The Empirical Rule states that for bell-shaped distributions, about 68% of the values fall within 1 standard deviation of the mean. The heights of women at a large university are approximately bell-shaped, with a mean of 67 inches and standard deviation of 3 inches. Use this information to answer the questions. (a) What is the probability that a randomly selected woman from this university is 70 inches or taller? (Give the answer to two decimal places.) (b) What is the probability that three randomly selected women from this university are all 70 inches or shorter? (Give the answer to four decimal places.) (c) What is the probability that of two randomly selected women, one is 70 inches or shorter and the other is 70 inches or taller? (Give the answer to four decimal places.)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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(b) What is the probability that three randomly selected women from this university are all 70 inches or shorter? (Give the answer to four decimal places.)
(c) What is the probability that of two randomly selected women, one is 70 inches or shorter and the other is 70 inches or taller? (Give the answer to four decimal places.)
(d) What is the probability that three randomly selected women are all 67 inches or taller? (Give the answer to two decimal places.)
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