How to Construct A 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 construct such intervals for sample means using the normal distribution.

What is a Confidence Interval?

A confidence interval provides an estimated range of values which is likely to contain the true population parameter. For a 90% confidence interval, we're 90% confident that the interval contains the true parameter.

Key points about confidence intervals:

They don't indicate the probability that the estimated interval contains the true parameter
90% confidence means that if we took 100 samples and calculated 90% confidence intervals for each, we would expect about 90 of them to contain the true parameter
The width of the interval depends on the sample size and the variability in the data

How to Calculate a 90% Confidence Interval

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

μ̄ ± z*(σ/√n)

Where:

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

When the population standard deviation is unknown, you can use the sample standard deviation (s) and the t-distribution:

μ̄ ± t*(s/√n)

The degrees of freedom for the t-distribution are n-1.

Note: This calculator uses the normal distribution (z-score) for simplicity. For small sample sizes (n < 30), you should use the t-distribution for more accurate results.

Worked Example

Suppose we want to estimate the average height of adult males in a city. We take a random sample of 50 men and find:

Sample mean height (μ̄) = 175 cm
Sample standard deviation (s) = 10 cm

Using the t-distribution for n=50 (df=49), the t-score for 90% confidence is approximately 1.677.

The 90% confidence interval is calculated as:

175 ± 1.677*(10/√50) ≈ 175 ± 2.68 ≈ (172.32, 177.68)

We're 90% confident that the true average height of adult males in the city is between 172.32 cm and 177.68 cm.

Interpreting the Results

When interpreting a 90% confidence interval:

It's not a probability statement about the interval
It's a statement about the method used to calculate the interval
If you took 100 different samples and calculated 90% confidence intervals for each, you would expect about 90 of them to contain the true parameter
The interval width depends on the sample size and variability in the data

Common mistakes to avoid:

Assuming that a 90% confidence interval means there's a 90% probability that the true parameter is in the interval
Using a confidence interval to make probability statements about future observations
Assuming that a narrower interval is always better - it depends on the context and what you're trying to estimate

FAQ

What does a 90% confidence interval mean?

A 90% confidence interval means that if we took many samples and calculated 90% confidence intervals for each, about 90% of those intervals would contain the true population parameter.

How does sample size affect the confidence interval?

Larger sample sizes generally result in narrower confidence intervals because the estimate of the population parameter becomes more precise. The width of the interval is inversely proportional to the square root of the sample size.

Can I use a 90% confidence interval to make probability statements about future observations?

No, a confidence interval is not a probability statement about future observations. It's a statement about the method used to estimate the population parameter from a sample.

What's the difference between a confidence interval and a prediction interval?

A confidence interval estimates the range of values that is likely to contain the true population parameter. A prediction interval estimates the range of values that is likely to contain a future observation.

How do I know if my sample size is large enough for a 90% confidence interval?

There's no strict rule, but sample sizes of 30 or more are generally considered adequate for using the normal distribution approximation. For smaller samples, you should use the t-distribution.