Question

The results of a recent survey indicate that the average new car costs $23,000 with a standard deviation on $3,500. The prices of the car are normally distributed. a. If someone buys a car for $32,000, what proportion of cars cost an equal amount or more than this? b. If someone buys a car for $16,000, what proportion of cars cost an equal amount or more than this?

Answer 1

Answer:

(a) 0.0051

(b) 0.9773

Step-by-step explanation:

Let X = prices of the car.

It is provided that $X\sim N(23,000, 3,500^{2})$.

(a)

Compute the value of P (X ≥ 32000) as follows:

$P(X\geq 32000)=P(\frac{X-\mu}{\sigma}>\frac{32000-23000}{3500})$

                       $=P(Z>2.57)\\=1-P(Z<2.57)\\=1-0.99492\\=0.00508\\\approx 0.0051$

Thus, the proportion of cars cost an equal amount or more than $32,000 is 0.0051.

(b)

Compute the value of P (X ≥ 16000) as follows:

$P(X\geq 16000)=P(\frac{X-\mu}{\sigma}>\frac{16000-23000}{3500})$

                       $=P(Z>-2)\\=P(Z<2)\\=0.97725\\\approx 0.9773$

Thus, the proportion of cars cost an equal amount or more than $16,000 is 0.9773.