Question

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.8 ppm and standard deviation 1.6 ppm. 7 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 more than 10.7 ppm pollutants?
- For the 7 cities, find the probability that the average amount of pollutants is more than 10.7 ppm.
- For part d), is the assumption that the distribution is normal necessary? YesNo
- Find the IQR for the average of 7 cities.

Q1 = ppm

Q3 = ppm

IQR: ppm

Answer #1

**Answer:**

a)

Given,

X ~ N(9.8 , 1.6)

b)

Xbar ~ N(9.8 , 1.6/sqrt(7))

Xbar ~ N(9.8 , 0.605)

c)

P(X > 10.7) = P(x-u)/s > (10.7 - 9.8)/1.6)

= P(z > 0.56)

= 0.2877397 [since from z table]

**=** 0.2877

d)

P(Xbar > 10.7) = P((x-u)/(s/sqrt(n)) > (10.7 - 9.8)/(1.6/sqrt(7)))

= P(z > 1.49)

= 0.0681121 [since from z table]

= 0.0681

e) Yes

f)

P(Z < z) = 0.25

since from standard normal table

z = - 0.67

we know,

z = (x - u)/(s/sqrt(n))

- 0.67 = (x - 9.8)/(1.6/sqrt(7))

x = 9.3948

Now,

Q3 = P(Z < z) = 0.75

since from standard normal table

z = 0.67

we know,

z = (x - u)/(s/sqrt(n))

0.67 = (x - 9.8)/(1.6/sqrt(7))

x = 10.2052

IQR = Q3-Q1

= 10.2052 - 9.3948

= 0.8104

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