Section 5.2 ~ Properties of the Normal Distribution
Objective: In this section you will learn how to interpret the normal distribution in terms of the 68-95-99.7 rule, standard scores, and percentiles.
Essential questions:
1. What percentage of values fall within 1 standard deviation? 2 deviations? 3
deviations?
2. When are values considered unusual?
Recall the following notations:
The population mean is represented using the Greek letter mu ______
The population standard deviation is represented using the Greek letter sigma ______
The following diagram is an example of a distribution that represents the replacement time (in years) for a TV.
Knowing the mean and standard deviation of a distribution allows us to say a lot about where the data values lie
Ex. ~ Suppose that the mean TV replacement time is 8.2 years with a standar
deviation of 1.1 years
About two-thirds of all values fall within one standard deviation, so the replacement time for about two-thirds of the TV’s is between 7.1 years and 9.3 years (8.2 - 1.1 = 7.1 and 8.2 + 1.1 = 9.3)
About 95% of the data fall within two standard deviations of the mean, so the replacement time for 95% of the TV’s in this sample is between 6.0 years and 10.4 years (8.2 – 2.2 = 6.0 and 8.2 + 2.2 10.4)
68-95-99.7 rule:
Example 1:
If there are 600 total values. Approximately, how many values occur within 1 standard of deviation from the mean? 2 standards of deviation? 3 standards of deviation?
Example 2:
Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal U.S. quarters have a normal distribution with a mean of 5.67 grams and a standard deviation of 0.0700 gram. If a vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what percentage of legal quarters will be rejected by the machine?
Unusual values:
Example 3:
Suppose your friend is pregnant and is trying to determine if she should schedule an important business meeting two weeks before her due date. Data suggests that the number of days between the birth date and due date is normally distributed with a mean of μ = 0 days and a standard deviation of σ = 15 days. How would you help your friend make the decision? Would a birth two weeks before the due date be considered “unusual”?
Example 4:
Suppose you measure your heart rate at noon every day for a year and record the data. You discover that the data has a normal distribution with a mean of 66 and a standard of deviation of 4. On how many days was your heart rate below 58 beats per minute?
Example 5:
Suppose that your fourth-grade daughter is told that her height is 1 standard deviation above the mean for her age and sex. What is her percentile for height? Assume that heights of fourth-grade girls are normally distributed.
Standard scores:
Ex.’s ~
o A value that is 2 standard deviations above the mean has a standard score of:
o A value that is 2 standard deviations below the mean has a standard score of :
o The standard score of the mean will be:
z-score formula:
Example 6:
The Stanford-Binet IQ test is scaled so that scores have a mean of 100 and a standard deviation of 16. Find the standard scores for IQs of 85, 100, and 125.
The standard score table in appendix A on p.446 in your book gives the percentage of values that fall below a certain value with a known z-score.
Ex. ~ A value that has a standard score of .15 has a
Example 7:
Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.
a. What is the percentile for a 20-year old man with a cholesterol level of 190?
b. What cholesterol level corresponds to the 90th percentile, the level at which
treatment may be necessary?
Example 8:
The heights of American women aged 18 to 24 are normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches. In order to serve in the U.S. Army, women must be between 58 inches and 80 inches tall. What percentage of women are ineligible to serve based on their height?