583 / 762
Chapter 13: Confidence Intervals – Small Samples
583
confidence intervals for the mean. In addition, for the two population case,
we will consider both independent and dependent samples.
13-2 The
t-
Distribution
When we discussed the assumptions in Chapter 11 for the large sample
confidence interval for the population mean, we assumed that the population
standard deviation was known, or when it was unknown, we assumed that
the sample size was large (
30). In the latter case, we estimated the
population standard deviation with the sample standard deviation. In both
cases, the standard normal distribution was used to find the confidence
interval for the mean and the variable of interest was assumed to be normally
distributed. In many cases however, the population standard deviation is
unknown and the sample size is small (
< 30). Again we can replace the
population standard deviation with the sample standard deviation, but in this
case we use the
-distribution
and not the standard normal distribution.
The
t
distribution has some properties that are similar to and some that are
different from the properties of the standard normal distribution. Properties
of the
distribution are listed below.
Properties which are similar to those of the
z
distribution
It is bell-shaped.
It is symmetric about the mean.
The mean, median, and the mode are all equal to zero.
The curve never touches the
x
axis (horizontal axis).
Properties which are different from those of the
z
distribution
The variance is greater than 1.
The shape of the distribution depends on the sample size or on the
concept of degrees of freedom (
).
As the sample size gets larger, the
t
distribution converges to the
standard normal
distribution (
-
distribution).
Note:
The
degrees of freedom
(
) are the number of values in the final
calculation of a statistic that are free to vary. For example, if the mean of
10 values is 3, then 9 of the 10 values are free to vary. However, once 9