Statistics / Unit 8: Sampling Distributions and Confidence Intervals
8.4 Confidence Intervals for a Proportion
Notes
A confidence interval gives a range of plausible values for a parameter, using a statistic and a margin of error.
- Interval $= \hat{p} \pm \text{margin of error}$, where margin of error $= z^* \cdot \text{SE}$ and $\displaystyle \text{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
- For $95\%$ confidence, $z^* \approx 2$ (more exactly $1.96$). In these problems use $z^* = 2$ unless told otherwise.
- Conditions: a random sample, and at least 10 successes and 10 failures in the sample.
- Example: $\hat{p} = 0.5$ and $n = 100$ give $\text{SE} = 0.05$, margin of error $= 0.10$ and interval $0.40$ to $0.60$.
Practice
In a random sample of 400 voters, 320 favor a proposal. Use $z^* = 2$. The $95\%$ confidence interval is:
$\hat{p} = 0.8$, $\displaystyle \text{SE} = \sqrt{\frac{0.8 \cdot 0.2}{400}} = 0.02$, margin of error $= 0.04$.
A confidence interval for a proportion is $0.42$ to $0.58$. The sample proportion and margin of error are:
$\hat{p}$ is the midpoint, 0.50, and the margin is half the width: $\displaystyle \frac{0.58 - 0.42}{2} = 0.08$.
Which condition is needed to use the confidence interval for a proportion?
A random sample and enough successes and failures make the normal model appropriate.
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