Sports shirts are frequently classified as S, M, L, XL for small, medium, large and extra-large neck sizes. S fits a neck circumference of less than 37cm, M fits between 37 and 40.5cm and L fits between 40.5cm and 44cm while XL fits necks over 44cm in circumference. The neck circumference of adult males has a normal distribution with µ = 40cm and σ = 2cm. (i) What proportion of shirts should be manufactured in each category? (ii) If you wanted to define categories S, M, L, XL so that the categories con- tained 20%, 30%, 30% and 20% respectively, of the total population of adult males, what neck sizes must you assign to each of the categories?
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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