Answer to Question #323847 in Statistics and Probability for Bless

Question

According to the Insurance Institute of America, a family of four spends between Ghc 400 and Ghc 3,800 per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts.

(a) What is the mean amount spent on insurance?

(b) What is the standard deviation of the amount spent?

(c) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year?

(d) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year? (e) What is the probability a family spends more than Ghc 3,000 per year?

Answer 1

We have a uniform distribution, $a=400, b=3800.$

(a) The mean:

$\mu=\cfrac{b+a}{2}=\cfrac{3800+400}{2}=2100\text{ Ghc}.$

(b) The standard deviation:

$\sigma=\cfrac{b-a}{\sqrt{12}}=\cfrac{3800-400}{\sqrt{12}}=981.50\text{ Ghc}.$

(c), (d) The probability that X takes on a value between x₁ and x₂ can be found by the following formula:

$P(x_1<X<x_2)=\cfrac{x_2-x_1}{b-a}.$

A family spends less than Ghc 2000 means a sum from 400 to 2000. Let X be the amount spent on insurance,

$P(400<X<2000)=\cfrac{2000-400}{3800-400}=0.4706.$

(e) $P(3000<X<3800)=\cfrac{3800-3000}{3800-400}=0.2353.$