Game plan 9 · Unit 4

Statistical inference

One formula, used forwards and backwards. Every answer ends with a sentence in context.

Reach for it when

You see $\bar x$, $\mu$, $\sigma$, $s$, $\frac{\sigma}{\sqrt n}$, “sample mean”, “confidence interval”, “margin of error”, or “based on this sample”.

Part 1

Statistic or parameter?

Game plan 9.1

Statistic or parameter?

  1. 1Greek letters ($\mu$ and $\sigma$) are parameters: fixed, usually unknown properties of the population. Latin letters ($\bar x$ and $s$) are statistics, worked out from the sample.
  2. 2Interpretation questions depend on this difference. A confidence interval estimates a parameter using statistics.
PropertyPopulation (parameter)Sample (statistic)
Mean$\mu$$\bar x$
Standard deviation$\sigma$$s$
Variance$\sigma^2$$s^2$
Size$N$$n$

A third object

$\bar X$ is a random variable in its own right, with its own mean and spread: $\mu_{\bar X}=\mu$ and $\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$. Write them with these labels, as the QCAA marking guides do. That isn’t the sample column above. Telling the two apart is Game plan 9.3, and mixing them up is the error in the “Watch for” box below. To earn interpretation marks, say which column a symbol comes from.

In the Topic Guides

Part 2

The distribution of sample means

Game plan 9.2

Matching a sample distribution to its population

  1. 1When you’re asked which distribution best represents samples drawn from a population, check three things in order.
  2. 2Shape: should the sample means be roughly normal? If the population is normal, they are normal for any sample size. If it is not, they are approximately normal once $n\ge30$, by the central limit theorem.
  3. 3Centre: the sample means should sit close to the population mean.
  4. 4Spread: the distribution of sample means must be narrower than the population. That is the giveaway, and the one most often missed.
0122436Individual valuesthe population: wide, and any shape0122436Sample meansnarrower, and roughly normal once n ≥ 30
Averaging shrinks the spread by a factor of $\sqrt n$. That is why the standard deviation you enter is $\frac{\sigma}{\sqrt n}$, not $\sigma$.
Game plan 9.3

Probabilities and quantiles for a sample mean

  1. 1State the sampling distribution: $\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right)$, so $\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$. Give the reason it is normal: the population is normal (then any $n$ will do), or $n\ge30$ by the central limit theorem, whatever the shape of the population. Write this, naming $n$, before you use the normal distribution. It is worth a mark.
  2. 2Standardise with the sample-mean $z$-score: $z=\frac{\bar x-\mu}{\sigma/\sqrt n}$.
  3. 3Paper 2: use normal CDF or inverse normal on your calculator, entering the standard deviation as $\frac{\sigma}{\sqrt n}$, not $\sigma$.
  4. 4Paper 1: use the empirical rule. About 68% is within one standard deviation, 95% within two and 99.7% within three. Questions are built around these values.

Watch for

Entering $\sigma$ instead of $\frac{\sigma}{\sqrt n}$ into the calculator is the most common inference error.

In the Topic Guides

Part 3

Constructing a confidence interval

Game plan 9.4

Constructing a confidence interval

  1. 1Pick out the four inputs: $\bar x$, $s$ (or $\sigma$, if the question tells you to use the population standard deviation), $n$, and the $z$-value (given in the question on Paper 1, from the calculator on Paper 2). Write all four down before you substitute. The first mark is often for getting them right.
  2. 2Work out the interval $\left(\bar x-z\frac{s}{\sqrt n},\ \bar x+z\frac{s}{\sqrt n}\right)$, which is on the sheet.
  3. 3Interpret it in context: an interval estimate for the population mean, at the stated confidence level.
x̄ − Ex̄x̄ + E EE claimed μ inside: consistentoutside: evidence against
The width is $2E$ and the centre is $\bar x$. Every working-backwards question is this picture rearranged, and judgement questions only ask whether the claimed $\mu$ lands inside the band.

In the Topic Guides

Part 4

Working backwards

Game plan 9.5

Working backwards

  1. 1The QCAA often asks you to use the confidence interval formula backwards. Given a margin of error, an interval width or a probability, solve for $n$, $s$ or $z$.
  2. 2Anchor on $E=z\frac{s}{\sqrt n}$ (on the sheet, with $\sigma$ in place of $s$ when the question says so) and note that the width is $2E$. The interval’s centre is $\bar x$ and its half-width is $E$.
  3. 3Rearrange for the unknown. When you solve for $n$, remember to square, then round to a whole number with care. For the smallest sample size that achieves a stated precision, round up. To recover the sample size behind a given interval, round to the nearest whole number: 75.02 is 75, as in 2025 Paper 2 Question 17.

In the Topic Guides

Part 5

Judgement questions

Game plan 9.6

Judgement questions

  1. 1“Does the data support the claim?” Check whether the claimed value of $\mu$ lies inside the confidence interval.
  2. 2Answer with a decision and the reason. Inside means the data are consistent with the claim. Outside means the sample provides evidence against it, at this confidence level.
  3. 3Use inference language. The interval is a point estimate with a margin of error either side, giving an interval estimate for $\mu$ at the stated level of confidence. Intervals vary between samples, and most but not all of them contain $\mu$. It does not “contain 95% of the data”.

Habit

Every inference answer ends in context: the population, the units and the confidence level.

In the Topic Guides

Formulas

Formula panel

Each formula is labelled with how to treat it in the exam.

The mean and standard deviation of the sample mean, $\mu_{\bar X}=\mu$ and $\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$
On the sheet
The margin of error $E=z\frac{s}{\sqrt n}$ and the confidence interval for $\mu$
On the sheet
The distribution statement $\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right)$ and the large-sample condition $n\ge30$
Memorise
The sample-mean $z$-score $z=\frac{\bar x-\mu}{\sigma/\sqrt n}$, used forwards and backwards
Memorise
The empirical rule, 68–95–99.7%: the only technology-free route to probabilities
Memorise
$z$-values: given on Paper 1, from the calculator on Paper 2. Know that 90% goes with 1.645, 95% with 1.96 and 99% with 2.576 for judgement questions. Marking guides also accept 1.64 or 1.65, and 2.58. A technology-free question will give you a friendlier value, often 2.
Supplied

QCAA language

What the question says, and what it means

Question typeWhat you’ll seeWhat it means
Distribution“The mean of a random sample of $n$ …”Game plan 9.3. State the distribution of $\bar X$ first
Construct“Determine an approximate confidence interval”Game plan 9.4. Four inputs, then the sheet formula
Working backwards“Determine the smallest sample size”, “the margin of error is …”Game plan 9.5. Rearrange, then round (up for a smallest sample size)
Judgement“State whether …”, “justify using mathematical reasoning”Game plan 9.6. Inside or outside, plus the reason

Back to the External Game plan or the Topic Guides.

Wording quoted from past QCAA Specialist Mathematics external assessments is © State of Queensland (QCAA), licensed under CC BY 4.0. Tangent Tuition is not affiliated with the QCAA.

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