3.27. Problem. (Section 11.5) The following are applications ofTheorem 11.6 or the Central Limit Theorem.
(a) Determine the distribution of (1/5)X1 + (2/5)X2 + (2/5)X3 if X1,X2 and X3 are independent normaldistributions with µ = 2 and
? = 3.
(b) The weight (kg) of a StarBrite watermelon harvested undercertain environmental conditions is normally distributed with amean of 8.0 with standard deviation of 1.9. Suppose 24 StarBritewatermelons grown in these conditions are harvested; compute theprobability that the average weight of all 24 watermelons is lessthan 7.8 kg/fruit
(c) A study of elementary school students reports that the meanage at which children begin reading is 5.7 years with a standarddeviation of 1.1 years. If 55 elementary school students areselected at random, approximate the probability that the averageage at which these 55 children begin reading is at least 6.
(d) Let the random variable X be defined as the number of pipsthat show up when a fair, six-sided die is rolled. The mean andstandard deviation of X can be shown to be µX = 3.5 and ?X = 1.71,respectively. If 100 fair, six-sided dice are rolled, aproximatethe probability that the mean of number of pips on the 100 dice isless than 3.25
