6-4 The Central Limit Theorem 289 0 z mx 5 98.6 26.64 x 5 98.2 0.0001 FIGURE 6-24 Means of Body Temperatures from Samples of Size n = 106 Table A-2: If we use Table A-2 to find the shaded area in Figure 6-24, we must first convert the sample mean of x = 98.2°F to the corresponding z score: z = x - m x s x = 98.2 - 98.6 0.0602197 = -6.64 Referring to Table A-2, we find that z = -6.64 is off the chart, but for values of z below -3.49, we use an area of 0.0001 for the cumulative left area up to z = -3.49. We therefore conclude that the shaded region in Figure 6-24 is 0.0001. YOUR TURN. Do Exercise 9 “Safe Loading of Elevators.” INTERPRETATION The result shows that if the mean of our body temperatures is really 98.6°F, as we assumed, then there is an extremely small probability of getting a sample mean of 98.2°F or lower when 106 subjects are randomly selected. The sample mean of 98.2°F is significantly low. University of Maryland researchers did obtain such a sample mean, and after confirming that the sample is sound, there are two feasible explanations: (1) The population mean really is 98.6°F and their sample represents a chance event that is extremely rare; (2) the population mean is actually lower than the assumed value of 98.6°F and so their sample is typical. Because the probability is so low, it is more reasonable to conclude that the common belief of 98.6°F for the mean body temperature is a belief that is incorrect. Based on the sample data, we should reject the belief that the mean body temperature is 98.6°F. Not Exactly, but “At Least as Extreme” In Example 3, we assumed that the mean body temperature is m = 98.6°F, and we determined that the probability of getting the sample mean of x = 98.2°F or lower is 0.0001, which suggests that the true mean body temperature is actually less than 98.6°F. In the context of Example 3, the sample mean of 98.2°F is significantly low not because the probability of exactly 98.2°F is low, but because the probability of 98.2°F or lower is small. (See Section 5-1 for the discussion of “Not Exactly, but At Least as Extreme.”) Correction for a Finite Population In applying the central limit theorem, our use of s x = s>1n assumes that the population has infinitely many members. When we sample with replacement, the
RkJQdWJsaXNoZXIy NjM5ODQ=