Levels of Confidence & Building a Confidence Interval
Once we have gathered information about samples, how do we relate them to the population? Think about a paycheck that varies. You have a low-end amount you receive when you work your least amount of hours, versus a high-end amount when you work more hours. These totals surround the average amount. So, to describe your pay on average, you can say you make between the low and high amount, give-or-take from the middle average. You can use this same method with many types of sample data. These are called Confidence intervals.
Levels of Confidence
Each test performed may have a different “level of confidence.” Some popular choices are 90%, 95%, and 99% confidence in your desired results. This confidence is built into the formula using a Critical Value. A critical value with the standard normal distribution is notated as: Zα/2 or Tα/2
Recall α = alpha
α = 100% − level of confidence
Remember to divide that by 2!
For instance, suppose you only wanted to be 61% confident. Then, you could find alpha by subtracting from 100%
Written as a decimal, this is 0.195, which is represented in the blue area shown on the symmetric curve here. So, you would be looking for the critical value of Z0.195 or T0.195
Building a Confidence Interval
A confidence interval consists of 2 numbers surrounding the point estimate, the center value that has been researched and found in a study. To build the surrounding numbers, you must use the Margin of Error, a buffer around the point estimate, to find numbers to describe the full population. Once calculated, these are used to have a confident estimate in the point estimate (mean or proportion) to relate the sample to the population studied.
Margin of error uses Critical Values from Topic 1, as well as the Standard Error
Point Estimate + (Critical Value * Margin of Error)
Point Estimate – (Critical Value * Margin of Error)
In Stat Crunch
Proportions: Select Stat-Proportion Stats to find different options for proportion confidence intervals.
- Choose between One sample, or Two sample to perform
- Next, you must decide if you have a summary of data, or information typed into your columns.
- This screenshot shows the entry screen for a one sample proportion test
For Means:
- Select Stat-T Stats to find different options for mean confidence intervals
- You will choose between One sample, Two sample, or Paired to perform This depends on how many groups you have and their relationship
- Decide if you have a summary of data, or information typed into your columns
- This screenshot shows the entry screen for a one sample mean test
*Note: StatKey, Excel, and Python will compute confidence intervals with a data set. If you have summary information, using the formula or a program like StatCrunch is useful
In Excel
- Select Data Tab at the top, then click Data Analysis Add On, then select Descriptive Statistics
- Under Input Range, select all data/numbers with your mouse
- Check the Summary Statistics Box
- Click “OK”
This will automatically populate a new sheet in Excel with various descriptive statistics data including the confidence interval about a set of data regarding means
In StatKey
- Edit the data to enter your own.
- Then, generate samples
- Change the middle proportion to the level of confidence you need. This will show the upper and lower on the bottom of the screen. The lower bound is on the left. The upper bound is on the right
Now, You Try! Practice Problems
Example One
Lyra has decided to take a sample regarding the favorite item of apparel she owns. She discovers that out of her sample of 90 pieces of clothing, 68 were T-shirts Lyra decides to make a confidence interval. First, she tries a 90% confidence interval, then a 95% confidence interval
- Perform these confidence intervals to see Lyra’s results
Example Two
Hyacinth wants to create a 99% confidence interval about the number of video games owned on average in the local area’s population. She takes a sample of 50 random people and finds an average of 56 video games owned, with a standard deviation of 3 games
- Perform this 99% confidence interval
Example 1
Lyra’s sample is a proportion problem. A proportion is part of the whole. Lyra is measuring T-shirts out of the total amounts of clothing.
We find 0.67 for the lower and 0.84 for the upper. I am 95% confident that the proportion of T-shirts that represents a favorite piece of clothing falls between 0.67 and 0.84.
Example 2
Hyacinth’s example is a problem about a mean, because it is looking at the average video games owned. Means are calculated or countable averages.
The lower bound is 54.86 and upper bound is 7.14. I am 99% confident that the mean number of video games owned falls between 54.86 and 57.14
Need More Help?
Click here to schedule a 1:1 with a tutor, coach, and or sign up for a workshop. *If this link does not bring you directly to our platform, please use our direct link to "Academic Support" from any Brightspace course at the top of the navigation bar.