Game plan 6 · Unit 4

Sample statistics

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Sample proportions, margin of error and confidence intervals.

Reach for it when

The First Question found statistics, and the question mentions a confidence interval, sample proportion, margin of error or sample size.

Part 1

Question types

There are two families of question. Sample proportion questions are about how $\hat p$ changes from sample to sample. Confidence interval questions use one sample to estimate $p$.

Sample proportions

Confidence intervals

Interpreting or evaluating bias

  • Random sampling is best, then systematic, then self-selected.
  • Increasing the sample size makes the margin of error smaller.
  • Increasing the level of confidence makes the margin of error larger, and lowering it makes the margin of error smaller.
90%95%99%0.4
The same sample at 90%, 95% and 99% confidence. More confidence means a bigger $z$, so a wider interval.

In the Topic Guides

Part 2

Calculating a value

Every calculation uses one of three formulas. Most of the work is deciding which one the question wants.

Game plan 6.1

Calculating a value

  1. 1Identify what you are given: the sample proportion $\hat p$, the sample size $n$, and the confidence level with its z-score.
  2. 2Decide what you are asked for, then use the matching formula.
  3. 3You can also find a confidence interval with technology: MENU → STAT → INTR → Z → 1-PROP.
  4. 4You may need to work backwards to find an unknown, often $n$, from the margin of error or confidence interval formula.
EE0.3220.40.478
The interval is $\hat p\pm E$. Here $\hat p=0.4$ and $n=150$ at 95% confidence, so $E\approx0.078$.

Confidence level → z

90%$z=1.65$95%$z=1.96$99%$z=2.58$
σp
Sample proportions centre on $p$. A bigger $n$ makes $\sigma_{\hat p}$ smaller, so the curve is narrower.

Mean and SD of $\hat p$

$$\mu_{\hat p}=p\qquad\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$$

Margin of error

$$E=z\sqrt{\frac{\hat p(1-\hat p)}{n}}$$

Confidence interval for a proportion

$$\left(\hat p-z\sqrt{\frac{\hat p(1-\hat p)}{n}},\ \ \hat p+z\sqrt{\frac{\hat p(1-\hat p)}{n}}\right)$$

In the Topic Guides

Part 3

Common QCAA language: Sample statistics

Question typeWhat the question says
Analysing bias in samples“Which sampling method would best minimise bias”, “most appropriate / fairest sampling method”.Your move The fairest method gives every member the same chance of being picked. That is usually the random number generator.
Mean and SD of $\hat p$“The distribution of the sample proportion has a mean of … and a standard deviation of …”, “determine the mean and standard deviation of $\hat p$”.Your move This is about how $\hat p$ behaves across samples. Its mean is $p$ and its SD is $\sqrt{p(1-p)/n}$.
Sample size from mean and SDThe mean and SD of $\hat p$ are given, with “determine the sample size” or “the value of $n$”.Your move Both the mean and SD of $\hat p$ are given, and you need $n$. Solve the SD formula for $n$.
Interpreting a confidence interval“Which statement correctly interprets this”, “can be X% confident that the proportion …”, “what does 95% confidence mean”.Your move The right statement talks about taking many samples. Cross out any that say there is a probability that $p$ is in this one interval.
Calculating a confidence interval“Determine the approximate confidence interval for the proportion of …”, with $n$, $\hat p$ and a confidence level (or $z$) supplied.Your move Everything is given: $n$, $\hat p$ and $z$. Put them straight into the formula.
Solving for an unknown“Determine the margin of error”, “the lower / upper end of the interval is …”, “a margin of error of 3% … determine $n$”, “if the confidence level increases from 95% to 99% …”.Your move One value is missing. Rearrange the formula to get it.
Modelling with confidence intervalsAn applied context with “evaluate the reasonableness of the claim”, “is the claim justified”, “how large a sample is required to …” or “compare the intervals”.Your move Work out the interval first. Then see whether the claim fits inside it, or solve for the sample size you need.

In the Topic Guides

Previous: Game plan 5, The normal distribution. This is the last game plan. Back to the External Game plan.

Wording quoted from past QCAA Mathematical Methods 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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