Let the random variable X denote the length of time (in seconds) required to complete a certain task. Suppose that the completion times are normally distributed with a mean of 216 seconds and a standard deviation of 50 seconds. (i) Find the probability that the time needed to complete the task is between 260 and 290 seconds. (ii) Find the value of k if P(X < k) = 0.025 . (iii) A random sample of 25 similar tasks is selected. Find the probability that the mean completion time will be more than 200 seconds.
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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