Standard Units and the Areas Under the Standard Normal Distribution Section 8.2

Objectives Given  and , convert raw data to z scores. Given  and , convert z scores to raw data. Graph the standard normal distribution, and find areas under the standard normal curve.

We need to take raw data and make it to something we can compare with other similar data. The z-score is a measure of the distance in standard deviations of a sample from the mean. We can use a simple formula to compute the number (z) of standard deviations between a measurement (x) and the mean () of a normal distribution with standard deviation ().

How to find the z-score from the raw score

When x is bigger than the mean the z-score is positive. When x is smaller than the mean the z-score is negative. x Values and Corresponding z Values

Example – Standard score A pizza parlor franchise specifies that the average (mean) amount of cheese on a large pizza should be 8 ounces and the standard deviation only 0.5 ounce. An inspector picks out a large pizza at random in one of the pizza parlors and finds that it is made with 6.9 ounces of cheese. The amount of cheese is normally distributed. If the amount of cheese is below the mean by more than 3 standard deviations, the parlor will be in danger of losing its franchise.

Example – Standard score

When given the z-score, it is possible to find the raw score (x) corresponding to z

The Standard Normal Distribution ( = 0,  = 1)

Example – Standard normal distribution table Use Table 5 of Appendix II to find the described areas under the standard normal curve. (a) Find the area under the standard normal curve to the left of z = –1.00. Area to the Left of z = –1.00

Example – Solution Excerpt from Table 5 of Appendix II Showing Negative z Values

Example – Standard normal distribution table (b) Find the area to the left of z = 1.18

Using a Standard Normal Distribution Table 1) Areas to left of z-value: Use table as is 2) Areas to right of z-value: 1 – table value 3) Areas between z-values:

Example – Using table to find areas Find the area between z = 1.00 and z = 2.70. Area between two z values, subtract table entries. (Area between 1.00 and 2.70) = (Area left of 2.70) – (Area left of 1.00) = 0.9965 – 08413 = 0.1552

Example– Using table to find areas Find the area to the right of z = 0.94. (Area to right of 0.94) = (Area under entire curve) – (Area to left of 0.94) = 1.000 – 0.8264 = 0.1736

Example – Standard score The average score on the test was 80 percent with a standard deviation of 8 percent. Suppose the scores are normally distributed. Find the percentage of student who scored between 70 and 100 percent.

8.2 Standard Units and Areas Under the Standard Normal Distribution Summarize Notes Read section 8.2 Homework Worksheet