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Old 07-28-2007   #3
Hobart
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a) To find the probability of a variable, X, between two limits, a and b when the variable is normally distributed:P(a < X < b) = Φ[(b-µ)/σ] - Φ[(a-µ)/σ] where the values within the square brackets are standard normal z-values. Phi (Φ) is and operator that means "find the probability associated with the Z-values in the square brackets". So evaluating:P(550 < X < 650) = Φ[(650-550)/80] - Φ[(550-550)/80] P(550 < X < 650) = Φ[(650-550)/80] = Φ[1.25] - Φ[0] = 0.8944 - 0.50 = 0.3944b) You want to eliminate the bottom 20%. So the Z-value associated with the BOTTOM 20% is -0.84, and the limit for the bottom 20% is found by:-0.84 = (x - 550)/80; solving for x = 482.8. If an integer is required, then the lowest acceptable score would be 483.c) Find the probability that the sample mean is less than 650:Z = (650-550)/(80/√50) = 100/35.78) = 2.79Then Φ[2.79] - Φ[0] = 0.9974 -0.50 = 0.4974
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