How to Calculate The 90 Percent Confidence Interval

A 90% confidence interval is a range of values that is likely to contain the true population parameter with 90% probability. This guide explains how to calculate it, its meaning, 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 statistics to quantify the uncertainty of sample estimates. They provide a range of plausible values for a population parameter, such as a mean or proportion, based on sample data.

90% Confidence Interval Formula

The formula for a 90% confidence interval depends on the type of data you're working with. Here are the most common formulas:

For a Mean (Z-Interval)

CI = x̄ ± z*(σ/√n)

Where:

CI = Confidence Interval
x̄ = Sample mean
z = Z-score for 90% confidence (approximately 1.645)
σ = Population standard deviation
n = Sample size

For a Proportion (Z-Interval)

CI = p̂ ± z*√(p̂*(1-p̂)/n)

Where:

CI = Confidence Interval
p̂ = Sample proportion
z = Z-score for 90% confidence (approximately 1.645)
n = Sample size

The z-score for a 90% confidence interval is approximately 1.645. This value comes from standard normal distribution tables and represents the point that cuts off the top 5% of the distribution (since 100% - 90% = 10%, and we split this equally between the two tails).

How to Calculate the 90% Confidence Interval

Calculating a 90% confidence interval involves several steps:

Determine the sample mean or proportion
Identify the appropriate z-score for 90% confidence
Calculate the standard error of the mean or proportion
Multiply the z-score by the standard error
Add and subtract this value from the sample mean or proportion to get the confidence interval

Note: For small sample sizes (n < 30), you should use the t-distribution instead of the normal distribution. The t-distribution accounts for additional uncertainty in small samples.

Example Calculation

Let's calculate a 90% confidence interval for the mean height of a sample of 50 people, with a sample mean of 170 cm and a population standard deviation of 10 cm.

Step 1: Identify the values

Sample mean (x̄) = 170 cm
Population standard deviation (σ) = 10 cm
Sample size (n) = 50
Z-score for 90% confidence = 1.645

Step 2: Calculate the standard error

Standard error (SE) = σ/√n = 10/√50 ≈ 1.414 cm

Step 3: Calculate the margin of error

Margin of error = z * SE = 1.645 * 1.414 ≈ 2.326 cm

Step 4: Calculate the confidence interval

Lower bound = x̄ - margin of error = 170 - 2.326 ≈ 167.674 cm

Upper bound = x̄ + margin of error = 170 + 2.326 ≈ 172.326 cm

90% Confidence Interval: (167.67 cm, 172.33 cm)

This means we're 90% confident that the true population mean height falls between approximately 167.67 cm and 172.33 cm.

Interpreting the Results

When you calculate a 90% confidence interval, you're making a statement about the range of plausible values for the population parameter. Here's how to interpret the results:

The confidence interval provides a range of values that is likely to contain the true population parameter
The 90% confidence level means that if you were to take many samples and calculate a 90% confidence interval for each, about 90% of those intervals would contain the true population parameter
It's important to note that this doesn't mean there's a 90% probability that the true parameter is within the calculated interval - the parameter is either within the interval or it isn't

Remember: A 90% confidence interval means that if you conducted the same study many times, 90% of the calculated intervals would contain the true population parameter. It does not mean there's a 90% probability that the true parameter is within the specific interval you calculated.

Common Mistakes

When calculating confidence intervals, there are several common mistakes to avoid:

Using the wrong z-score: Always use the correct z-score for your desired confidence level
Assuming the population standard deviation is known: In practice, you often estimate it from the sample
Misinterpreting the confidence level: Remember that the confidence level refers to the method, not the specific interval
Using the wrong formula: Make sure to use the appropriate formula for your data type (mean or proportion)

FAQ

What does a 90% confidence interval mean?: A 90% confidence interval means that if you were to take many samples and calculate a 90% confidence interval for each, about 90% of those intervals would contain the true population parameter.
How do I know if my sample size is large enough?: A general rule of thumb is that your sample size should be at least 30 for the normal distribution to approximate the t-distribution well. For smaller samples, you should use the t-distribution.
Can I calculate a confidence interval for any type of data?: Confidence intervals can be calculated for various types of data, including means, proportions, and differences between groups. The specific formula depends on the type of data you're analyzing.
What if my data is not normally distributed?: For small sample sizes (n < 30) with non-normal data, you should use the t-distribution. For larger samples, the Central Limit Theorem often ensures that the sampling distribution is approximately normal.
How does sample size affect the confidence interval?: Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population parameter. Smaller sample sizes lead to wider intervals, reflecting greater uncertainty.