# Margin of error

### Margin of error

#### Lessons

• Introduction
The maximum difference from $\hat{p}$ to $p$ is $E$

$\cdot$ $E=Z_\frac{\alpha}{2} \sqrt{ \frac{ \hat{p} (1-\hat{p})}{n}}$
$\cdot$ $\hat{p}$: the point estimate, a sample estimate
$\cdot$ $p$: the population proportion (this is the data we are concerned with ultimately finding)
$\cdot$ $n$: the sample size

• 1.
Finding the Margin of Error
A sample of $n=750$ is polled from a population. The sample has a critical value of $Z_\frac{\alpha}{2}=1.75$ with a point estimate of $\hat{p}=0.44$. What is the margin of error for estimating the population proportion?

• 2.
A ski mountain (Whistler) sees 25,000 visitors a day. The company running the ski mountain wishes to estimate the number of snowboarders who visit this mountain daily. The company surveys 100 people who are visiting the mountain and finds that 43 of them are snowboarders. If the company desires a confidence level of 0.90 in their calculations, what is their margin of error?

• 3.
A survey is done linking the number of concussions a season to hockey players. The NHL consists of $700$ players and we wish to have a confidence level of $0.95$ (corresponding to a critical value of $Z_\frac{\alpha}{2}=1.96$). Let us assume that the point estimate will be $\hat{p}=0.0125$ for every sample taken.
a)
What is the margin of error, if 20 players are sampled?

b)
What is the margin of error, if 100 players are sampled?

c)
What is the margin of error, if all 700 players in the NHL are polled?