Statistics Calculator for Confidence Interval with 98 Confidence

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain an unknown population parameter. The interval has an associated confidence level which indicates the probability that the interval contains the true parameter value.

For a 98% confidence interval, we can say that if we were to take 100 different samples and compute a 98% confidence interval for each, approximately 98 of those intervals would contain the true population parameter.

How to Calculate a Confidence Interval

The formula for calculating a confidence interval depends on the type of data and the distribution of the population. For large samples (typically n > 30) from a normally distributed population, the confidence interval for the mean is calculated as:

Where:

• x̄ is the sample mean
• z is the z-score corresponding to the desired confidence level (for 98% confidence, z ≈ 2.326)
• σ is the population standard deviation
• n is the sample size

If the population standard deviation is unknown and the sample size is small (n ≤ 30), you should use the t-distribution instead of the normal distribution. The formula becomes:

Where:

• t is the t-score corresponding to the desired confidence level and degrees of freedom (df = n - 1)
• s is the sample standard deviation

Interpreting the Results

When you calculate a 98% confidence interval, you can interpret the result as follows: "We are 98% confident that the true population parameter lies within this interval."

It's important to note that the confidence level does not indicate the probability that a particular confidence interval contains the true parameter. Instead, it refers to the long-run frequency of intervals that contain the true parameter.

For example, if you calculate a 98% confidence interval for the mean height of adults in a population, you can say that you are 98% confident that the true mean height lies within the calculated interval. This does not mean that there is a 98% probability that any individual interval contains the true mean.

Worked Example

Let's say you have a sample of 50 people and you want to estimate the mean height of the population with 98% confidence. The sample mean height is 170 cm and the sample standard deviation is 10 cm.

Since the sample size is greater than 30, we can use the normal distribution. The z-score for 98% confidence is approximately 2.326.

The margin of error is calculated as:

The 98% confidence interval is then:

This means we are 98% confident that the true population mean height lies between 166.72 cm and 173.28 cm.