The amount of pollutants that are found in waterways near large cities is normally distributed with mean 8.6 ppm and standard deviation 1.4 ppm. 17 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.What is the distribution of XX? XX~ N(,) What is the distribution of ¯xx¯? ¯xx¯~ N(,) What is the probability that one randomly selected city's waterway will have less than 8.5 ppm pollutants? For the 17 cities, find the probability that the average amount of pollutants is less than 8.5 ppm. For part d), is the assumption that the distribution is normal necessary? YesNo

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Asked Nov 15, 2019
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The amount of pollutants that are found in waterways near large cities is normally distributed with mean 8.6 ppm and standard deviation 1.4 ppm. 17 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.
What is the distribution of
X
X
?
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 8.5 ppm pollutants?
For the 17 cities, find the probability that the average amount of pollutants is less than 8.5 ppm.
For part d), is the assumption that the distribution is normal necessary? YesNo

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Expert Answer

Step 1

It is given that the amount of pollutants follows normal distribution with mean 8.6 and standard deviation 1.4.

That is, µ= 8.6, σ = 1.4.

Thus, X~ N (8.6,1.4).

Step 2

Central Limit Theorem for mean:

If a random sample of size n is taken from a population having mean µ and standard deviation σ then, as the sample size increases, the sample mean approaches the normal distribution with mean µ and standard deviation σ/sqrt(n).

Finding the distribution of X-bar:

It is given that, μ = 8.6 and σ = 1.4.  

Random samples of 17 (n) cities is considered.

By central limit theorem for mean, the average amount of pollutants follows a normal distribution with mean μ = 8.6 and standard deviation 1.4/sqrt (17) = 0.3395.

Thus, the distribution of X-bar ~ N(8.6,0.3395).

Step 3

Finding the probability that the amount of pollutants in a randomly selected city is less than 8.5:

Denote the random variable X as the amount of pollutants in a randomly selec...

P(X <8.5) P(X8.5)
using EXCEL formula
= 0.4715
"=NORM.DIST(8.5,8.6,1.4,1)"
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P(X <8.5) P(X8.5) using EXCEL formula = 0.4715 "=NORM.DIST(8.5,8.6,1.4,1)"

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