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5.5 – Z-Scores
Foundations of Math 11
1. Properties of a Standard Normal Distribution:
a. The mean is µ and the standard deviation is σ.
b. The graph is symmetrical about the mean.
c. Almost all the data lie within 3 standard deviations of the mean. (Recall the 6895-98 Rule from last lesson)
d. The curve approaches the x-axis, but does not cross it.
2. Using the Standard Normal Distribution Curve:
a. We use the letter z to represent numbers along the x-axis. These values are called zscores, and they represent the number of standard deviations from the mean.
Ex. A z-score of 2.3 means 2.3 standard deviations to the _________ of the mean.
Ex. A z-score of – 1.9 means 1.9 standard deviations to the _________ of the mean.
b. We can calculate z-scores as long as we have the following information:
𝒛=
𝒙−𝝁
𝝈
μ – mean
σ – standard deviation
x – given data vale
c. Using z-scores and the Standard Normal Distribution allows us to calculate
probabilities for data that does not lay exactly 1, 2, or 3 standard deviations from
the mean. It also allows us to compare data.
Homework:
Do: P264: 5-11, 14, 18, 21
5.5 – Z-Scores
Ex1.
Foundations of Math 11
The math scores in a class are normally distributed. The mean is 80, with a standard
deviation of 5.
a. What percentage of the class scored 80% or lower?
b. What percentage of the class scored above 85%?
c. What percentage of the class scored between 70% and 90%?
d. Callie is in Ramone’s class. She scored an 83% on the test. How did she do compared
to the rest of the class? (hint: use your z-score tables on page 294!!)
Homework:
Do: P264: 5-11, 14, 18, 21
5.5 – Z-Scores
Ex2.
Foundations of Math 11
Recall the example from last class: A soft drink company produces soft drinks in cans,
whose volume reads 355-mL. Every 30 minutes, 5 cans are taken from the production
line. The volume in each of these cans is measured and recorded. Some of the results
are listed below.
Time
Volume (mL)
10:00
355.5
353.8
356.1
351.0
352.8
10:30
353.1
353.7
355.2
354.1
351.9
11:00
354.8
353.2
351.9
355.6
355.1
11:30
354.8
355.0
354.2
356.2
351.8
Now we can use this data to estimate the probability that a randomly selected can will
have a volume of 355.0-mL or less. (Even though this value is not exactly 1, 2, or 3
standard deviations from the mean):
Homework:
Do: P264: 5-11, 14, 18, 21
5.5 – Z-Scores
Ex3.
Foundations of Math 11
Using the formula 𝒛 =
𝒙−𝝁
𝝈
, find the following:
a. x, given that z = 1.5, σ = 2, μ = 15.5
b. σ, given that z = – 2.3, x = 14.8, μ = 18.7
c. μ, given that z = 2.1, x = 75, σ = 4.5
Ex4.
In the general population, the IQ scores of dogs are normally distributed with a mean of
70 and a standard deviation of 8. If a large group of dogs is tested:
a. What proportion of dogs will be expected to have IQs between 62 and 78 inclusive?
Homework:
Do: P264: 5-11, 14, 18, 21
5.5 – Z-Scores
Foundations of Math 11
b. What is the probability that a dog in the group has an IQ of greater than or equal to
82?
c. What IQ is necessary to be within the top 5% of the sample group?
Homework:
Do: P264: 5-11, 14, 18, 21
5.5 – Z-Scores
Foundations of Math 11
Ex5.
The mean lifespan for a non-stick frying pan is two years, with a standard deviation of
0.5 years. Karnie is an avid cook who wants to replace her frying pan after 65% of its
natural life. When should she replace her frying pan?
Ex6.
Every student who applies to a particular university must write an entrance exam. Last
year, Susan scored 753 on her exam. The mean was 500 and the standard deviation was
105. This year, Julian scored 715 on his exam. The mean was 525 and the standard
deviation was 95. Who scored better on the exam?
Homework:
Do: P264: 5-11, 14, 18, 21