6-3 Sampling Distributions and Estimators 279 Estimators: Unbiased and Biased The preceding examples show that sample proportions, means, and variances tend to target the corresponding population parameters. More formally, we say that sample proportions, means, and variances are unbiased estimators. See the following two definitions. DEFINITIONS An estimator is a statistic used to infer (or estimate) the value of a population parameter. An unbiased estimator is a statistic that targets the value of the corresponding population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the corresponding population parameter. Unbiased Estimators These statistics are unbiased estimators. That is, they each target the value of the corresponding population parameter (with a sampling distribution having a mean equal to the population parameter): ■ Proportion p n ■ Mean x ■ Variance s 2 Biased Estimators These statistics are biased estimators. That is, they do not target the value of the corresponding population parameter: ■ Median ■ Range ■ Standard deviation s Important Note: The sample standard deviations do not target the population standard deviation s, but the bias is relatively small in large samples, so s is often used to estimate S even though s is a biased estimator of s. Sampling Distribution of the Sample Range EXAMPLE 5 As in Example 2, consider samples of size n = 2 randomly selected from the population 54, 5, 96. a. List the different possible samples along with the probability of each sample, then find the range for each sample. b. Describe the sampling distribution of the sample range in the format of a table summarizing the probability distribution. c. Based on the results, do the sample ranges target the population range, which is 9 - 4 = 5? d. What do these results indicate about the sample range as an estimator of the population range? SOLUTION a. In Table 6-4 on the next page we list the nine different possible samples of size n = 2 selected with replacement from the population 54, 5, 96. The nine samples are equally likely, so each has probability 1>9. Table 6-4 also shows the range for each of the nine samples. continued
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