Question

A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, 7% had exactly three, 11% exactly two, and 21% only one defect. Find the expected number of appearance defects in a new car and the standard deviation.

Answer 1

The expected number of appearance defects in a new car = 0.64

standard deviation = 0.93(approximately)

Explanation:

Given -

7% had exactly three, 11% exactly two, and 21% only one defect in cars.

Let X be no of defects in a car

P(X=3) = .07 , P(X=2) = .11 , P(X=3) = .21

P(X=0 ) = 1 - .07 - .11 - .21 = 0.61

The expected number of appearance defects in a new car =

E(X) =
`\\u` =
`= \sum X P(X) =3* 0.07 + 2*0.11 + 3*0.21 + 0*0.61` = 0.64

`\sigma^(2)` = variation of E(X) =
`\sum (X - \\u) ^(2)P(X)` =
`(3 - 0.64) ^(2)*0.07 + (2 - 0.64) ^(2)*0.11 + (1 - 0.64) ^(2)*0.21 + (0 - 0.64) ^(2)*0.61`

=
`(2.36) ^(2)*0.07 + (1.36) ^(2)*0.11 + (.36) ^(2)*0.21 + ( 0.64) ^(2)*0.61` = 0.8704

standard deviation =
`\sqrt{\sigma^(2)}` =
`\sqrt{0.8704^(2)}` = 0.93(approximately)