What is a Confidence Interval?
Statisticians use a
confidence interval to describe the amount of uncertainty
associated with a sample estimate of a population
How to Interpret Confidence Intervals
Suppose that a 90% confidence interval states that
the population mean is greater than 100 and less than 200.
How would you interpret this statement?
Some people think this means there is a 90% chance that the
population mean falls between 100 and 200. This is incorrect.
Like any population
the population mean is a constant, not a
random variable. It does not change. The
probability that a constant falls within any given range
is always 0.00 or 1.00.
describes the uncertainty associated
with a sampling method.
Suppose we used the same sampling method to select
different samples and to compute a different interval estimate
for each sample.
Some interval estimates would include the true population
parameter and some would not. A 90% confidence level means
that we would expect 90% of the interval estimates to include
the population parameter; A 95% confidence level means that
95% of the intervals would include the parameter; and so on.
Confidence Interval Data Requirements
To express a confidence interval, you need three pieces of
Given these inputs, the range of the confidence interval is
defined by the sample statistic +
margin of error. And the uncertainty associated with
the confidence interval is specified by the confidence level.
Often, the margin of error is not given; you must calculate it.
Previously, we described
how to compute the margin of error.
How to Construct a Confidence Interval
There are four steps to constructing a confidence interval.
The sample problem in the next section applies the above four steps
to construct a 95% confidence interval for a mean score. The next
few lessons discuss this topic in greater detail.
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Test Your Understanding of This Lesson
Suppose we want to estimate the average weight of an adult male in
Dekalb County, Georgia. We draw a random sample of 1,000 men from a
population of 1,000,000 men and weigh them. We find that the average
man in our sample weighs 180 pounds, and the standard deviation of
the sample is 30 pounds. What is the 95% confidence interval.
(A) 180 + 1.86
(B) 180 + 3.0
(C) 180 + 5.88
(D) 180 + 30
(E) None of the above
The correct answer is (A). To specify the confidence interval, we work
through the four steps below.
- Identify a sample statistic. Since we are trying to estimate
the mean weight in the population, we choose the mean weight
in our sample (180) as the sample statistic.
- Select a confidence level. In this case, the confidence level
is defined for us in the problem. We are working with a 95%
- Find the margin of error. Previously, we described
how to compute the margin of error.
The key steps are shown below.
- Find standard error. The standard error (SE) of the
SE = s / sqrt( n ) = 30 / sqrt(1000) = 30/31.62 = 0.95
- Find critical value. The critical value is a factor used to
compute the margin of error. To express the critical value
(t*), follow these steps.
- Compute alpha (α): α = 1 - (confidence level / 100) = 0.05
- Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
- Find the
degrees of freedom (df): df = n - 1 = 1000 - 1 = 999
- The critical value is
the t score having 999 degrees of freedom and a
equal to 0.975. From the
t Distribution Calculator,
we find that the critical value is 1.96.
Note: We might also have expressed the critical value as a
Because the sample size is large, a z score analysis produces
the same result - a critical value equal to 1.96.
- Compute margin of error (ME): ME = critical value * standard error
= 1.96 * 0.95 = 1.86
- 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, this 95% confidence interval says that the population mean falls
within the interval 180 + 1.86.