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+ Chapter 6: Random Variables Section 6.2 Transforming and Combining Random Variables The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE + Combining Normal Random Variables One of the skills you need to learn from this section is combining two independent normal random variables and finding probabilities. Find the combined mean and standard deviation, and then work the problem as you would any normal curve probability (find the Z-score). + Example Suppose women’s heights are normally distributed with a mean of 64” and a standard deviation of 2.5”. Suppose men’s heights are normally distributed with a mean of 69” and a standard deviation of 2.25”. What is the probability that the difference in height between a randomly chosen male and a randomly chosen female is more than 12 inches? Normal Random Variables Mr. Starnes likes between 8.5 and 9 grams of sugar in his hot tea. Suppose the amount of sugar in a randomly selected packet follows a Normal distribution with mean 2.17 g and standard deviation 0.08 g. If Mr. Starnes selects 4 packets at random, what is the probability his tea will taste right? Let X = the amount of sugar in a randomly selected packet. Then, T = X1 + X2 + X3 + X4. We want to find P(8.5 ≤ T ≤ 9). 8.5 8.68 9 8.68 1.13 and+2.17 z = 8.68 2.00 µT = µX1 + µX2 + µX3 + µzX4 = 2.17 + 2.17 + 2.17 0.16 0.16 2 2 2 2 2 T2 X2 X2 X2 P(-1.13 0.0256 ≤ Z≤(0.08) 2.00) =(0.08) 0.9772 –(0.08) 0.1292 = 0.8480 X (0.08) There is about an 85% chance Mr. Starnes’s T 0.0256 0.16 tea will taste right. 1 2 3 4 Transforming and Combining Random Variables Example + Combining