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2. The sampling distribution of X has standard deviation (6.3) T This is called the standard error of X. 3. If the sample size, n, is not small compared to the population size, N, then the standard error of X is as follows: IN 1 ox (6.4) N-1 4. If the parent population distribution is normal and, thus, the sampling distribution of the sample means is normal, then the random variable X-M Z= (6.5) Tx has a standard normal distribution with a mean of O and a variance of 1. Figure 6.3 shows the sampling distribution of the sample means for sample sizes n = 25 and 1 = 100 from a normal distribution. Each distribution is centered on the mean, but as the sample size increases, the distribution becomes concentrated more closely around the population mean because the standard error of the sample mean de- creases as the sample size increases. Thus, the probability that a sample mean is a fixed distance from the population mean decreases with increased sample size. n = 100 37 im 5 1 = 25 97 98 99 100 101 102 103 I Example 6.2 Executive Salary Distributions (Normal Probability) Suppose that, based on historical data, we believe that the annual percentage salary increases for the chief executive officers of all midsize corporations are normally dis- tributed with a mean of 12.2% and a standard deviation of 3.6%. A random sample of nine observations is obtained from this population, and the sample mean is computed. What is the probability that the sample mean will be greater than 14.4%