In a survey of a group of​ men, the heights in the​ 20-29 age group were normally​ distributed, with a mean of 67.9 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts​ (a) through​ (d) below. ​(a) Find the probability that a study participant has a height that is less than 67 inches.   The probability that the study participant selected at random is less than 67 inches tall is nothing. ​(Round to four decimal places as​ needed.) ​(b) Find the probability that a study participant has a height that is between 67 and 70 inches.   The probability that the study participant selected at random is between 67 and 70 inches tall is nothing. ​(Round to four decimal places as​ needed.) ​(c) Find the probability that a study participant has a height that is more than 70 inches.   The probability that the study participant selected at random is more than 70 inches tall is nothing. ​(Round to four decimal places as​ needed.) ​(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.     A. The events in parts (a), (b), and (c) are unusual because all of their probabilities are   less than 0.05.   B. The events in parts (a) and (c) are unusual because its probabilities are less than 0.05.   C. There are no unusual events because all the probabilities are greater than 0.05.   D. The event in part (a) is unusual because its probability is less than 0.05.

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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!