Constructing a Confidence Interval for the Mean

Overview

Confidence intervals come into play when we want to do better approximations for what the true value of a parameter is. In this module, we will discuss the confidence interval for the sample mean. Up to this point, we have created point estimates, which is what we get when we compute the sample mean. This approximation is almost surely incorrect, so we can be better suited using an interval estimate, in this case the confidence interval.

The concept here is we buffer our prediction of the mean using a margin of error, which uses the Z or T distribution, as well as a level of confidence, c. Common confidence intervals we create are 80%, 90%, 95%, and 99% confidence intervals. The approach of a confidence interval is this: If we collect sample data and run this approach over and over again, then approximately 100*(1-c)% of the confidence intervals will contain the true value of the parameter. So, if we construct 95% confidence intervals, we would expect that approximately 95% of the intervals we create will contain the true value of the parameter of interest.

The common formula we use when construction confidence intervals for the mean is this: xEwhere E is our margin of error. This is the value that will change depending on which distribution we are using.

If we are using the Z-distribution, then E=Zc/n where Zc is our critical Z value. If we are using the T-distribution, then E=Tcsn where Tcis our critical T value. Now we have to figure out what our critical Z and T values are.

Critical Z values will never change and are as follows:

80% confidence interval: Zc=1.28

90% confidence interval: Zc=1.645

95% confidence interval: Zc=1.96

99% confidence interval: Zc=2.576

When it comes to finding critical T-values, we also need to account for the degrees of freedom. The degrees of freedom for our sample will be equal to one less than the sample size, or df=n1. Here is the process we follow to find critical t-values: Take the confidence level as a decimal and subtract it from 1. Then divide the resulting value by 2. Use this as our "alpha" value on our T-table. We then look up the critical T-value on our T-table for the associated degrees of freedom.

For example, if we wanted to construct a 95% confidence interval for the mean with a sample size n of 20, we would find the critical T-value as follows: 1 - 0.95 = 0.05. 0.05/2 = 0.025. This is our alpha value. Then we compute the degrees of freedom: df = 20 - 1 = 19. Now we go to our T-tables and find the value that corresponds to 19 degrees of freedom and an alpha value of 0.025, which will be 2.093

So, lets walk through a confidence interval calculation using the Z distribution: Suppose we have a sample of data with a mean of 50, a population standard deviation of 10, and a sample size of 64. We want to create a 95% confidence interval for this sample:

E=Zc/n=1.9610/64=2.45

Lower bound: xE=502.45=47.55

Upper bound: x+E=50+2.45=52.45

Then we write our final answer as such: (47.55, 52.45). We can then say that we are 95% confident the true value of the population mean falls between 47.55 and 52.45.

Now let's walk through a confidence interval calculation using the T distribution: Suppose we have a sample of data with a mean of 25, a sample standard deviation of 6, and a sample size of 16. We want to create a 95% confidence interval for this sample:

E=Tcs/n=2.1316/16=3.1965

Lower bound: xE=253.1965=21.8035

Upper bound: x+E=25+3.1965=28.1965

Then we write our final answer as such: (21.8035, 28.1965). We can then say that we are 95% confident the true value of the population mean falls between 21.8035 and 28.1965

Instructions

For this discussion post, we are going to construct a confidence interval based on a collection of sample data. Read the following:

We are interested in the average wait estimated time of our local ER at 7 PM on Friday nights. So, we sample 18 estimated wait times (in minutes) at 7 PM on Friday nights over the last 2 years and found the following:

3, 8, 25, 47, 61, 25, 10, 32, 31, 20, 10, 15, 7, 62, 48, 51, 17, 30

Using these ER wait times, construct a 90% confidence interval for the mean ER wait times for Friday nights at 7 PM

Discussion Prompts

Answer the following questions in your initial post:

1. What is the sample mean and sample standard deviation of this data set?
2. Should we be using the Z or T distribution? Explain why
3. Find the Critical Z or T value for this problem.
4. Compute the Margin of Error, E
5. Write out the confidence interval
6. The ER claims its average wait time on Friday nights will be less than 35 minutes. Based on our confidence interval, does this seem like a valid claim?

Supporting Resources:

https://www.youtube.com/watch?v=DT-fPG0Hff8

https://www.youtube.com/watch?v=MUD390jtgQs