Question

According to the U.S. Department of Labor, the average American household spends $639 on household supplies per year. Suppose annual expenditures on household supplies per household are uniformly distributed between the values of $251 and $1,027.

(a) What is the standard deviation of this distribution?
(b) What is the height of this distribution?
(c) What proportion of households spend more than $870 per year on household supplies?
(d) What proportion of households spend more than $1,290 per year on household supplies?
(e) What proportion of households spend between $380 and $490 on household supplies?

Answer 1

a)
`Var(X) = ((b-a)^2)/(12) = ((1027 -251)^2)/(12)= 50181.333`

And the deviation would be:

`Sd(X) = √(50181.333)= 224.012`

b)
`h = (1)/(b-a) =(1)/(1027-251)=0.00129`

c)
`P(X>870) = 1-P(X<870) = 1- (870-251)/(1027-251)= 0.2023`

d)
`P(X>1290) = 1-P(X<1290) = 1- (1027-251)/(1027-251)= 0`

e)
`P(380<X<490) = P(X<490)-P(X<380) = (490-251)/(1027-251)- (380-251)/(1027-251)= 0.3080-0.1662= 0.1418`

Explanation:

For this case we define the random variable X as "American household spend" and we know that the distribution for X is given by:

`X \sim Unif (a=251, b =1027)`

Part a

We can calculate the variance first with this formula:

`Var(X) = ((b-a)^2)/(12) = ((1027 -251)^2)/(12)= 50181.333`

And the deviation would be:

`Sd(X) = √(50181.333)= 224.012`

Part b

For this case the height represent the individual probability for any value in the interval and is given by:

`h = (1)/(b-a) =(1)/(1027-251)=0.00129`

Part c

For this case we can use the cumulative distribution function given by:

`F(x) = (x-a)/(b-a) , a \leq X \leq b`

We want this probability:

`P(X>870) = 1-P(X<870) = 1- (870-251)/(1027-251)= 0.2023`

Part d

For this case we can use the cumulative distribution function given by:

`F(x) = (x-a)/(b-a) , a \leq X \leq b`

We want this probability:

`P(X>1290) = 1-P(X<1290) = 1- (1027-251)/(1027-251)= 0`

Part e

We want this probability:

`P(380<X<490) = P(X<490)-P(X<380) = (490-251)/(1027-251)- (380-251)/(1027-251)= 0.3080-0.1662= 0.1418`