Question

II. The results of a recent survey indicate that the average new car costs $23,000, with a standard deviation of $3,500. The price of cars is normally distributed. a. What is a Z score for a car with a price of $33,000? b. What is a Z score for a car with a price of $30,000?

Answer 1

a) Z = 2.86

b) Z = 2

Explanation:

Z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 23000, \sigma = 3500`

a. What is a Z score for a car with a price of $33,000?

This is Z when X = 33000. So

`Z = (X - \mu)/(\sigma)`

`Z = (33000 - 23000)/(3500)`

`Z = 2.86`

b. What is a Z score for a car with a price of $30,000?

This is Z when X = 30000. So

`Z = (X - \mu)/(\sigma)`

`Z = (30000 - 23000)/(3500)`

`Z = 2`