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 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.
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