Confidence intervals
From looking at a sample, we would like to know about the population from which
it came. Although we would like to know everything about the population
including the mean, median, variance, quartiles, etc.; in the present course
we shall only inquire about the mean (and we shall assume we know the
standard deviation of the population). x-bar is an unbiased estimator of the
mean of the population, but to measure how accurate it is we shall use a
confidence interval.
A (1-*alpha*) confidence interval is an interval of
the form
(x-bar - z_(*alpha*/2) × *sigma*/(n^.5), x-bar + z_(*alpha*/2) × *sigma*/(n^.5) )
where
- x-bar
- (an overscored lower case x) is the mean of the sample.
- z_(*alpha*/2)
- (a lowercase z with *alpha*/2 as a subscript) is the z-value beyond which
the area in the tail is equal to *alpha*/2.
- *sigma*
- is the standard deviation of the population.
- n
- is the size of the sample.
Note that *sigma*/(n^.5) will often be denoted as *sigma* with the subscript
x-bar.
Example: If x-bar = 145, *sigma* = 25, and n = 50, what is the 95% confidence
interval for the mean?
-
x-bar = 145
- 1-*alpha* = .95, *alpha* = .05, *alpha*/2 = .025, looking for .975 in the
body of the normal table, one finds that z_.025 = 1.96.
- *sigma*/(n^.5) = 25/(50^.5) = 3.54.
- Therefore the 95% confidence interval is (138.07, 151.93).
- A confidence interval is a random variable because x-bar (its center)
is a random variable. It depends on the particular sample which produced
x-bar.
- It is an interval which, by its method of construction, you are confident
contains the mean. 95% of the time that you construct a 95% confidence
interval, the mean of the population (µ) will be in that interval.
- 1-*alpha* is called the level of confidence.
- The radius of the confidence interval, z_(*alpha*/2) × *sigma*/(n^.5),
is also called the margin of error.
- For a fixed level of confidence, the radius of the confidence interval
decreases as the sample size (n) increases. This means that the estimate of
µ is more precise for larger n.
- For fixed n, the radius of the confidence interval increases as the level
of confidence (1-*alpha*) increases. This is because you need a bigger interval
to be more confident it contains the mean.
N.B.: As (1-*alpha*) increases, *alpha* decreases.
Applets: The relation between data points and confidence intervals for the mean (including dependence on the level of confidence (1-alpha) and whether it contains the mean of the population) is illustrated by Charles Stanton. An alternative by R. Webster West illustrates how the size of the confidence intervals change with (1-alpha) and the extent to which they include the mean.
Competencies: If x-bar = 27 based on a sample of size n=60 from a populaton with standard deviation 5, what is the 90% confidence interval?
Reflection: How does the size (radius, margin of error) of a confidence interval change when the sample size increases? How does the size (radius, margin of error) of a confidence interval change when the level of confidence increases? How does the level of confidence of an inter val change when the sample size changes if the size (radius, margin of error) does not change?
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Questions?