| 11.3 | Standard Errors of Estimates |
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- Key Idea 1: What the standard error (SE) of an estimate measures...
Discussion It is very important to be able to calculate a quantity that measures the error associated with an estimate.
One important way to measure this error is called the standard error (SE) of an estimate.
Recall that the size of the standard error is interpreted as the size of a typical error-that is, the typical distance between the sample estimate and the population parameter.
Example - Key Idea 2: Two distinct ways of finding the standard error of a particular sample estimate...
The first method is to repeatedly simulate the drawing of a random sample from the population, following our five-step approach, and each time compute the estimate from the sample. Then you compute the standard deviation of this collection of estimates. This is an estimate of the needed standard error.
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The second method is to use a theoretical formula for the standard error of the estimate, as in fact we already did for the standard error of the sample mean of a set of measurements in Section 9.3.
Discussion
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| Example and Simulated Experiment |
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It requires that the general shape of the probability law of the population be known, in particular that its spread, which is usually specified by the population standard deviation, be known. Usually this is not the case. One approach that gets around this problem is known as " bootstrapping." |
When we do not know the general shape (probability law) of the population, the bootstrap method uses the shape of the sample to provide us with an estimate of the unknown shape of the population.
In other words, the shape of the sample data supplies an estimated population distribution that, because the sample is a random sample, should be shaped approximately like the unknown population distribution.
We then sample with replacement from this artificial population created from the sample in order to repeatedly simulate obtaining the estimate whose standard error we seek.
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| Example and Simulated Experiment |
For example, as we stated in Chapter 9, the standard error of a sample mean is
where s is the population standard deviation and n is the sample size.
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The standard error of a sample proportion is
where p is the population proportion and n is the sample size. For example, in the Key Problem the standard error is estimated to be
Here we have substituted the sample proportion
in the formula for the population proportion p (
is read "p
hat"). We know that since is a reasonable
estimate of p, will be close to p and
therefore this estimate of the true standard error should be close
to the true standard error.
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In many cases proven theories enable us to calculate standard errors of estimates. (The details of some of these theories will be discussed in Chapter 14.) |
In summary, the second method is to use a theoretical formula for the standard error of our estimate, possibly requiring us to estimate a population parameter by using the random sample.
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Note that this method requires no repeated actual or simulated random samples from the population; the single obtained random sample suffices.However, we often need the standard error of an estimate for which there does not exist an easy method 2 formula like the two illustrated above. (The sample median, used to estimate the population median, is one such estimate.)
In such situations we have to use the first method.
Methods 1 and 2 are both important in statistical practice for determining the standard errors of estimates.
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