5-1 Probability Distributions 209 CAUTION An expected value need not be a whole number, even if the different possible values of x might all be whole numbers. The expected number of girls in five births is 2.5, even though five particular births can never result in 2.5 girls. If we were to survey many couples with five children, we expect that the mean number of girls will be 2.5. Meta-Analysis The term metaanalysis refers to a technique of conducting a study that essentially combines results of other studies. It has the advantage that separate smaller samples can be combined into one big sample, making the collective results more meaningful. It also has the advantage of using work that has already been done. Meta-analysis has the disadvantage of being only as good as the studies that are used. If the previous studies are flawed, the “garbage in, garbage out” phenomenon can occur. The use of meta-analysis is currently popular in medical research and psychological research. As an example, a study of migraine headache treatments was based on data from 46 other studies. (See “Meta-Analysis of Migraine Headache Treatments: Combining Information from Heterogeneous Designs,” by Dominici et al., Journal of the American Statistical Association, Vol. 94, No. 445.) t di It h th Finding the Mean, Variance, and Standard Deviation EXAMPLE 3 Table 5-2 on page 206 describes the probability distribution for the number of females in two births. Find the mean, variance, and standard deviation for the probability distribution described in Table 5-2 from Example 1. In Table 5-4, the two columns at the left describe the probability distribution given earlier in Table 5-2. We create the two columns at the right for the purposes of the calculations required. Using Formulas 5-1 and 5-2 and the table results, we get Mean: m = Σ3x # P1x24 = 1.0 Variance: s 2 = Σ31x - m2 2 # P1x24 = 0.5 The standard deviation is the square root of the variance, so Standard deviation: s = 20.5 = 0.707107 = 0.7 1rounded2 Rounding: In Table 5-4, we use m = 1.0. If m had been the value of 1.23456, we might round m to 1.2, but we should use its unrounded value of 1.23456 in Table 5-4 calculations. Rounding in the middle of calculations can lead to results with errors that are too large. SOLUTION TABLE 5-4 Calculating m and s for a Probability Distribution x P(x) x~ P1x2 1x − M2 2 ~ P1x2 0 0.25 0# 0.25 = 0.00 10 - 1.022 # 0.25 = 0.25 1 0.50 1# 0.50 = 0.50 11 - 1.022 # 0.50 = 0.00 2 0.25 2# 0.25 = 0.50 12 - 1.022 # 0.25 = 0.25 Total 1.00 c m = Σ3x # P1x24 0.50 c s 2 = Σ31x - m2 2 # P1x24 INTERPRETATION In two births, the mean number of females is 1.0 female, the variance is 0.5 female2, and the standard deviation is 0.7 female. Also, the expected value for the number of females in two births is 1.0 female, which is the same value as the mean. If we were to collect data on a large number of trials with two births in each trial, we expect to get a mean of 1.0 female. YOUR TURN. Do Exercise 15 “Mean and Standard Deviation.” Making Sense of Results: Significant Values We present the following two different approaches for determining whether a value of a random variable x is significantly low or significantly high.
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