Confidence Interval: Sample Mean
This lesson describes how to construct a
confidence interval
around a sample mean, x.
Estimation Requirements
The approach described in this lesson is valid whenever the
following conditions are met:
Generally, the sampling distribution will be approximately
normally distributed when the sample size is greater than or equal to 30.
The Variability of the Sample Mean
To construct a
confidence interval
for a sample mean, we need to know the variability of
the sample mean. This means we need to know
how to compute the
standard deviation
or the
standard error
of the
sampling distribution.
Note: In real-world analyses, the standard deviation of the
population is seldom known. Therefore, the standard error is used
more often than the standard deviation.
Alert
The Advanced Placement Statistics
Examination only covers the "approximate" formulas for the standard
deviation and standard error.
σ_{x} =
σ / sqrt( n )
SE_{x} =
s / sqrt( n )
However, students are expected to be
aware of the limitations of these formulas; namely, the
approximate formulas should only be used when the population
size is at least 20 times larger than the sample size.
How to Find the Confidence Interval for a Mean
Previously, we described
how to construct confidence intervals. For convenience, we
repeat the key steps below.
- Identify a sample statistic. Use the sample mean to
estimate the population mean.
- Select a confidence level. The confidence level describes the
uncertainty of a sampling
method. Often, researchers choose 90%, 95%, or 99% confidence
levels; but any percentage can be used.
- Find the margin of error. Previously, we showed
how to compute the margin of error.
- Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic +
margin of error. And the uncertainty is denoted
by the confidence level.
In the next section, we work through a problem that shows how to use
this approach to construct a confidence interval to
estimate a population mean.
Sample Planning Wizard
As you may have noticed, the four steps required to specify a confidence
interval for a sample mean can involve many time-consuming computations. Stat Trek's
Sample Planning Wizard does this work for you - quickly, easily, and
error-free. In addition to constructing a confidence interval, the Wizard
creates a summary report that lists key findings and documents analytical
techniques. Whenever you need to construct a confidence interval, consider
using the Sample Planning Wizard. The
wizard is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
Sample Planning Wizard
Test Your Understanding
Problem 1
Suppose a simple random sample of 150 students is drawn
from a population of 3000
college students. Among sampled students, the average IQ score is
115 with a standard deviation of 10. What is the 99%
confidence interval for the students' IQ score?
(A) 115 + 0.01
(B) 115 + 0.82
(C) 115 + 2.1
(D) 115 + 2.6
(E) None of the above
Solution
The correct answer is (C). The approach that we used to solve this
problem is valid when the following conditions are met.
Since the above requirements are satisfied, we can use the following
four-step approach to construct a confidence interval.
- Identify a sample statistic. Since we are trying to estimate
a population mean, we choose the sample mean
(115) as the sample statistic.
- Select a confidence level. In this analysis, the confidence level
is defined for us in the problem. We are working with a 99%
confidence level.
- Find the margin of error. Elsewhere on this site, we show
how to compute the margin of error when the sampling
distribution is approximately normal. The key steps are
shown below.
- Find standard deviation or standard error. Since we do not
know the standard deviation of the population, we cannot compute the
standard deviation of the sample mean; instead, we compute the standard
error (SE). Because the sample size is much smaller than the
population size, we can use the "approximate" formula for the
standard error.
SE = s / sqrt( n ) = 10 / sqrt(150)
SE = 10 / 12.25 = 0.82
- Find critical value. The critical value is a factor used to
compute the margin of error. For this example, we'll express the critical
value as a
t score.
To find the critical value, we take these steps.
- Compute alpha (α):
α = 1 - (confidence level / 100)
α = 1 - 99/100 = 0.01
- Find the critical probability (p*):
p* = 1 - α/2 = 1 - 0.01/2 = 0.995
- Find the
degrees of freedom (df):
df = n - 1 = 150 - 1 = 149
- The critical value is
the t statistic having 149 degrees of freedom and a
cumulative probability
equal to 0.995. From the
t Distribution Calculator,
we find that the critical value is 2.61.
Note: We might also have expressed the critical value as a
z-score.
Because the sample size is fairly large, a z-score analysis produces
a similar result - a critical value equal to 2.58.
- Compute margin of error (ME):
ME = critical value * standard error
ME = 2.61 * 0.82 = 2.1
- Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic +
margin of error. And the uncertainty is denoted
by the confidence level.
Therefore, the 99% confidence interval is 112.9 to 117.1. That is, we are 99%
confident that the true population mean is in the range
defined by 115 + 2.1.