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.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the probability that one randomly selected city's waterway will have more than 10.7 ppm pollutants?
4. For the 7 cities, find the probability that the average amount of pollutants is more than 10.7 ppm.
5. For part d), is the assumption that the distribution is normal necessary? YesNo
6. 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