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5.130. (a) 78.7 (b) 0.0090 CHAPTER 12 CHAPTER 6
Estimation Theory
Unbiased Estimates and Efficient Estimates
As we remarked in 5 (see page 158), a statistic is called an unbiased estimator of a population parameter if the mean or expectation of the statistic is equal to the parameter. The corresponding value of the statistic is then called an unbiased estimate of the parameter. # EXAMPLE 6.1 The mean X and variance S 2 as defined on pages 155 and 158 are unbiased estimators of the popula^ # tion mean m and variance s2, since E(X) m, E(S 2) s2. The values x and ^2 are then called unbiased estimates. Hows # ^ ^ ever, S is actually a biased estimator of s, since in general E(S) 2 s. If the sampling distributions of two statistics have the same mean, the statistic with the smaller variance is called a more efficient estimator of the mean. The corresponding value of the efficient statistic is then called an efficient estimate. Clearly one would in practice prefer to have estimates that are both efficient and unbiased, but this is not always possible. EXAMPLE 6.2 For a normal population, the sampling distribution of the mean and median both have the same mean, namely, the population mean. However, the variance of the sampling distribution of means is smaller than that of the sampling distribution of medians. Therefore, the mean provides a more efficient estimate than the median. See Table 51, page 160. Point Estimates and Interval Estimates. Reliability
An estimate of a population parameter given by a single number is called a point estimate of the parameter. An estimate of a population parameter given by two numbers between which the parameter may be considered to lie is called an interval estimate of the parameter. EXAMPLE 6.3 If we say that a distance is 5.28 feet, we are giving a point estimate. If, on the other hand, we say that the distance is 5.28 0.03 feet, i.e., the distance lies between 5.25 and 5.31 feet, we are giving an interval estimate. A statement of the error or precision of an estimate is often called its reliability.
Confidence Interval Estimates of Population Parameters
Let mS and sS be the mean and standard deviation (standard error) of the sampling distribution of a statistic S. Then, if the sampling distribution of S is approximately normal (which as we have seen is true for many statistics if the sample size n 30), we can expect to find S lying in the intervals mS sS to mS sS, mS 2sS to mS 2sS or mS 3sS to mS 3sS about 68.27%, 95.45%, and 99.73% of the time, respectively. Equivalently we can expect to find, or we can be confident of finding, mS in the intervals S sS to S sS, S 2sS to S 2sS or S 3sS to S 3sS about 68.27%, 95.45%, and 99.73% of the time, respectively. Because of this, we call these respective intervals the 68.27%, 95.45%, and 99.73% confidence intervals for estimating mS (i.e., for estimating the population parameter, in the case of an unbiased S). The end numbers of these intervals (S sS, S 2sS, S 3sS) are then called the 68.27%, 95.45%, and 99.73% confidence limits. Similarly, S 1.96sS and S 2.58sS are 95% and 99% (or 0.95 and 0.99) confidence limits for mS. The percentage confidence is often called the confidence level. The numbers 1.96, 2.58, etc., in the confidence limits are called critical values, and are denoted by zc. From confidence levels we can find critical values, and conversely.

