T Test Confidence Interval How to Calculate

What is a t-test confidence interval?

A t-test confidence interval provides a range of values that is likely to contain the true population mean. It's commonly used in hypothesis testing when the sample size is small (typically less than 30) or when the population standard deviation is unknown.

The confidence interval is calculated based on the sample mean, sample standard deviation, sample size, and the t-distribution critical value. The width of the confidence interval depends on the level of confidence chosen (typically 90%, 95%, or 99%) and the variability in the sample data.

Formula for t-test confidence interval

The formula for a t-test confidence interval is:

Where:

• Sample Mean - The average of your sample data
• t-critical - The critical value from the t-distribution table based on your degrees of freedom and confidence level
• Standard Error - A measure of the variability of the sample mean
• Sample Standard Deviation - A measure of how spread out the numbers in your sample are
• Sample Size - The number of observations in your sample

How to calculate a t-test confidence interval

To calculate a t-test confidence interval, follow these steps:

1. Calculate the sample mean (x̄)
2. Calculate the sample standard deviation (s)
3. Determine the sample size (n)
4. Calculate the degrees of freedom (df = n - 1)
5. Find the t-critical value from the t-distribution table based on your confidence level and degrees of freedom
6. Calculate the standard error (SE = s / √n)
7. Calculate the margin of error (ME = t-critical × SE)
8. Calculate the confidence interval (x̄ ± ME)

You can use our interactive calculator above to perform these calculations quickly and accurately.

Worked example

Let's calculate a 95% confidence interval for a sample with the following data:

• Sample mean (x̄) = 50
• Sample standard deviation (s) = 10
• Sample size (n) = 25

Step-by-step calculation:

1. Degrees of freedom (df) = n - 1 = 25 - 1 = 24
2. t-critical value (for 95% confidence, df=24) ≈ 2.064
3. Standard error (SE) = s / √n = 10 / √25 = 2
4. Margin of error (ME) = t-critical × SE = 2.064 × 2 = 4.128
5. Confidence interval = x̄ ± ME = 50 ± 4.128 = (45.872, 54.128)

This means we are 95% confident that the true population mean falls between 45.872 and 54.128.

How to interpret the results

When interpreting a t-test confidence interval, consider the following:

• The confidence interval provides a range of plausible values for the population mean
• A wider confidence interval indicates more uncertainty about the population mean
• A narrower confidence interval indicates more precision in estimating the population mean
• The confidence level (e.g., 95%) represents the probability that the interval contains the true population mean

If the confidence interval does not include the hypothesized population mean (e.g., zero in a test of no effect), this provides evidence against the null hypothesis.