The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.5 ppm and standard deviation 1.3 ppm. 38 randomly selected large cities are studied. Round all answers to 4 decimal places where possible. What is the distribution of X ? X ~ N(,) What is the distribution of ¯x ? ¯x ~ N(,) What is the probability that one randomly selected city's waterway will have less than 8.6 ppm pollutants?
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!
The amount of pollutants that are found in waterways near large cities is
- What is the distribution of X
? X
- ~ N(,)
- What is the distribution of ¯x
? ¯x
- ~ N(,)
- What is the
probability that one randomly selected city's waterway will have less than 8.6 ppm pollutants? - For the 38 cities, find the probability that the average amount of pollutants is less than 8.6 ppm.
- For part d), is the assumption that the distribution is normal necessary? YesNo
- Find the IQR for the average of 38 cities.
Q1 = ppm
Q3 = ppm
IQR: ppm
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