Bill contends that people spend between 0 and 80 minutes each day listening to music, and he insists that no range of times is any more likely than any other. Jane argues that it doesn’t make sense to believe that because people are just as likely to spend more than two hours listening to music as they are to spend between 30 and fifty minutes and proposes a normal distribution instead. According to Bill's conjecture, what would be the 95 th percentile of the amount of time that people spend listening to music in a day?
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!
Bill contends that people spend between 0 and 80 minutes each day listening to music, and he insists that no range of times is any more likely than any other. Jane argues that it doesn’t make sense to believe that because people are just as likely to spend more than two hours listening to music as they are to spend between 30 and fifty minutes and proposes a
According to Bill's conjecture, what would be the 95 th percentile of the amount of time that people spend listening to music in a day?
As per Bill's conjecture, people spend between 0 to 80 minutes listening to music. He has defined this range and as per his conjecture, values beyond this range are impossible. He also insists that no time is more likely than any other. This just means that within the range, all values are equally likely. We can safely assume that this is a Uniform distribution.
To determine the percentile, we'll first draw the distribution.
We need to find the value of X for which the shaded area is 95% of the entire distribution.
We know that the distribution is uniform. Therefore,
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