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Estimation

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Let's return to our example of the random sample of 200 USC undergraduates. Remember that this is both a large and a random sample, and therefore the Central Limit Theorem applies to any statistic that we calculate from it. We ask these 200 randomly-selected USC students to tell us their grade point average (GPA). We calculate the mean GPA for the sample and find it to be 2.58 . Next, we calculate the standard deviation for these self-reported GPA values and find it to be 0.44 . How can we use these two simple univariate statistics from our random sample to estimate the probable GPA for the entire USC student body (i.e, the statistical universe)?
Recall that the standard error is the standard deviation of the sampling distribution . The Central Limit Theorem tells us how to estimate it: N s Y = σ ˆ

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The standard error is estimated by dividing the standard deviation of the sample by the square root of the size of the sample. In our example, 200 44 . 0 ˆ = σ 142 . 14 44 . 0 ˆ = σ 031 . 0 ˆ = σ
First, we let our sample mean (Y-bar), 2.58 , be our estimate of the mean of the sampling distribution of all possible mean GPAs from random samples of 200. Since the Central Limit Theorem applies in this situation, we know that the mean of the sampling distribution is exactly equal to the (unknown) mean GPA for the universe. Thus, 2.58 is the first element in our estimate of the mean of all USC undergraduate GPAs. This is called the point estimate . However, we do not stop here.

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Estimation consists of inferring TWO values. These are known as the upper and lower confidence levels (UCL and LCL). We have no reason to expect that the universe (i.e., actual) GPA is identical to the sample GPA. However, we do know the likelihood of GPA values above and below 2.58. For example, we know that 95.44 percent of the probable true GPAs lie between z = – 2.00 and z = + 2.00. What we need to do is to CONVERT our known z-values (i.e., – 2.00 and + 2.00)
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