The results of a recent survey indicate that the average new

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

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

At what percentile rank is a car that sold for $30,000?

Solution

Z scores are based on the normal distribution or the normal curve. The z score of the arithmetic mean or average on the normal curve is zero (0), a positive z score indicates a value above the average and a negative z score indicates a value below the average. One standard deviation above the mean is given the value of +1.0, two standard deviations above the mean would result in a value of +2.0 and so on. One standard deviation below the mean is given the value of -1.0, two standard deviations below the mean is given the value of -2.0 and so on.

Referring to a normal curve table will give you the percent of the population that would fall above and below a particular z score. Thus, z = 0 would indicate the percent below and above this value would be at the 0.5 percentile since this would be at the midpoint of the population.

In your example, the average new car cost of $23,000 would result in a z score of zero (0). Since the standard deviation is $3,500, a multiple or fraction of a multiple of this value added to or subtracted from the average would result in a positive or negative z score respectively. Therefore, to determine how many standard deviations would be represented by a price of $33,000 simply determine the difference between this value and the average cost then divide the difference by $3,500. Example: 33,000 - 23,000 = 10,000; 10,000 / 3,500 = 2.857.