Question

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 8.8 ppm and standard deviation 1.8 ppm. 37 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.

1. What is the distribution of X? X ~ N(Correct,Correct)  
2. What is the distribution of ¯x? ¯x ~ N(,)
3. What is the probability that one randomly selected city's waterway will have less than 9.6 ppm pollutants?
4. For the 37 cities, find the probability that the average amount of pollutants is less than 9.6 ppm.
5. For part d), is the assumption that the distribution is normal necessary? YesNo
6. Find the IQR for the average of 37 cities.
   Q1 =  ppm
   Q3 =  ppm
   IQR:  ppm

Answer 1

Note: Denoting Normal distribution as N(μ,σ²)

a)

Distribution of X: N(8.8, 1.8² )

b)

Concept: The central limit theorem tells us that if we take repeated samples from an unknown population, the distribution of sample means would follow a normal distribution. We call this as sampling distribution and has a mean equal to the population mean and standard deviation equal to population standard deviation divided by square root of sample size

Distribution of : N(8.8, 1.8/Sqrt(37)² )

Distribution of : N(8.8, 0.296² )

c)

We need to find the probability that one randomly selected city's waterway will have less than 9.6 ppm pollutants, so we need to take the distribution of X

d)

We need to find the probability that the average amount of pollutants is less than 9.6 ppm for 37 cities, we have to take the distribution of

e)

No, as discussed above as per CLT, if n is greater than or equal to 30, we can assume the sample distribution to be normal

f)

Q1: We need to find "a" such that P( < a) = 0.25

P( < a) = 0.25

We also know that, Pr(Z<−0.675)=0.25

So, Q1 = 8.6 ppm

Q3: We need to find "b" such that P( < b) = 0.75

P( < b) = 0.75

We also know that Pr(Z<0.675)=0.75

So, Q3 = 9 ppm

IQR: Q3 - Q1 = 9 - 8.6 = 0.4 ppm

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