Finding the Z-Score from a Given Area

Normal Distributions: Finding Values

Objectives:

• How to find a z-score given the area under the normal curve

• How to transform a z-score to an x-value

• How to find a specific data value of a normal distribution given the probability

Find the z-score that corresponds to a cumulative area of 0.3632.

Find the z-score that has 10.75% of the distribution’s area to its right.

Multiple Choice

Find the z-score that has 90.66% of the distribution’s area from the left.

• 1.3

• 1.31

• 1.32

• 1.34

Multiple Choice

Find the z-score that has 14.46% of the distribution’s area to its right.

• -1.06

• 1.26

• 1.06

Find the z-score that corresponds to P₁₈.

Find the z-score that corresponds to P₃₂.

Multiple Choice

Find the z-score that corresponds to P₄₇.

• 0.08

• 0.07

• -0.08

• -0.07

Transforming a z-Score to an x-Score

To transform a standard z-score to a data value x in a given population, use the formula

x = μ + zσ

Example:

A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z-scores of 1.96, –0.44, and 0. 

Example 2:

Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency?

Multiple Choice

The annual per capita consumption of fresh apples (in pounds) in the US can be approximated by a normal distribution, with a mean of 9.5 pounds and a standard deviation of 2.8 pounds. What is the smallest annual per capita consumption of apples that can be in the top 25% of consumption?

• 11.39 pounds

• 12.27 pounds

• 6.59 pounds

Multiple Choice

The annual per capita consumption of fresh apples (in pounds) in the US can be approximated by a normal distribution, with a mean of 9.5 pounds and a standard deviation of 2.8 pounds. What is the largest annual per capita consumption of apples that can be in the bottom 15% of consumption?

• 11.39 pounds

• 12.27 pounds

• 6.59 pounds