Two students are throwing water balloons at a target. Accuracy is measured as how close the balloon is from the center of the target. Tory’s distances from the center of the target are approximately Normally distributed with a mean of 123 mm and a standard deviation of 31 mm. Adam’s distances from the center is approximately Normally distributed with a mean of 108 mm and a standard deviation of 47 mm. If 5 attempts for Tory and 3 attempts for Adam are randomly selected, what is the probability that the mean distance from the center of the target for Tory is more than for Adam?
The difference between the sample means, Tory’s minus Adam’s, is Normally distributed with mean 123 − 108 = 15 mm. Its standard deviation is √(31²/5 + 47²/3) ≈ 30.47 mm, so the chance that this difference is above 0 is Φ(15/30.47) ≈ Φ(0.49) = 0.6879. Thus the probability is 0.6879, choice A.
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