Question

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.7 ppm and standard deviation 1.6 ppm. 38 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 less than 10.4 ppm pollutants?
4. For the 38 cities, find the probability that the average amount of pollutants is less than 10.4 ppm.
5. For part d), is the assumption that the distribution is normal necessary? NoYes
6. Find the IQR for the average of 38 cities.
   Q1 =  ppm
   Q3 =  ppm
   IQR:  ppm

Answer 1

a) X ~ N(9.7, 1.6)

b) $\mu_{\bar x}$ = 9.7

   $\sigma_{\bar x} = \sigma/\sqrt n$ = 1.6/ = 0.2596

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$\bar x$ ~ N(

c) P(X < 10.4)

= P((X - $\mu$)/$\sigma$ < (10.4 - $\mu$)/$\sigma$)

= P(Z < (10.4 - 9.7)/1.6)

= P(Z < 0.44)

= 0.6700

d) P($\bar x$ < 10.4)

= P(($\bar x$ - $\mu$)/(
o/n
) < (10.4 - $\mu$)/(
o/n
))

= P(Z < (10.4 - 9.7)/(1.6/))

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= P(Z < 2.70)

= 0.9965

e) No, since the sample size is greater than 30, so it is not necessary to assume that the distribution is normak.

f) P($\bar x$ < x) = 0.25

or, P(($\bar x$ - $\mu$)/(
o/n
) < (x - $\mu$)/(
o/n
)) = 0.25

or, P(Z < (x - 9.7)/(1.6/)) = 0.25

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or, (x - 9.7)/(1.6/) = -0.67

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or, x = -0.67 * (1.6/) + 9.7

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or, x = 9.526

Q1 = 9.526

P($\bar x$ < x) = 0.75

or, P(($\bar x$ - $\mu$)/(
o/n
) < (x - $\mu$)/(
o/n
)) = 0.75

or, P(Z < (x - 9.7)/(1.6/)) = 0.75

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or, (x - 9.7)/(1.6/) = 0.67

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or, x = 0.67 * (1.6/) + 9.7

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or, x = 9.874

Q3 = 9.874

IQR = Q3 - Q1 = 9.874 - 9.526 = 0.348