What is a Confidence Interval?

A confidence interval describes the amount of uncertainty associated with a sample estimate of a population parameter. Every confidence interval consists of two parts: a confidence level and a range of values.

For example, based on sample data, we might assert with 90% confidence that a population mean is greater than 100 and less than 200. Here, the confidence level is 90%, and the range of values is 100 to 200.

How to Interpret a Confidence Interval

Suppose that a 90% confidence interval states that the population mean is greater than 100 and less than 200. How would you interpret this statement?

Some people think it means there is a 90% chance that the population mean falls between 100 and 200. That is incorrect. Like any population parameter, the population mean is a constant, not a random variable. It does not change. The probability that a constant falls within any given range is always 0.00 or 1.00 - never a value in between, like 0.90.

Here is what a confidence interval means in statistics. Suppose we used the same sampling plan to select different samples and computed a different confidence interval for each sample. Some confidence intervals would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the confidence intervals to include the population parameter; a 95% confidence level means that we would expect 95% of the confidence intervals to include the parameter; and so on.

Prerequisites

This lesson assumes that you are familiar with the standard error of a sampling distribution and the margin of error, topics that we covered in previous lessons. If you are not familiar with these topics, click one or both of the links below for a quick review:

• Lesson 49: What is the Standard Error?
• Lesson 50: Margin of Error

How to Construct a Confidence Interval

There are five steps to constructing a confidence interval around a statistic.

1. Choose the confidence level. The confidence level describes the uncertainty of a sampling plan. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.
2. Compute the standard deviation or standard error. Formulas for computing the standard deviation of a sampling distribution and the standard error appear in the lesson on standard error. In future lessons, we'll note the standard error formula for particular statistics, as they come up.
3. Find the critical value. The critical value will be a z-score or a t-score. The value of the z-score or t-score depends on the confidence level. Common z-score critical values are 1.645 for a 90% confidence level, 1.96 for a 95% confidence level, and 2.576 for a 99% confidence level. Instructions for finding z-score values for other confidence levels and for finding t-score values are provided in the lesson on margin of error.
4. Find the margin of error. You can compute the margin of error, based on one of the following equations.

   Margin of error = Critical value * Standard deviation of statistic

   Margin of error = Critical value * Standard error of statistic

5. Specify the confidence interval. The uncertainty is denoted by the confidence level. And the range of the confidence interval is defined by the following equation.

   Confidence interval = Sample statistic ± Margin of error

   Note: A shorthand way to describe a confidence interval is to list the endpoints of the interval within parentheses. For example, suppose the sample mean is 10 and the margin of error is 2. We might say the confidence interval is 10 ± 2. Or we might use shorthand notation and say the confidence interval is (8, 12).

You can use the same five-step recipe to construct a confidence interval around any sample statistic, as you will see in the next six lessons:

• Confidence Interval: Proportions
• Confidence Interval: Difference Between Proportions
• Confidence Interval: Sample Mean
• Confidence Interval: Difference Between Means
• Confidence Intervals with Paired Data
• Regression Slope: Confidence Interval

Test Your Understanding

Problem 1

Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men from a population of 300,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 95% confidence interval.

(A) 180 + 1.86
(B) 180 + 3.0
(C) 180 + 5.88
(D) 180 + 30
(E) None of the above

Solution

The correct answer is (A). To specify the confidence interval, we work through the five steps below.

1. Choose the confidence level. In this case, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
2. Compute the standard deviation or standard error. The standard error (SE) of the mean is:

   SE = s / sqrt( n )

   SE = 30 / sqrt(1000) = 30/31.62 = 0.95

   where s is the sample standard deviation and n is the sample size.

3. Find the critical value. Instructions for expressing the critical value as a t-score appear in the lesson on margin of error. We follow those instructions below:
   • Compute alpha (α):

     α = 1 - (confidence level / 100) = 0.05

   • Find the critical probability (p*):

     p* = 1 - α/2 = 1 - 0.05/2 = 0.975

   • Find the degrees of freedom (df):

     df = n - 1 = 1000 - 1 = 999

   • The critical value is the t-score having 999 degrees of freedom and a cumulative probability equal to 0.975. From the t Distribution Calculator, we find that the critical value is about 1.96.
   

   Note: We might also have expressed the critical value as a z-score. Because the sample size is large, a z-score analysis produces the same result - a critical value equal to about 1.96.

4. Find the margin of error (ME).

   ME = critical value * standard error

   ME = 1.96 * 0.95 = 1.86

5. Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level. Therefore, this 95% confidence interval is 180 + 1.86.