Introduction to Confidence Intervals


Suppose we want to know the mean height of a student at a school that has 1,000 students. Since it would take too long to measure the height of every student, we instead take a simple random sample of 100 students and calculate the mean height of students in this sample. 

In this example, the mean height of students in the sample is our statistic and the actual mean height of students in the entire population is the parameter we are trying to estimate using our sample. 

The sample mean is known as a point estimate, which is a single value used to estimate a population parameter. And while this value might be a good estimate of the true population mean, there is no guarantee that the sample mean is exactly equal to the population mean.

For example, the mean height of students in our sample might be 66 inches while the actual mean height of students at this school may be 68 inches. In other words, our point estimate doesn’t account for any uncertainty we may have.

One way to account for uncertainty is to use a confidence interval, which is a range of values that we believe contains the true population parameter.

A confidence interval consists of three parts:

A sample statistic (often this is a sample mean or sample proportion)

A standard error (sample standard deviation divided by sample size)

A critical value (determined by the confidence level we choose – the table below shows the critical values associated with various confidence levels)

The formula to create a confidence interval is:

Confidence Interval = sample statistic  +/-  (critical value * standard error)

How to Interpret Confidence Intervals & Confidence Levels

To create a confidence interval, we must choose a confidence level. Common choices are 90%, 95%, and 99%.

The confidence level refers to how often this type of interval will contain the parameter we’re interested in.

The confidence interval gives us a range of possible values for the parameter we’re interested in. 

Here are a few examples of how to interpret confidence levels and confidence intervals.

Example 1: A researcher plans to ask a random sample of 500 residents in a certain city whether or not they support a new law. The researcher will take these results and construct a 90% confidence interval for the true proportion of all residents in this city who support the new law. 

How to interpret this confidence level: If the researcher repeats this process many times, then about 90% of the intervals he produces will capture the true proportion of residents in this city who support the new law.

Example 2: A botanist wants to know what the mean height of tomato plants are in a certain region. He records the height (in inches) of a random sample of 100 plants and constructs the following 99% confidence interval for the mean height: (11, 15)

How to interpret this confidence interval: The researcher can be 99% confident that the interval (11,15) captures the true mean height of tomato plants in this region.

A Note on Confidence Interval Widths

The higher the confidence level, the wider the confidence interval. 

Recall the formula to construct a confidence interval:

Confidence Interval = sample statistic  +/-  (critical value * standard error)

When we choose a higher confidence level, we use a larger critical value:


This means the quantity we add and subtract from the sample statistic is larger, which makes our confidence interval wider:

Confidence Interval = sample statistic  +/-  (critical value * standard error)

In the previous example, if the botanist recorded the heights of a random sample of 100 plants and constructed a 90%, 95%, and 99% confidence interval for the mean height, the 90% confidence interval would be the narrowest and the 99% confidence interval would be the widest:

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