15. ACT scores The Mathematics section of the ACT test had a mean of 20.9 and an SD of5.3 for the years 2013-2015. If these are well modeled by a Normal distribution, about what percent of students scored a) over 31? b) under 18? c) between 18 and 31?
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!
Given mean, mu= 20.9
Standard deviation, sigma= 5.3
Z score is given by,
Z= (X-mu)/sigma
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