What do we mean when we say we are 95% confident that an interval contains the population proportion?

By Dr. Saul McLeod, published June 10, 2019, updated 2021

The confidence interval (CI) is a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed as a % whereby a population mean lies between an upper and lower interval.

What does a 95% confidence interval mean?

The 95% confidence interval is a range of values that you can be 95% confident contains the true mean of the population. Due to natural sampling variability, the sample mean (center of the CI) will vary from sample to sample.

The confidence is in the method, not in a particular CI. If we repeated the sampling method many times, approximately 95% of the intervals constructed would capture the true population mean.

Therefore, as the sample size increases, the range of interval values will narrow, meaning that you know that mean with much more accuracy compared with a smaller sample.

We can visualize this using a normal distribution (see the below graph).

For example, the probability of the population mean value being between -1.96 and +1.96 standard deviations (z-scores) from the sample mean is 95%.

Accordingly, there is a 5% chance that the population mean lies outside of the upper and lower confidence interval (as illustrated by the 2.5% of outliers on either side of the 1.96 z-scores).

Why do researchers use confidence intervals?

It is more or less impossible to study every single person in a population so researchers select a sample or sub-group of the population.

This means that the researcher can only estimate the parameters (i.e. characteristics) of a population, the estimated range being calculated from a given set of sample data.

Therefore, a confidence interval is simply a way to measure how well your sample represents the population you are studying.

The probability that the confidence interval includes the true mean value within a population is called the confidence level of the CI.

You can calculate a CI for any confidence level you like, but the most commonly used value is 95%. A 95% confidence interval is a range of values (upper and lower) that you can be 95% certain contains the true mean of the population.

How do I calculate a confidence interval?

To calculate the confidence interval, start by computing the mean and standard error of the sample.

Remember, you must calculate an upper and low score for the confidence interval using the z-score for the chosen confidence level (see table below).

Confidence Interval Formula

Where:

• X is the mean
• Z is the chosen Z-value (1.96 for 95%)
• s is the standard error
• n is the sample size

For the lower interval score divide the standard error by the square root on n, and then multiply the sum of this calculation by the z-score (1.96 for 95%). Finally, subtract the value of this calculation from the sample mean.

An Example

• X (mean) = 86
• Z = 1.960 (from the table above for 95%)
• s (standard error) = 6.2
• n (sample size) = 46

Lower Value: 86 - 1.960 × 6.2 √46 = 86 - 1.79 = 84.21

Upper Value: 86 + 1.960 × 6.2 √46 = 86 + 1.79 = 87.79

So the population mean is likely to be between 84.21 and 87.79

How can we be confident the population mean is similar to the sample mean?

The narrower the interval (upper and lower values), the more precise is our estimate.

As a general rule, as a sample size increases the confident interval should become more narrow.

Therefore, with large samples, you can estimate the population mean with more precision than you can with smaller samples, so the confidence interval is quite narrow when computed from a large sample.

How to report a confident interval APA style

The APA 6 style manual states (p.117):

“ When reporting confidence intervals, use the format 95% CI [LL, UL] where LL is the lower limit of the confidence interval and UL is the upper limit. ”

For example, one might report: 95% CI [5.62, 8.31].

Confidence intervals can also be reported in a table

How to reference this article:

McLeod, S. A. (2019, June 10). What are confidence intervals in statistics? Simply psychology: https://www.simplypsychology.org/confidence-interval.html

Home | About Us | Privacy Policy | Advertise | Contact Us

Simply Psychology's content is for informational and educational purposes only. Our website is not intended to be a substitute for professional medical advice, diagnosis, or treatment.

© Simply Scholar Ltd - All rights reserved

report this ad

Home>AP statistics>This page

Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.

How to Interpret Confidence Intervals

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 this means there is a 90% chance that the population mean falls between 100 and 200. This 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.

The confidence level describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter; a 95% confidence level means that 95% of the intervals would include the parameter; and so on.

Confidence Interval Data Requirements

To express a confidence interval, you need three pieces of information.

• Confidence level
• Statistic
• Margin of error

Given these inputs, the range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty associated with the confidence interval is specified by the confidence level.

Often, the margin of error is not given; you must calculate it. Previously, we described how to compute the margin of error.

Advertisement

How to Construct a Confidence Interval

There are four steps to constructing a confidence interval.

• Identify a sample statistic. Choose the statistic (e.g, sample mean, sample proportion) that you will use to estimate a population parameter.
• Select a confidence level. As we noted in the previous section, the confidence level describes the uncertainty of a sampling method. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.
• Find the margin of error. If you are working on a homework problem or a test question, the margin of error may be given. Often, however, you will need to 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

  For guidance, see how to compute the margin of error.
• 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

The sample problem in the next section applies the above four steps to construct a 95% confidence interval for a mean score. The next few lessons discuss this topic in greater detail.

As you may have guessed, the four steps required to specify a confidence interval can involve many time-consuming computations. Stat Trek's Sample Size Calculator does this work for you - quickly, easily, and error-free. In addition to constructing a confidence interval, the calculator creates a summary report that lists key findings and documents analytical techniques. Whenever you need to construct a confidence interval, consider using the Sample Size Calculator. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

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 1,000,000 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 four steps below.

• Identify a sample statistic. Since we are trying to estimate the mean weight in the population, we choose the mean weight in our sample (180) as the sample statistic.
• Select a confidence level. In this case, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
• Find the margin of error. Previously, we described how to compute the margin of error. The key steps are shown below.
  • Find 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.

  • Find critical value. The critical value is a factor used to compute the margin of error. To express the critical value as a t score (t*), follow these steps.
    • 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 statistic 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.

  • Compute margin of error (ME):

    ME = critical value * standard error

    ME = 1.96 * 0.95 = 1.86

• 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.

If you would like to cite this web page, you can use the following text:

Berman H.B., "What is a Confidence Interval", [online] Available at: https://stattrek.com/estimation/confidence-interval URL [Accessed Date: 8/20/2022].