Question

HW5 Continuous and Normal Random Variables Part1: Probl Previous Problem Problem List Next Problem (1 point) A new car that is a gas- and electric-powered hybrid has recently hit the market. The distance travelled on 1 gallon of tuel is normally distnibuted with a mean of 55 miles and a standard deviation of 7 miles. Find the probability of the following events A The car travels more than 62 miles per gallon. Probability B. The car travels less than 47 miles per gallion Probability C. The car travels between 50 and 59 miles per gallon Probability =

Answer 1

Solution:

Given: The distance travelled on 1 gallon of fuel is Normally distributed with a mean of 55 miles and a standard deviation of 7 miles.

That is: Mean = $\mu = 55$ and Standard Deviation = $\sigma = 7$

Part A) Find :P( X > 62 ) = ......?

Find z score for x = 62:

z score formula:

$z = \frac{x-\mu}{\sigma}$

$z = \frac{62-55}{7}=\frac{7}{7}=1.00$

Thus we get:

P( X > 62 ) = P( Z > 1.00)

P( X > 62 ) = 1 - P( Z < 1.00)

Look in z table for z = 1.0 and 0.00 and find corresponding Area.

Thus from z table we get: P( Z < 1.00) = 0.8413

Thus

P( X > 62 ) = 1 - P( Z < 1.00)

P( X > 62 ) = 1 - 0.8413

P( X > 62 ) = 0.1587

Part B) Find : P( X < 47) = ........?

Find z score for x = 47

$z = \frac{47-55}{7}=\frac{-8}{7}=-1.14$

Thus we get:

P( X < 47) = P( Z < -1.14)

Look in z table for z = -1.1 and 0.04 and find area.

Thus from z table, we get:

P( Z < -1.14) = 0.1271

Thus

P( X < 47) = P( Z < -1.14)

P( X < 47) = 0.1271

Part C) Find: P( 50 < X < 59) = .........?

Find z scores for x = 50 and x = 59

$z = \frac{50-55}{7}=\frac{-5}{7}=-0.71$

$z = \frac{59-55}{7}=\frac{4}{7}=0.57$

P( 50 < X < 59) = P( -0.71 < Z < 0.57)

P( 50 < X < 59) = P( Z < 0.57) - P( Z < -0.71)

Look in z table , for z = 0.5 and 0.07 as well as for z = -0.7 and 0.01 and find area.

Thus from z table , we get:

P( Z < 0.57) = 0.7157   and P( Z < -0.71) = 0.2389

Thus we get:

P( 50 < X < 59) = P( Z < 0.57) - P( Z < -0.71)

P( 50 < X < 59) = 0.7157 - 0.2389

P( 50 < X < 59) = 0.4768