How to Find 90 Confidence Interval Calculator

Calculating a 90% confidence interval is essential in statistics for estimating population parameters from sample data. This guide explains the process step-by-step, including the formula, assumptions, and practical applications.

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 a 90% confidence interval, we're 90% confident that the true parameter falls within the calculated range.

Confidence intervals are used in various fields including medicine, social sciences, engineering, and quality control to provide a measure of uncertainty around estimates.

90% Confidence Interval Formula

The formula for a 90% confidence interval for a population mean (μ) when the population standard deviation (σ) is known is:

Confidence Interval = x̄ ± z*(σ/√n) Where: x̄ = sample mean z = z-score for 90% confidence (1.645) σ = population standard deviation n = sample size

For small samples where σ is unknown, you would use the t-distribution instead of the z-score, but this guide focuses on the z-score method for simplicity.

How to Calculate a 90% Confidence Interval

Step 1: Gather Your Data

You'll need:

The sample mean (x̄)
The population standard deviation (σ)
The sample size (n)

Step 2: Find the Z-Score

For a 90% confidence interval, the z-score is 1.645. This value comes from standard normal distribution tables.

Step 3: Calculate the Margin of Error

The margin of error (ME) is calculated as:

ME = z*(σ/√n)

Step 4: Determine the Confidence Interval

Add and subtract the margin of error from the sample mean to get the confidence interval:

Lower Bound = x̄ - ME Upper Bound = x̄ + ME

Assumptions

This method assumes:

The sample is randomly selected
The population is normally distributed or the sample size is large (n ≥ 30)
The population standard deviation is known

Worked Example

Let's calculate a 90% confidence interval for a sample with:

Sample mean (x̄) = 50
Population standard deviation (σ) = 10
Sample size (n) = 64

Step 1: Find the Z-Score

For 90% confidence, z = 1.645

Step 2: Calculate the Margin of Error

ME = 1.645*(10/√64) = 1.645*(10/8) = 2.056

Step 3: Determine the Confidence Interval

Lower Bound = 50 - 2.056 = 47.944 Upper Bound = 50 + 2.056 = 52.056

Therefore, the 90% confidence interval is from 47.944 to 52.056. This means we're 90% confident that the true population mean falls within this range.

Interpreting the Results

A 90% confidence interval means that if we were to take many samples and calculate a 90% confidence interval for each, approximately 90% of those intervals would contain the true population parameter.

Common interpretations include:

We can be 90% confident that the true value lies within the calculated range
The interval provides a range of plausible values for the population parameter
As the sample size increases, the confidence interval becomes narrower

It's important to note that a 90% confidence interval doesn't mean there's a 90% probability that the true value is in the interval. Instead, it reflects the reliability of the estimation method over many repetitions.

FAQ

What does a 90% confidence interval mean?: It means that if we were to take many samples and calculate a 90% confidence interval for each, approximately 90% of those intervals would contain the true population parameter.
How does sample size affect the confidence interval?: As the sample size increases, the confidence interval becomes narrower, providing a more precise estimate of the population parameter.
Can I use this calculator for other confidence levels?: This calculator specifically calculates 90% confidence intervals. For other confidence levels, you would need to adjust the z-score accordingly.
What if my sample size is small?: For small samples (n < 30), you should use the t-distribution instead of the z-score, as the population standard deviation is typically unknown.
How do I know if my sample is normally distributed?: You can check for normality using statistical tests like the Shapiro-Wilk test or by examining a histogram of your data. For large samples (n ≥ 30), the Central Limit Theorem often ensures approximate normality.