Confidence intervals

Part 1

What a confidence interval is

From Topic 4, $\hat{p}$ is approximately normal with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$. We do not know $p$, so we use the next best thing, $\hat{p}$, in the standard deviation:

$$\sigma_{\hat{p}}\approx\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

A $z$-score says how many standard deviations $\hat{p}$ is from $p$. Rearranging $z=\frac{\hat{p}-p}{\sigma_{\hat{p}}}$ gives $p=\hat{p}\pm z\,\sigma_{\hat{p}}$, which is the approximate confidence interval on your formula sheet:

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

The interval is centred on $\hat{p}$. The bit you add and subtract is the margin of error, $E$:

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

What "95% confident" means

The 95% describes the method. It does not describe any one interval. If you took lots of samples and built an interval from each, about 19 in every 20 would contain the true $p$. Any one interval either contains $p$ or it does not. You just cannot tell which. The wording QCAA have used in the past is "we can be 95% confident that the population proportion is between $a$ and $b$". Write it in the context of the question.

Part 2

Margin of error, confidence and sample size

The confidence level sets the $z$-value. These are the ones to know. For any other level, use inverse normal on your calculator.

Read the margin of error formula and you can see both relationships:

For a fixed sample, you cannot have it both ways: a smaller margin of error means a lower confidence level. The only way to get both is a bigger sample.

Part 3

Calculating a confidence interval

This is the most common question type. In Technology-free questions the numbers are chosen so the square root comes out nicely. In Technology-active questions, your calculator's 1-PROP interval does the whole thing, but I would still write the formula with your numbers substituted so the marker can see your method.

Part 4

Working backwards from an interval

If you are given an interval $(a,b)$, two things fall straight out of it because it is centred on $\hat{p}$:

$$\hat{p}=\frac{a+b}{2}\qquad E=\frac{b-a}{2}$$

Then put what you know into the margin of error formula and solve for what is missing, usually $n$. Squaring both sides gets rid of the square root:

$$n=\frac{\hat{p}(1-\hat{p})}{\left(\frac{E}{z}\right)^{2}}$$

If the answer is not a whole number and the question is planning a survey, round up, so the margin of error is no bigger than required.

Part 5

Using intervals to judge a claim

Modelling questions usually end with a claim to evaluate. The interval is your list of believable values for $p$. If the claimed value is inside it, the data support the claim (or at least do not contradict it). If it is outside, the data do not support it. For a claim like "more than 75%", check whether the whole interval, or only part of it, sits above 0.75.

Always finish with a sentence in context, and mention a limitation if there is an obvious one: the sample might not be random, or another sample could give a different interval.

Practice

Practice questions

Eight questions of my own, covering every skill. Each one is labelled with the skill it practises, so if one goes wrong you know which part of the guide to go back to.

Keep practising

Ready for something harder?

Three modelling questions from this topic have their own complex unfamiliar walkthroughs. Try them from a blank page first.

Game plan

Your game plan for confidence intervals

1. 1Interpreting? "We can be 95% confident that the proportion of [population] who [feature] is between $a$ and $b$."
2. 2What changes the width? Higher confidence makes it wider, a bigger sample makes it narrower.
3. 3Calculating? Find $n$, $\hat{p}$ and $z$, then $\hat{p}\pm E$ by hand or 1-PROP.
4. 4Given an interval? Midpoint is $\hat{p}$, half the width is $E$. Square to solve for $n$.
5. 5Judging a claim? Inside the interval supports it, outside does not. Make the decision in context.