Often in statistics we’re interested in measuring population parameters – numbers that describe some characteristic of an entire population.
For example, we might be interested in measuring the mean height of males in a certain country.
Since it’s too costly and time-consuming to collect data on the height of every male in the country, we would instead collect data on a simple random sample of males. We would then use the mean height of males in this sample to estimate the mean height of all males in the country.
Unfortunately, the mean height of males in the sample is not guaranteed to exactly match the mean height of males in the whole population. For example, we might just happen to pick a sample full of shorter men or perhaps a sample full of taller men.
In order to capture our uncertainty around our estimate of the true population mean, we can create a confidence interval.
Confidence Interval: A range of values that is likely to contain a population parameter with a certain level of confidence.
A confidence interval is calculated using the following general formula:
Confidence Interval = (point estimate) +/- (critical value)*(standard error)
For example, the formula to calculate a confidence interval for a population mean is as follows:
Confidence Interval = x +/- z*(s/√n)
where:
- x: sample mean
- z: the z critical value
- s: sample standard deviation
- n: sample size
The z critical value that you will use in the formula is dependent on the confidence level that you choose.
Confidence Level: The percentage of all possible samples that are expected to include the true population parameter.
The most common choices for confidence levels include 90%, 95%, and 99%.
The following table shows the z critical value that corresponds to these popular confidence level choices:
| Confidence Level | z critical value |
|---|---|
| 0.90 | 1.645 |
| 0.95 | 1.96 |
| 0.99 | 2.58 |
For example, suppose we measure the heights of 25 men and find the following:
- Sample size n = 25
- Sample mean height x = 70 inches
- Sample standard deviation s = 1.2 inches
Here is how to find calculate a confidence interval for the true population mean height using a 90% confidence level:
90% Confidence Interval: 70 +/- 1.645*(1.2/√25) = [69.6052, 70.3948]
This means that if we used the same sampling method to select different samples and calculated a confidence interval for each sample, we would expected the true population mean height to fall within the interval 90% of the time.
Now suppose we instead calculate a confidence interval using a 95% confidence level:
95% Confidence Interval: 70 +/- 1.96*(1.2/√25) = [69.5296, 70.4704]
Notice that this confidence interval is wider than the previous one. This is because the higher the confidence level, the wider the confidence interval.
The higher the confidence level, the wider the confidence interval.
This should make sense intuitively: A wider confidence level has a higher probability of containing a true population parameter.
Summary
In summary:
A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence. It uses the following basic formula:
Confidence Interval = (point estimate) +/- (critical value)*(standard error)
The confidence level determines the critical value to use in that formula. The higher the confidence level, the larger the critical value and thus the wider the confidence interval.
Additional Resources
Introduction to Confidence Intervals
Introduction to Hypothesis Testing
What is a Point Estimate?