Question

The American Housing Survey reported the following data on the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months.

	Number of Houses (1000s)
Number of Times	Owner Occupied		Renter Occupied
0	549		23
1	5,012		542
2	6,100		3,734
3	2,544		8,690
4 times or more	558		3,783

Do not round intermediate calculations. Round your answers to two decimal places.

a. Define a random variable x = number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let x = 4 represent 4 or more times.)

x	f(x)
0
1
2
3
4
Total

b. Compute the expected value and variance for x.

	Total
E(x)
Var(x)

c. Define a random variable y = number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let y = 4 represent 4 or more times.)

y	f(y)
0
1
2
3
4
Total

d. Compute the expected value and variance for y.

	Total
E(y)
Var(y)

Answer 1

(a) Let, X be a random variable number of times that owner occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months.

A probability distribution for the random variable. (Let x = 4 represent 4 or more times.)

X	Number of Houses (1000s) Owner Occupied	f(x)
0	549	0.037188
1	5012	0.339497
2	6100	0.413195
3	2544	0.172323
4	558	0.037797
total	14763	1

The f(x) column has been constructed by dividing the number of Houses (1000s) Owner Occupied for each X by the total number of Houses (1000s) Owner Occupied

(b) Expected value of X and variance of X:

Necessary calculation is shown in the table beloiw:

X	f(x)	x*f(x)	x2*f(x)
0	0.037188	0	0
1	0.339497	0.339497	0.339497
2	0.413195	0.82639	1.652781
3	0.172323	0.516968	1.550904
4	0.037797	0.151189	0.604755
total	1	1.834045	4.147937

E(X)=1.834045

Var(X)=4.147937-(1.834045)2

=0.7842159

(c) Let: Y= number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months .

A probability distribution for the random variable. (Let y = 4 represent 4 or more times.)

Y	Renter Occcupied	f(y)
0	23	0.001371
1	542	0.032316
2	3734	0.222633
3	8690	0.518125
4	3783	0.225554
Total	16772	1

The f(y) column has been constructed by dividing the number of Houses (1000s) Renter Occupied for each Y by the total number of Houses (1000s) Renter Occupied.

(d) Expected value of Y and variance of Y:

Necessary calculation is shown in the table beloiw:

Y	Renter Occcupied	f(y)	y*f(y)	y^2*f(y)
0	23	0.001371	0	0
1	542	0.032316	0.032316 0.58546	0.032316
2	3734	0.222633	0.445266	0.890532
3	8690	0.518125	1.554376	4.663129
4	3783	0.225554	0.902218	3.608872
Total	16772	1	2.934176	9.194849

E(X)=2.934176

Var(X)=9.194849-(2.934176)2

=0.58546

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