Confidence Intervals for the Population Proportion (VCE SSCE Mathematical Methods): Revision Notes

Confidence Intervals for the Population Proportion

Introduction

When we collect data from a sample, our goal is to learn something about the larger population from which the sample was drawn. We know what's in our sample, and we want to use this information to make educated guesses about the population.

Point estimates

A point estimate is a single value used to estimate an unknown population parameter.

The sample proportion $\hat{p}$ can be used as a point estimate for the population proportion $p$. Since this gives us just one value, it's called a point estimate.

Interval estimates and confidence intervals

Why we need interval estimates

The sample proportion $\hat{p}$ changes from sample to sample. Sometimes it will be close to the population proportion $p$, and sometimes it won't be. Using a single value to estimate $p$ can be risky.

What we need is an interval that we are reasonably confident contains the true value of $p$.

Definition: An interval estimate for the population proportion $p$ is called a confidence interval for $p$.

The mathematical foundation

When the sample size $n$ is large, the sample proportion $\hat{P}$ has an approximately normal distribution with:

• Mean: $\mu = p$
• Standard deviation: $\sigma = \sqrt{\frac{p(1-p)}{n}}$

By standardising, the distribution of $\frac{\hat{P} - p}{\sqrt{\frac{p(1-p)}{n}}}$ is approximately standard normal (denoted $Z$).

Deriving a 95% confidence interval

For a 95% confidence interval, we want symmetric values around $\hat{p}$. Using the inverse normal function:

• Upper critical value: $c_2 = 1.9600$ (inverse normal of 0.975)
• Lower critical value: $c_1 = -1.9600$ (by symmetry)

Therefore: $\text{Pr}(-1.9600 < Z < 1.9600) = 0.95$

This means:

$$\text{Pr}\left(-1.9600 < \frac{\hat{P} - p}{\sqrt{\frac{p(1-p)}{n}}} < 1.9600\right) \approx 0.95$$

After algebraic manipulation, we get:

$$\text{Pr}\left(\hat{P} - 1.9600\sqrt{\frac{p(1-p)}{n}} < p < \hat{P} + 1.9600\sqrt{\frac{p(1-p)}{n}}\right) \approx 0.95$$

Making it practical

Since we don't know $p$ (that's what we're trying to estimate!), we substitute $\hat{p}$ for $p$ in the standard deviation formula. This gives us an approximate 95% confidence interval:

$$\left(\hat{p} - 1.9600\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + 1.9600\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)$$

General formula for any confidence level

For a $C\%$ confidence interval:

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

where $k$ is chosen such that $\text{Pr}(-k < Z < k) = \frac{C}{100}$

Worked example: Comparing confidence intervals

Interpreting confidence intervals correctly

Common misunderstanding

Correct interpretation

The proper interpretation is: if we repeated this sampling process many times and calculated a 95% confidence interval each time, approximately 95% of those intervals would contain the true population proportion $p$.

Whether any particular confidence interval contains $p$ is unknown to us.

Visualising the concept

This diagram shows confidence intervals from 20 different samples taken from the same population:

• Each horizontal line represents one confidence interval
• The dots show the sample estimates $\hat{p}$
• The vertical dashed line shows the true population proportion $p$
• Notice that 19 out of 20 intervals (95%) capture the true value of $p$
• One interval missed (the expected result for 95% confidence)

Using a calculator to find confidence intervals

Worked example: University students

Margin of error

Definition

The margin of error ($M$) is the distance between the sample estimate and the endpoints of the confidence interval.

For example, we might say: "We predict 52% ± 2% of people will vote Labor in the next election." Here, the sample estimate is 52%, and the margin of error is 2%.

Formulas for margin of error

For a 90% confidence interval:

$$M = 1.6449\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

For a 95% confidence interval:

$$M = 1.9600\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

For a 99% confidence interval:

$$M = 2.5758\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Factors affecting margin of error

Determining required sample size

Worked example: Sample size calculation

General formula for sample size

Usually, we don't know $\hat{p}$ before taking the sample. Instead, we use an estimated value $p^*$ based on:

• Prior knowledge of the population
• Results from a pilot study

For a 90% confidence interval:

$$n = \left(\frac{1.6449}{M}\right)^2 p^*(1-p^*)$$

For a 95% confidence interval:

$$n = \left(\frac{1.9600}{M}\right)^2 p^*(1-p^*)$$

For a 99% confidence interval:

$$n = \left(\frac{2.5758}{M}\right)^2 p^*(1-p^*)$$

where:

• $M$ is the desired margin of error
• $p^*$ is an estimated value for the population proportion $p$

Remember!