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), zcrit, has value
In the two-tailed case, we have two critical values: +zcrit and –zcrit, and so we have
Definition 1: The confidence interval is the interval
where zcrit is the critical value of z on the right tail, i.e. where zcrit > 0.
The term zcrit ∙ std err is the called the margin of error. This term depends on the value of zcrit 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 1 – α % confident that the population mean will occur in this interval.
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
And so 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.