Confidence Intervals Using the t-Distribution

In statistics, a confidence interval is a range of values that is likely to contain the true value of a population parameter. For example, a 95% confidence interval for the average height of women in the United States might be 5'4" to 5'6". This means that we are 95% confident that the true average height of women in the United States falls somewhere between 5'4" and 5'6".

Confidence intervals are often used in hypothesis testing. For example, we might want to test the hypothesis that the average height of women in the United States is 5'5". To do this, we could calculate a confidence interval for the average height and see if 5'5" falls within that interval. If it does, then we would fail to reject the null hypothesis. If it does not, then we would reject the null hypothesis.

The t-distribution is a probability distribution that is used to calculate confidence intervals when the population standard deviation is unknown. The t-distribution is similar to the normal distribution, but it has heavier tails. This means that there is a higher probability of observing values that are far from the mean in the t-distribution than in the normal distribution.

Steps for Calculating a Confidence Interval Using the t-Distribution

The following steps can be used to calculate a confidence interval using the t-distribution:

1. Calculate the sample mean (x̄).
2. Calculate the sample standard deviation (s).
3. Determine the degrees of freedom (df). The degrees of freedom are equal to the sample size minus 1 (n - 1).
4. Look up the t-value for the desired confidence level and degrees of freedom in a t-table. For example, for a 95% confidence level and 10 degrees of freedom, the t-value is 2.228.
5. Calculate the margin of error (ME). The margin of error is equal to the t-value multiplied by the sample standard deviation divided by the square root of the sample size. ME = t * s / √n.
6. Calculate the confidence interval. The confidence interval is equal to the sample mean plus or minus the margin of error. CI = x̄ ± ME.

Example

Let's say we want to calculate a 95% confidence interval for the average height of women in the United States. We take a sample of 20 women and find that their average height is 5'5" and the sample standard deviation is 2 inches.

The following steps can be used to calculate the confidence interval:

1. x̄ = 5'5"
2. s = 2 inches
3. df = n - 1 = 20 - 1 = 19
4. t = 2.093 (from a t-table for a 95% confidence level and 19 degrees of freedom)
5. ME = t * s / √n = 2.093 * 2 / √20 = 0.936 inches
6. CI = x̄ ± ME = 5'5" ± 0.936 inches = 5'4.06" to 5'5.94"

Therefore, the 95% confidence interval for the average height of women in the United States is 5'4.06" to 5'5.94".

Conclusion

The t-distribution is a useful tool for calculating confidence intervals when the population standard deviation is unknown. By following the steps outlined above, you can calculate a confidence interval that will help you to estimate the true value of a population parameter.