Confidence Interval for the Mean

Prerequisites
Areas Under Normal Distributions, Sampling Distribution of the Mean, Introduction to Sampling Distributions

When you compute a confidence interval, you compute the mean of a sample in order to estimate the mean of the population. Clearly, if you already knew the population mean, there would be no need for a confidence interval. However, to explain how confidence intervals are constructed, we are going to work backwards and begin by assuming characteristics of the population. Then we will show how sample data can be used to construct a confidence interval.

Assume that the weights of 10-year old children are normally distributed with a mean of 90 and a standard deviation of 36. What is the sampling distribution of the mean for a sample size of 9? Recall from the section on the sampling distribution of the mean, that the mean of the sampling distribution is m and the standard error of the mean is

For the present example, the mean of the sampling distribution of the mean is 90 and the standard deviation is 36/3 = 12. Figure 1 shows this distribution. The shaded area represents the middle 95% of the distribution and stretches from 66.52 to 113.48. These limits were computed by adding and subtracting 1.96 standard deviations to/from the mean of 90 as follows:

 

90 - (1.96)(12) = 66.52
90 + (1.96)(12) =113.48

Figure 1. The sampling distribution of the mean for N =9. The middle 95% of the distribution is shaded.

Figure 1 shows that 95% of the means are within 1.96 standard deviatio

 

The calculation of a confidence on the mean is based on the sampling distribution of the mean and assumes that the sampling distribution of the mean is normally distributed.