What Is The 90 Confidence Interval Calculator

A 90% confidence interval is a range of values that is likely to contain the true population parameter with 90% probability. This calculator helps you compute confidence intervals for sample means when the population standard deviation is known.

What Is a 90% Confidence Interval?

In statistics, a confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. A 90% confidence interval means that if we took many samples and computed a 90% confidence interval for each, about 90% of these intervals would contain the true population parameter.

Confidence intervals are different from confidence levels. A 90% confidence level means that if the study were repeated many times, 90% of the intervals would contain the true parameter.

Key Concepts

Sample Mean (x̄): The average of the sample data.
Population Standard Deviation (σ): The standard deviation of the entire population.
Sample Size (n): The number of observations in the sample.
Z-Score: The critical value from the standard normal distribution for the desired confidence level.

How to Calculate a 90% Confidence Interval

The formula for a 90% confidence interval when the population standard deviation is known is:

Confidence Interval = x̄ ± (Z * (σ / √n))

Where:

x̄ = sample mean
Z = Z-score for 90% confidence (approximately 1.645)
σ = population standard deviation
n = sample size

The Z-score for a 90% confidence interval is approximately 1.645. This value comes from the standard normal distribution and represents the number of standard deviations from the mean that contains 90% of the data.

Steps to Calculate

Calculate the sample mean (x̄).
Identify the population standard deviation (σ).
Determine the sample size (n).
Find the Z-score for 90% confidence (1.645).
Calculate the margin of error: (Z * (σ / √n)).
Compute the lower bound: x̄ - margin of error.
Compute the upper bound: x̄ + margin of error.

Example Calculation

Suppose you have a sample of 30 people with an average height of 170 cm and a known population standard deviation of 10 cm. Calculate the 90% confidence interval for the population mean height.

Given:

x̄ = 170 cm
σ = 10 cm
n = 30
Z = 1.645

Margin of Error = 1.645 * (10 / √30) ≈ 1.645 * 1.826 ≈ 3.03

Lower Bound = 170 - 3.03 ≈ 166.97 cm

Upper Bound = 170 + 3.03 ≈ 173.03 cm

90% Confidence Interval = (166.97 cm, 173.03 cm)

This means we are 90% confident that the true population mean height falls between 166.97 cm and 173.03 cm.

Interpreting the Results

When you calculate a 90% confidence interval, you are making a probabilistic statement. It means that if the same study were repeated many times, 90% of the calculated intervals would contain the true population parameter.

Common Misinterpretations

Not the probability of the parameter: The confidence interval does not indicate the probability that the true parameter lies within the interval. The parameter is either in the interval or not.
Not the probability of the data: The confidence level does not indicate the probability that the observed data occurred. It refers to the long-run success rate of the method.

Practical Implications

A 90% confidence interval provides a range of plausible values for the population parameter. If the interval is wide, it indicates more uncertainty about the true value. If it is narrow, it suggests more precision in the estimate.

FAQ

What does a 90% confidence interval mean?

A 90% confidence interval means that if the same study were repeated many times, 90% of the calculated intervals would contain the true population parameter.

How do I calculate a 90% confidence interval?

Use the formula: x̄ ± (Z * (σ / √n)), where Z is 1.645 for 90% confidence.

What is the difference between confidence level and confidence interval?

The confidence level is the percentage that the interval will contain the true parameter (e.g., 90%). The confidence interval is the range of values calculated from the sample data.

When would I use a 90% confidence interval instead of a 95% one?

Use a 90% confidence interval when you need a narrower interval and can accept slightly less certainty. A 95% interval provides more certainty but is wider.