# Confidence intervals to estimate population mean

### Confidence intervals to estimate population mean

#### Lessons

In previous sections we used a point estimate to estimate the range of where our population proportion might lie (with a specific level of confidence). In this section we will be doing the same thing, except with means. We will find our sample mean (similar in idea to our point estimate) and then use that sample mean to figure out the range of where our population mean might lie (with a specific level of confidence).

However in this section there are two different scenarios we have to consider. Either $\sigma$ is known, or $\sigma$ is unknown.

$\sigma$ is unknown: We will use t-scores, which will be explored in the next section
$\sigma$ is known: $E=Z_{\frac{\alpha}{2}}*\frac{\sigma}{\sqrt{n}}$

$\mu$: the population mean (what we are interested in finding)
$\overline{x}$: The sample estimate for $\mu$
• Introduction
How do we estimate population mean? ($\sigma$ is known)

• 1.
Determining a Confidence Interval for a Population Mean
At a wrecking yard 40 cars are weighed and found to have an average weight of 1500lbs. The standard deviation of the weight of all cars is 175lbs. With a critical value of $Z_{\frac{\alpha}{2}}=2$ what is the confidence interval for the weight of all cars?

• 2.
Byron’s company designs tugboats. During a particular month this company designs 70 tugboats, with an average length of 85 feet. All tugboats designed by his company have a standard deviation of 10 feet. With a 90% confidence level find the average length of tugboat designed by his company.

• 3.
André is a bartender who pours drinks for wedding parties. For a particular party he pours 50 glasses of champagne that have an average amount of 175mL. The standard deviation of every single glass he has ever poured and will ever pour is 5mL. With a 92% confidence level construct a confidence interval for the average amount of champagne that André pours.

• 4.
Determining the Sample Size with a given Margin of Error
The average person can bench press 75lbs. There is a standard deviation of 10lbs in the amount that the population can bench press. With a critical value of $Z_{\frac{\alpha}{2}}=1.96$ how large of a sample would I have to take such that my confidence interval is within a range of 2 lbs of the population mean?