An estimate with a margin of error

What you will be able to do

• give a point estimate and its margin of error;
• explain where the number $1.96$ comes from;
• turn an estimate and a margin into an interval, and read one in the news.

A poll about a park

A made-up town plans a new park. A polling team asks $1000$ adults, chosen at random, and $520$ say they back the plan. What share of all the adults back it?

Our best single guess is the sample share, $\hat p = \dfrac{520}{1000} = 0.52$. A single number used as a guess for a parameter is called a point estimate.

You know from the first course of this chapter that a second sample would give a different share. So how far from the truth could $0.52$ be?

How far off?

The standard error of a share tells us. We do not know $p$, so we use $\hat p$ in its place:

Sample shares follow a normal curve centred at the truth. About $95\%$ of the area of a normal curve lies within $1.96$ standard deviations of its centre. You met this number in the chapter on random variables and distributions. So in $95\%$ of samples, $\hat p$ lands within $1.96$ standard errors of $p$.

Turning it around

If $\hat p$ is within $1.96$ standard errors of $p$, then $p$ is within $1.96$ standard errors of $\hat p$. The distance is the same either way. So we can say

For the park, the margin is $1.96 \times 0.0158 \approx 0.031$. The 95% confidence interval runs from $0.52 - 0.031 = 0.489$ to $0.52 + 0.031 = 0.551$. A news report would say "$52\%$ back the park, with a margin of error of $3$ points".

You should see the bar shrink as the survey grows, and widen a little as the sample share moves towards $0.5$. A share near one half carries the most uncertainty.

Reading a margin in the news

Notice what the park interval says about a majority. It runs from just under $0.49$ to about $0.55$. So the data fit a true share a little below one half as well as one above it. The headline "a majority backs the park" goes further than the poll can.

Practise with new numbers

Try it in Python

The program turns the count of the park poll into the estimate, the margin of error and the interval.

Common mistakes

• Reporting the estimate alone. Without a margin, nobody can tell how much to trust it.
• Using the standard error as the margin. The margin for $95\%$ is $1.96$ standard errors, about twice the standard error.
• Forgetting that the margin covers chance only. A biased question or a biased sample adds an error that no margin shows.

Remember these

• A point estimate is a single number used to estimate a parameter, such as $\hat p$ for $p$.
• The margin of error for $95\%$ confidence is $1.96$ standard errors.
• The interval is estimate $\pm$ margin of error.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A made-up survey of $250$ students finds that $85$ play a musical instrument. What is the point estimate of the share of all the students who play one?

2. A standard error is $0.02$. What is the margin of error for $95\%$ confidence?

3. A report says that $38\%$ of drivers want a new bridge, with a margin of error of $3$ points. Give the interval.

4. Why does the margin of error use $1.96$ and not some other number?

5. A poll of $1000$ people has a margin of $3$ points. Does it also cover the error made by a leading question?

References

• OpenStax, Introductory Statistics, Chapter 8, section on a population proportion.