# 2.1 - The Situation

2.1 - The SituationPoint estimates, such as the sample proportion (\(\hat{p}\)), the sample mean (\(\bar{x}\)), and the sample variance (\(s^2\)) depend on the particular sample selected. For example:

- We might know that \(\hat{p}\) , the proportion of a sample of 88 students who use the city bus daily to get to campus, is 0.38. But, the bus company doesn't want to know the sample proportion. The bus company wants to know population proportion \(p\), the proportion of
*all*of the students in town who use the city bus daily. - We might know that \(\bar{x}\), the average number of credit cards of 32 randomly selected American college students is 2.2. But, we want to know \(\mu\), the average number of credit cards of
*all*American college students.

## The Problem

- When we use the sample mean \(\bar{x}\) to estimate the population mean \(\mu\), can we be confident that \(\bar{x}\) is close to \(\mu\)? And, when we use the sample proportion \(\hat{p}\) to estimate the population proportion \(p\), can we be confident that \(\hat{p}\) is close to \(p\)?
- Do we have any idea as to how close the sample statistic is to the population parameter?

### A Solution

Rather than using just a point estimate, we could find an interval (or range) of values that we can be really confident contains the actual unknown population parameter. For example, we could find lower (\(L\)) and upper (\(U\)) values between which we can be really confident the population mean falls:

\(L<\mu<U\)

And, we could find lower (\(L\)) and upper (\(U\)) values between which we can be really confident the population proportion falls:

\(L<p<U\)

An interval of such values is called a **confidence interval**. Each interval has a **confidence coefficient** (reported as a proportion):

\(1-\alpha\)

or a **confidence level** (reported as a percentage):

\((1-\alpha)100\%\)

Typical confidence coefficients are 0.90, 0.95, and 0.99, with corresponding confidence levels 90%, 95%, and 99%. For example, upon calculating a confidence interval for a mean with a confidence level of, say 95%, we can say:

"We can be 95% confident that the population mean falls between \(L\) and \(U\)."

As should agree with our intuition, the greater the confidence level, the more confident we can be that the confidence interval contains the actual population parameter.