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Confidence Interval for the Mean
Prerequisites
Areas
Under Normal Distributions, Sampling
Distribution of the Mean, Introduction
to Estimation, Introduction
to Confidence Intervals
Learning Objectives
- Use the inverse normal distribution calculator to find the value of
z to use for a confidence interval
- Compute a confidence interval on the mean when σ is
known
- Determine whether to use a t distribution or a normal distribution
- Compute a confidence interval on the mean when σ is estimated
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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 μ and the standard
error of the mean is
For the present example, the sampling distribution
of the mean has a mean of 90 and a standard deviation of 36/3
= 12. Note that the standard deviation of a sampling distribution
is its standard error. Figure 1 shows this distribution. The shaded
area represents the middle 95% of the distribution and stretches
from 66.48 to 113.52. These limits were computed by adding and
subtracting 1.96 standard deviations to/from the mean of 90 as
follows:
90 - (1.96)(12) = 66.48
90 + (1.96)(12) = 113.52
The value of 1.96 is based on the fact that 95%
of the area of a normal distribution is within 1.96 standard deviations
of the mean; 12 is the standard error of the mean.
Figure 1 shows that 95% of the means are no more
than 23.52 units (1.96 standard deviations) from the mean of
90. Now consider the probability that a sample mean computed
in a random sample is within 23.52 units of the population mean
of 90. Since 95% of the distribution is within 23.52 of 90,
the probability that the mean from any given sample will be
within 23.52 of 90 is 0.95. This means that if we repeatedly
compute the mean (M) from a sample, and create an interval ranging
from M - 23.52 to M + 23.52, this interval will contain the population
mean 95% of the time. In general, you compute
the 95% confidence interval for the mean with the following formula:
Lower limit = M - Z.95σm
Upper limit = M + Z.95σm
where Z.95 is the number of standard
deviations extending from the mean of a normal distribution required
to contain 0.95 of the area and σm
is the standard error of the mean.
If you look closely at this formula for a confidence
interval, you will notice that you need to know the standard deviation (σ)
in order to estimate the mean. This may sound unrealistic, and
it is. However, computing a confidence interval when σ is
known is easier than when σ has to be estimated, and serves
a pedagogical purpose. Later in this section we will show how
to compute a confidence interval for the mean when σ has
to be estimated.
Suppose the following five were sampled
from a normal distribution with a standard deviation of 2.5:
2, 3, 5, 6, and 9. To compute the 95% confidence interval, start
by computing the mean and standard error:
M = (2 + 3 + 5 + 6 + 9)/5 = 5.
σm =  =
1.118.
Z.95 can be found using the normal
distribution calculator and specifying that the shaded area
is 0.95 and indicating that you want the area to be between the
cutoff points. As shown below, the value is 1.96. If you had wanted
to compute the 99% confidence interval, you would have set the
shaded area to 0.99 and the result would have been 2.58.
Normal
Distribution Calculator
The confidence interval can then be computed as follows:
Lower limit = 5 - (1.96)(1.118)= 2.81
Upper limit = 5 + (1.96)(1.118)= 7.19
When the variance is not known but has to be estimated
from sample data you should use the t
distribution rather than
the normal distribution. When the sample size is large, say 100
or above, the t distribution is very similar to the standard
normal distribution. However, with smaller sample sizes, the
t distribution is leptokurtic,
which means it has relatively more scores in its tails than does
the normal distribution. As a result, you have to extend farther
from the mean to contain a given proportion of the area. Recall
that with a normal distribution, 95% of the distribution is within
1.96 standard deviations of the mean. Using the t distribution,
if you have a sample size of only 5, 95% of the area is within
2.78 standard deviations of the mean. Therefore, the standard
error of the mean would be multiplied by 2.78 rather than 1.96.
The values of t to be used in a confidence interval
can be looked up in a table of the t distribution. A small version
of such a table is shown in Table 1. The first column, df, stands
for degrees of freedom, and for confidence intervals on the mean,
df is equal to N -1, where N is the sample size.
You can also use the "inverse
t distribution" calculator to find the t values to use
in confidence intervals. You will learn more about the t distribution
in the next section.
Assume that the following five numbers are sampled
from a normal distribution: 2, 3, 5, 6, and 9 and that the standard
deviation is not known. The first steps are to compute the sample
mean and variance:
M = 5
s2 = 7.5
The next step is to estimate the standard error of the mean. If
we knew the population variance, we could use the following
formula:
Instead we compute an estimate of the standard error (sm):
= 1.225
The next step is to find the value of t. As
you can see from Table 1, the value for the 95% interval for
df = N - 1 = 4 is 2.776. The confidence interval is then computed
just as it is when σm. The only
differences are that sm and t rather
than σm and Z are used.
Lower limit = 5 - (2.776)(1.225)= 1.60
Upper limit = 5 + (2.776)(1.225)= 8.40
More generally, the formula for the 95% confidence
interval on the mean is:
Lower limit = M - (tCL)(sm)
Upper limit = M + (tCL)(sm)
where M is the sample mean, tCL
is the t for the confidence level desired (0.95 in the above example),
and sm is the estimated standard error
of the mean.
We will finish with an analysis of the Stroop
Data. Specifically, we will compute a confidence interval
on the mean difference score. Recall that 47 subjects named
the color of ink that color words were written in. The names conflicted
so that, for example, they would name the ink color of the word
"blue" written in red ink. The
correct response is to say "red" and ignore the fact
that the word is "blue." Each subject named the ink
color of 60 words and their times in seconds were recorded.
Table 2 shows the time difference between the interference
and color-naming conditions for 10 of the 47 subjects. The mean
time difference for all 47 subjects is 16.362 seconds and the
standard deviation is 7.470 seconds. The standard error of the
mean is 1.090. A t table shows the critical value of t for 47
- 1 = 46 degrees of freedom is 2.013 (for a 95% confidence interval). Therefore the confidence
interval is computed as follows:
Lower limit = 16.362 - (2.013)(1.090)= 14.17
Upper limit = 16.362 + (2.013)(1.090)= 18.56
Therefore, the interference effect (difference) for the whole
population is likely to be between 14.17 and 18.56 seconds.
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