# Making a confidence interval

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##### Intros

###### Lessons

- Confidence interval is given by: $\hat{p} -E$ < $p$ < $\hat{p}+E$

Or equivalently: $p=\hat{p} \pm E$

$\cdot$ $E=Z_\frac{\alpha}{2} \sqrt{ \frac{ \hat{p} (1-\hat{p})}{n}}$

$\cdot$ $\hat{p}$: 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

$\cdot$ $Z_\frac{\alpha}{2}$: the critical value

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##### Examples

###### Lessons

**Finding the Confidence Intervals**

Let $\hat{p} =0.65$, $n=250$, and $Z_\frac{\alpha}{2}=1.645$. What is the resulting confidence interval?

- It was found that from a sample of students from an introductory statistics course, which has 50 students enrolled, 15 of these students planned to take higher level statistics courses. There are several statistics courses with an entirety of 300 students in all these courses together. We want to estimate the number of students who are going to take higher level statistics courses with a 0.95 confidence level.
- It was found that from a sample of students from an introductory statistics course, which has 50 students enrolled, 15 of these students planned to take higher level statistics courses. There are several statistics courses with an entirety of 300 students in all these courses together. We want to estimate the number of students who are going to take higher level statistics courses with a 0.95 confidence level.

If we want to raise our confidence level from 0.95 to 0.98, what is the resulting confidence interval? **Determining the Sample Size**

If $\hat{p}=0.40$, and we want to be $95%$ certain that our population proportion lies within $0.025$ of $\hat{p}$ , then what size does our sample have to be?- A new start-up micro-brewing company called "Thirsty Brothers Brewing" is giving out free beer to the general populace. They discovered that out of the 75 free beers given out to customers; a total of 72 customers liked the beers. "Thirsty Brothers Brewing" wants to have a 99% confidence level for the confidence interval of the population that will like their beers. What is the confidence interval?