6-3 Sampling Distributions and Estimators 277 If we were to create a probability histogram from Table 6-2, it would not have the bell shape that is characteristic of a normal distribution, but that’s because we are working with such small samples. If the population of 54, 5, 96 were much larger and if we were selecting samples much larger than n = 2 as in this example, we would get a probability histogram that is much closer to being bell-shaped, indicating a normal distribution, as in Example 3. YOUR TURN. Do Exercise 11 “Sampling Distribution of the Sample Mean.” Sampling Distribution of the Sample Mean EXAMPLE 3 Consider repeating this process: Roll a die 5 times to randomly select 5 values from the population 51, 2, 3, 4, 5, 66, then find the mean x of the results. What do we know about the behavior of all sample means that are generated as this process continues indefinitely? SOLUTION Figure 6-19 illustrates a process of rolling a die 5 times and finding the mean of the results. Figure 6-19 shows results from repeating this process 10,000 times, but the true sampling distribution of the mean involves repeating the process indefinitely. Because the values of 1, 2, 3, 4, 5, 6 are all equally likely, the population has a mean of m = 3.5. The 10,000 sample means included in Figure 6-19 have a mean of 3.5. If the process is continued indefinitely, the mean of the sample means will be 3.5. Also, Figure 6-19 shows that the distribution of the sample means is approximately a normal distribution. Means Sample 1 Sample 2 Sample 3 (Population Mean is m 5 3.5) Distribution of Sample Means Sample Means Sample m 5 3.5 Sample means are approximately normal • • • 3.4 4.4 2.8 Sampling Procedure: Roll a die 5 times and find the mean x for each sample. FIGURE 6-19 Sample Means from 10,000 Trials Sampling Distribution of the Sample Variance Let’s now consider the sampling distribution of sample variances. DEFINITION The sampling distribution of the sample variance is the distribution of sample variances (the variable s2), with all samples having the same sample size n taken from the same population. (The sampling distribution of the sample variance is typically represented as a probability distribution in the format of a table, probability histogram, or formula.)
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