276 CHAPTER 6 Normal Probability Distributions Sampling Distribution of the Sample Mean EXAMPLE 2 A friend of the author has three children with ages of 4, 5, and 9. Let’s consider the population consisting of 54, 5, 96. (We don’t usually know all values in a population, and we don’t usually work with such a small population, but it works well for the purposes of this example.) If two ages are randomly selected with replacement from the population 54, 5, 96, identify the sampling distribution of the sample mean by creating a table representing the probability distribution of the sample mean. Do the values of the sample mean target the value of the population mean? SOLUTION If two values are randomly selected with replacement from the population 54, 5, 96, the leftmost column of Table 6-2 lists the nine different possible samples. The second column lists the corresponding sample means. The nine samples are equally likely with a probability of 1>9. We saw in Section 5-1 that a probability distribution is a description that gives the probability for each value of a random variable, as in the second and third columns of Table 6-2. The second and third columns of Table 6-2 constitute a probability distribution for the random variable representing sample means, so the second and third columns represent the sampling distribution of the sample mean. In Table 6-2, some of the sample mean values are repeated, so we combined them in Table 6-3. INTERPRETATION Because Table 6-3 lists the possible values of the sample mean along with their corresponding probabilities, Table 6-3 is an example of a sampling distribution of a sample mean. The value of the mean of the population 54, 5, 96 is m = 6.0. Using either Table 6-2 or 6-3, we could calculate the mean of the sample values and we get 6.0. Because the mean of the sample means (6.0) is equal to the mean of the population (6.0), we conclude that the values of the sample mean do target the value of the population mean. It’s unfortunate that this sounds so much like doublespeak, but this illustrates that the mean of the sample means is equal to the population mean m. TABLE 6-2 Sampling Distribution of Mean Sample Sample Mean x Probability 4, 4 4.0 1>9 4, 5 4.5 1>9 4, 9 6.5 1>9 5, 4 4.5 1>9 5, 5 5.0 1>9 5, 9 7.0 1>9 9, 4 6.5 1>9 9, 5 7.0 1>9 9, 9 9.0 1>9 TABLE 6-3 Sampling Distribution of Mean (Condensed) Sample Mean x Probability 4.0 1>9 4.5 2>9 5.0 1>9 6.5 2>9 7.0 2>9 9.0 1>9 HINT Read the last sentence of the above paragraph a few times until it makes sense.
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