Confidence Intervals for Sampling Distributions

Basic Concepts

Suppose we take a sample of size n from a normal population N(μ,σ) and ask whether the sample mean differs significantly from the overall population mean. As we have seen in Single Sample Hypothesis Testing

The exact point of rejection (at the right tail), z_crit, has the value

Thus,

In the two-tailed case, we have two critical values: +z_crit and –z_crit, and so we have

Definition 1: The confidence interval is the interval

i.e. the interval

where z_crit is the critical value of z on the right tail, i.e. where z_crit > 0.

The term z_crit ∙ std err is called the margin of error. This term depends on the value of z_crit which in turn depends on the value of α, and so we refer to this interval as the 1 – α confidence interval since we are 100(1 – α)% confident that the population mean will occur in this interval.

As we note in Confidence Interval, while this is a convenient way to think of a confidence interval, it is not accurate. In fact, a 1 – α confidence interval actually means that in 100(1 – α)% of similar random samples the population mean will lie in this interval.

Examples

Example 1: Find the confidence interval for Example 2 (two-tailed case) of Single Sample Hypothesis Testing.

As we have seen, the mean of the sample is 75 and the standard error is 2.58 (based on a sample of size 60). From Definition 1, the confidence interval is given by

Thus, we consider the 95% confidence interval to be (75 – 5.06, 75 + 5.06) = (69.94, 80.06).

If the null hypothesis is true, we are 95% confident that the population mean will be in this interval. Since indeed the population mean, 80, is within the interval, we retain the null hypothesis.

Example 2: Find the confidence interval for Example 2 (two-tailed case) of Single Sample Hypothesis Testing.

This time the sample size is larger (n = 100), and so the standard error is smaller (s.e. = 2.0), and so the confidence interval is also smaller:

Thus, the 95% confidence interval is (75 – 3.92, 75 + 3.92) = (71.08, 78.92). Since the population mean, 80, is outside the interval, we reject the null hypothesis.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf