Question

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.8 ppm and standard deviation 1.4 ppm. 18 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 8.8 ppm pollutants?

F or the 18 cities, find the probability that the average amount of pollutants is less than 8.8 ppm

For part d), is the assumption that the distribution is normal necessary?

Q1

Q3

IQR

all ppm

Answer 1

1)

X ~ N(9.8,1.4)

b)

¯x ~ N(9.8 , 0.33)

c)

Here, μ = 9.8, σ = 1.4 and x = 8.8. We need to compute P(X <= 8.8). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (8.8 - 9.8)/1.4 = -0.71

Therefore,
P(X <= 8.8) = P(z <= (8.8 - 9.8)/1.4)
= P(z <= -0.71)
= 0.2389

d)

Here, μ = 9.8, σ = 0.33 and x = 8.8. We need to compute P(X <= 8.8). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (8.8 - 9.8)/0.33 = -3.03

Therefore,
P(X <= 8.8) = P(z <= (8.8 - 9.8)/0.33)
= P(z <= -3.03)
= 0.0012

e)

yes