5.5 Repeated water samples. A nonprofit wants to understand the fraction of households that have elevated levels of lead in their drinking water. They expect at least 5% of homes will have elevated levels of lead, but not more than about 30%. They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called? (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. (c) If the proportions are distributed around 8%, what is the variability of the distribution?
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!
Statistics problem on distribution
![5.5 Repeated water samples. A nonprofit wants to understand the fraction of households that have
elevated levels of lead in their drinking water. They expect at least 5% of homes will have elevated levels of
lead, but not more than about 30%. They randomly sample 800 homes and work with the owners to retrieve
water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this
1,000 times and build a distribution of sample proportions.
(a) What is this distribution called?
(b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain
your reasoning.
(c) If the proportions are distributed around 8%, what is the variability of the distribution?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91195c14-8c67-42a0-b9d0-a3e8c2a59df0%2Fed7b2f52-e4d2-4121-9787-a96bc5b4842a%2F1gnnyab.png&w=3840&q=75)
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