How to Calculate Width of 95 Confidence Interval

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. For example, a 95% confidence interval suggests that if you were to take 100 different samples and compute a 95% confidence interval for each, approximately 95 of those intervals would contain the true population parameter.

The width of the confidence interval depends on several factors including the sample size, the standard deviation of the population, and the desired level of confidence. A narrower interval indicates a more precise estimate, while a wider interval suggests more uncertainty.

Formula for Confidence Interval Width

The width of a confidence interval for a population mean can be calculated using the following formula:

For a 95% confidence interval, the z-score is approximately 1.96. This value comes from the standard normal distribution and represents the number of standard deviations from the mean that contains 95% of the data.

Example Calculation

Let's say you want to estimate the average height of students in a school. You take a random sample of 50 students and find that the standard deviation of their heights is 3 inches. You want to calculate the width of a 95% confidence interval for the population mean height.

Using the formula:

This means the width of the 95% confidence interval for the population mean height is approximately 1.658 inches. In other words, you can be 95% confident that the true average height of all students in the school falls within 1.658 inches of your sample mean.

Interpreting the Result

The width of the confidence interval provides several important insights:

• Precision: A narrower interval indicates a more precise estimate of the population parameter. In our example, the interval width of 1.658 inches suggests a relatively precise estimate of the average height.
• Uncertainty: A wider interval indicates more uncertainty about the true population parameter. If the interval were much wider, it would suggest that the sample size is too small to make a precise estimate.
• Confidence Level: The 95% confidence level means that if you were to take many samples and compute a 95% confidence interval for each, approximately 95% of those intervals would contain the true population parameter.

It's important to note that the confidence interval does not indicate the probability that the true parameter lies within the interval. Instead, it reflects the long-run frequency of intervals that contain the true parameter.

Common Mistakes

When calculating the width of a confidence interval, it's easy to make a few common mistakes:

• Using the wrong z-score: Remember that the z-score for a 95% confidence interval is 1.96. Using a different z-score will result in an incorrect interval width.
• Confusing the confidence interval with the probability: The confidence interval does not indicate the probability that the true parameter lies within the interval. Instead, it reflects the long-run frequency of intervals that contain the true parameter.
• Assuming the population standard deviation is known: In practice, the population standard deviation is often unknown and must be estimated from the sample. Using the sample standard deviation instead of the population standard deviation can affect the interval width.