# In a poll of 1,000 randomly selected adults, 400 answered “yes” when asked if they planned to vote in the state election. What is the best point estimate of the population proportion of all adults who plan to vote in that election?

Mar 18, 2018

Between 36.96% and 43.04% plan to vote in the election.

#### Explanation:

If we want to know if an event will happen, we can't just use the poll and accept it as fact. We need to find the standard error. For a proportion, this is the equation:
$\sigma \hat{p} = \sqrt{\frac{\hat{p} \left(1 - \hat{p}\right)}{n}}$
where $\hat{p}$ is the sample proportion, and $n$ is the sample size. Using the information given, let's find the standard deviation. $\hat{p}$ is the number that said "yes" over the total number asked, which ends up being 0.40. $n$ is 1,000, since that's how many people were asked. so, let's solve.

$\sigma \hat{p} = \sqrt{\frac{0.40 \left(1 - 0.40\right)}{1000}}$
$\sigma \hat{p} = \sqrt{\frac{0.40 \cdot 0.60}{1000}}$
$\sigma \hat{p} = \sqrt{\frac{0.24}{1000}}$
$\sigma \hat{p} = \sqrt{0.00024}$
$\sigma \hat{p} = 0.0155$

Now that we have our standard error, we should find the margin of error. This can be used to find the range that the actual answer to this problem would be in. In statistics, we usually say that we want a 95% confidence level, meaning we want to be 95% confident that the actual answer is in this range. To find this range, we multiply the standard deviation (standard error) by the critical value. A critical value is the number we multiply the standard deviation by to find the margin of error. In the case of a 95% confidence level, the critical value is 1.960.

margin of error = $1.96 \cdot 0.0155$
margin of error = $0.0304$
margin of error = 3.04%

Now that we have the margin of error, we add it to or subtract it from the mean to find the range I mentioned earlier. The mean is the number of people who said "yes" over the total number of people asked.

mean = $\frac{400}{1000}$
mean = $0.40$
mean = 40%

mean - SD = 40% - 3.04%
mean - SD = 36.96%

mean + SD = 40% + 3.04%
mean + SD = 43.04%

So, our answer is between 36.96% and 40.04%.