Confidence Intervals for Proportions and Means: Quick Revision

A brief note on Confidence Intervals for Proportions and Means. Read it before a test. Try each question at the end before you open its answer.

Rules

• Point estimate: a single number used for a parameter, such as $\hat p$ for $p$ or $\bar x$ for $\mu$.
• Margin of error at 95%: $1.96$ standard errors.
• Interval for a proportion: $\hat p \pm 1.96\sqrt{\dfrac{\hat p(1 - \hat p)}{n}}$.
• Conditions: a random sample, at least $10$ successes and $10$ failures, a sample under a tenth of the population.
• Sample size: $n \ge \left(\dfrac{1.96}{m}\right)^2 \hat p(1 - \hat p)$, rounded up; use $\hat p = 0.5$ with no guess.
• Interval for a mean (large sample): $\bar x \pm 1.96\dfrac{s}{\sqrt n}$. Small samples use the Student t multiplier.
• Other levels: $1.645$ for $90\%$ and $2.576$ for $99\%$.

What the level means

• The method catches the true value in about $95\%$ of samples.
• It is not a $95\%$ chance for one interval, not a range for single values, and not a forecast of other samples.

A worked example

A made-up survey of $400$ people finds that $180$ read a paper every day.

1. Conditions. $180$ successes and $220$ failures, both at least $10$.
2. Estimate and margin. $\hat p = 0.45$, and the margin is $1.96\sqrt{\dfrac{0.45 \times 0.55}{400}} \approx 0.0488$.
3. Interval. From about $0.401$ to about $0.499$.

Practice with new numbers

A program to try

The program checks the conditions and works out the interval of the worked example.

Mistakes to avoid

• Using the standard error as the margin. Multiply by $1.96$.
• Rounding a sample size down. Always round up.
• Saying "a 95% chance" for one interval.
• Skipping the conditions, especially the $10$ successes and $10$ failures.

Check yourself

1. A sample share is $0.5$ from $100$ people. Find the margin of error at $95\%$.

2. An interval runs from $0.22$ to $0.30$. Find the estimate and the margin.

3. Which is wider for the same data, the 95% or the 90% interval?

4. A survey of $50$ people has $8$ successes. Can the formula be used?

5. A sample of $36$ has $s = 12$ g. Find the margin of the 95% interval for the mean.

Sources

• OpenStax, Introductory Statistics, Chapter 8, confidence intervals for a population mean and a population proportion.