Question

The results of a survey indicate that the average new car costs $23,000 with a standard deviation of $3,500. The price of cars is normally distributed. Table A.1 may be found beginning on page 410. a. If someone bought a car for $16,000, what percentage of cars cost an equal amount or more than this? Explain how you arrived at your answer, showing your work. You must show your work and explain how you arrived at your answer to earn credit. b. At what percentile rank is a car that sold for $16,000? Explain how you arrived at your answer. c. If there is a z score of 1, is that z score higher or lower than the average? d. What percentage of scores falls between the mean and a z score of 1? e. What percentage of the total distribution of scores fall below a z score of +1.0?

Answer 1

Let X be the new car cost

X~ N( 23000, 3500²)

1. P( X >=16000) = P( >= )

= P( Z >= -2)

= P( Z <2 )

= 0.97725

= 97.7%

2. P( X < 16000) = 1- P( X >=16000) = 1-0.97725 = 0.02275 i.e at 2nd percentile

3. A z score of 1 means higher than the average because a positive z score shows values above the mean.

4. Percentage of scores falls between the mean and a z score of 1= P( 0 <Z<1)

= P( Z <1) - P( Z <0)

=0.84134 - 0.5

=0.34134

5 P( Z <1) = 0.84134 = 84.134%