Question

I'm so confused on how to do the z-score...I was absent for this part of class and I need help figuring out how to even answer the following questions.

EPA fuel economy for automobiles was found to have a mean of 30 mpg and a standard deviation of 7 mpg. Assume that a Normal model can be applied. Use this information to answer the next four questions.

18. What is the standardized value for a car that got 20 mpg?

a. -1.428

b. 1.428

c. -10

d. 20

19. My car has a gas mileage that is 2.5 standard deviations below average. What is my car’s actual gas mileage?

a. 42.50 mpg

b. 27.50 mpg

c. 17.5 mpg

d. 12.5 mpg 5

20. In what interval would you expect the central 99.7% of autos to be found?

a. 16 mpg to 30 mpg

b. 23 mpg to 37 mpg

c. 16 mpg to 44 mpg

d. 9 mpg to 51 mpg

21. Which one of the following statements is false?

a. The median for our car example is 30 mpg.

b. The Standard Normal Curve has a mean of 1 and a SD of 0.

c. A z-score tells us how many SDs above or below the mean a value is.

d. When we convert a value to a z-score, the distribution is not automatically unimodal and symmetric.

Answer 1

here we use standard normal variate z=(x-mean)/sd=(x-30)/7

(18) right choice is a. -1.428

for x=20, z=(20-30)/7=-1.4286

(19) right choice is d. 12.5 mpg

2.5 standard deviations below average means z=-2.5

so (x-mean)/sd=-2.5

or, x=30-2.5*7=12.5

(20) right choice is d. 9 mpg to 51 mpg

here we want to find x1 and x2 such that P(x1<X<x2)=0.997( since X is normal so it is symmetric )

first we find z1 and z2 such that  P(z1<X<z2)=0.997

or, P(Z<z2)-P(Z<z1)=0.997

or, 1-P(Z<z1)-P(Z<z1)=0.997 ( since Z is symmetric)

or, 2P(Z<z1)=1-0.997=0.003

or, P(Z<z1)=0.0015

and z1=-2.97 and z2=2.97

so x1=30-2.97*7=9.21 (9 whole number approximation)

and x2=30+2.97*7=50.79 (51 whole number approximation)

(21) right choice is b. The Standard Normal Curve has a mean of 1 and a SD of 0.