level to suggest ...". Phrases that lose marks: "the mean is 498.4" (confuses sample with population); "we accept $H_0$H0​" (statisticians do not accept $H_0$H0​ — they fail to reject it); "the test proves ..." (statistical tests provide evidence, not proof); writing the conclusion only in symbolic terms with no contextual statement.

A specific AQA pattern to watch: questions phrased "test whether there is evidence that ..." with a direction in the stem (e.g. "evidence that the mean has increased") demand a one-tailed test. Questions phrased "test whether the mean has changed" demand a two-tailed test. Reading this distinction wrong inverts the critical value and usually inverts the decision — losing every single mark from Step 4 onwards.

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Grade-band model answers

3-mark question

Question: State, with reasons, whether each of the following hypothesis-test setups uses a one-tailed or a two-tailed test:

(i) A teacher tests whether the mean exam score has changed since last year.

(ii) An engineer tests whether a new alloy has greater tensile strength than the standard alloy.

(iii) A pharmacist tests whether a drug's mean dissolution time differs from the manufacturer's stated value.

Grade C response (~150 words):

(i) Two-tailed because "changed" goes either way.

(ii) One-tailed because we want greater than.

(iii) Two-tailed because "differs" means up or down.

Examiner commentary: Full marks (3/3) for correct identifications. The reasoning is brief but the key trigger words ("changed", "greater", "differs") are correctly mapped to the right tails. This is what a 3-mark question of this kind looks like at Grade C — efficient, accurate, no decoration.

Grade A response (~190 words):*

(i) Two-tailed. "Changed" admits both increase and decrease, so $H_1: \mu \neq \mu_0$H1​:μ=μ0​. The critical region is split: $|z| > z_{\alpha/2}$∣z∣>zα/2​.

(ii) One-tailed (upper). "Greater" specifies a single direction, so $H_1: \mu > \mu_0$H1​:μ>μ0​. The full $\alpha$α is in the upper tail; the critical value is $z_\alpha$zα​, not $z_{\alpha/2}$zα/2​.

(iii) Two-tailed. "Differs" carries the same logic as "changed" — the manufacturer's value could be low or high. $H_1: \mu \neq \mu_0$H1​:μ=μ0​.