The following general formula is used to calculate confidence intervals:
Confidence Interval = (point estimate) +/- (critical value)*(standard error)
This formula creates an interval with a lower bound and an upper bound, which likely contains a population parameter with a certain level of confidence.
Confidence Interval = [lower bound, upper bound]
The following examples provide several situations where confidence intervals are used in the real world.
Example 1: Biology
Confidence intervals are often used in biology to estimate the mean height, weight, width, diameter, etc. of different plant and animal species.
For example, a biologist may be interested in measuring the mean weight of a certain species of frog in Australia. Since it would take too long to go around and weigh thousands of individual frogs, the biologist may instead collect a simple random sample of 50 frogs and measure the mean and standard deviation of the frogs in the sample.
She could then use the sample mean and sample standard deviation to construct an interval for the true mean of the frogs in the entire population.
Example 2: Clinical Trials
Confidence intervals are often used in clinical trials to determine the mean change in blood pressure, heart rate, cholesterol, etc. produced by some new drug or treatment.
For example, a doctor may believe that a new drug is able to reduce blood pressure in patients. To test this, he may recruit 20 patients to participate in a trial in which they used the new drug for one month. At the end of the month, the doctor may record the mean decrease in blood pressure and the standard deviation of the decrease in each patient in the sample.
He could then use the sample mean and sample standard deviation to construct an interval for the true mean change in blood pressure that patients are likely to experience in the population.
Example 3: Advertising
Confidence intervals are often used by marketing departments within companies to determine if some new advertising technique, method, tactic, etc. produces significantly higher revenue.
For example, a marketing team at a grocery retailer may run two different advertising campaigns at 20 different stores each during one quarter and measure the average sales produced by each campaign at each store at the end of the quarter.
They could then use the sample mean and sample standard deviation of sales from each campaign to construct a confidence interval for the difference between mean sales. This will tell the marketing team if there is any meaningful difference in sales that occurs as a result of the two campaigns.
Example 4: Manufacturing
Confidence intervals are often used by engineers in manufacturing plants to determine if some new process, technique, method, etc. causes a meaningful change in the number of defective products produced by the plant.
For example, an engineer may believe that a new process will change the number of defective widgets produced per day, which is currently 50. To test this, he may implement the new process and record the number of defective products produced each day for one month at the plant.
He could then use the sample mean and sample standard deviation of the number of daily defects to construct a confidence interval for the true mean number of defective products produced by the new process.
If the confidence interval does not contain the value “50” then the engineer can be confident that the new process produces a different number of daily defective products compared to the current process.
An Introduction to Confidence Intervals for a Mean
An Introduction to Confidence Intervals for the Difference Between Means
An Introduction to Confidence Intervals for a Proportion
An Introduction to Confidence Intervals for the Difference in Proportions