Question

Exercise 2

When calculating probabilities associated with normal distributions, z-scores are used. A z-score for a particular value measures the number of standard deviations away from the mean.
A positive z-score corresponds to a value that is above the mean, and a negative z-score corresponds to a value that is below the mean. The letter z is used to represent
a variable that has a standard normal distribution where the mean is 0 and standard deviation is 1. This distribution was used to define a z-score. A z =z-score is calculated by
z=Value - mean/Standard deviation .
a. The prices of the printers in a store have a mean of $240 and a standard deviation of $50. The printer that you eventually choose costs $340.
i. What is the zz-score for the price of your printer?
ii. How many standard deviations above the mean was the price of your printer?
b. Ashish’s height is 63 inches. The mean height for boys at his school is 68.1 inches, and the standard deviation of the boys’ heights is 2.8 inches.
i. What is the z-score for Ashish’s height? (Round your answer to the nearest hundredth.)
ii. What is the meaning of this value?
c. Explain how a zz-score is useful in describing data.

Answer 1

a. z=2 , the printer chosen is 2 standard deviations above the mean

b. z=-1.82, Ashish’s height is -1.82 standard deviations below the mean

c. z-score measures the number of standard deviations away from the mean, by knowing the z-score we can have an idea where the value of a variable falls in the distribution of all values of that variable.

We can estimate its probability of observing (p-value) and within what percent of all values it falls.

Explanation:

z-score is calculated by

z= (Value - mean) / Standard deviation

a. Thus z-score of the printer chosen is:

z= ($340 - $240) / $50 = 2

• the printer chosen is 2 standard deviations above the mean

b. z-score for Ashish’s height is:

z= (63 - 68.1) / 2.8 = -1.82

Ashish’s height is -1.82 standard deviations below the mean