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Chapter 5: Statistics
Math 2201
Z-Scores
Limitations of the Normal Curve/Normal Distribution
▪ So far we have only found information regarding data that falls exactly at one on the
standard deviation lines on the normal curve
▪ If a data value is not at one of these locations we cannot determine any information
about as it relates to its location in relation to the mean or the percentage of data
above or below it.
▪ In the normal distribution of the weights of adult male husky dogs, we can
determine information about dogs with weights of 45.3, 47.7, 50.1, 52.5, 54.9, 57.3, or
59.7 kg.
▪ If we wanted to know what percent of dogs have a weight above or below 50.5 kg, or
between 45.5 to 55.5 kg, we would be unable to do so.
▪ To overcome this problem, z-scores are used.
Z-Scores
▪A z-score is a value that tell the number of standard deviations that a data value (x) is
above or below the mean
▪ A z-score is used in the standard normal distribution. This is a normal distribution
where the mean is zero and the standard deviation is 1.
▪ Z-scores are useful for several reasons:
▪ Data that have different means and standard deviations can be compared by
converting them to the standard distribution. The means become 0 and the
standard deviations become 1. Comparisons can be more easily made.
▪ If a z-score is positive then the data value is above the mean
▪ If a z-score is negative then the data value is below the mean
▪ If a z-score is 1 then the data value is 1 standard deviation above the mean
▪ If a z-score is 2 then the data value is 2 standard deviations above the mean
▪ If a z-score is 3 then the data value is 3 standard deviations above the mean
▪ If a z-score is -1 then the data value is 1 standard deviation below the mean
▪ If a z-score is -2 then the data value is 2 standard deviations below the mean
▪ If a z-score is -3 then the data value is 3 standard deviations below the mean
▪ A data value (X) converts to a z-score based on the mean and standard deviation for the
set of data that it is in.
▪ The formula for conversion is: z = x – 

▪ The answer for z is the number of standard deviations that x is from the mean
▪ To determine the percent of data above or below this data value we use the “z-score
tables” found in the textbook on pages 580-581.
▪ The value found in the table for a z-score is the decimal equivalent of the percent of data
to the left of or below the z-scores (and therefore to the left of, or below, the data value)
Page 1 of 3
Chapter 5: Statistics
Math 2201
Z-Scores
Fill in the blanks in each of the following statements.
1. For all z-scores the mean is __________ and the standard deviation is __________
(the answers are 2 numbers)
2. A student’s mark on a test is equal to a z-score of -1.75. This means that the student’s
marks is __________ standard deviations __________ the mean
(the first answer is a number; the second answer is “above” or “below”)
Calculating Z-Scores and Percentages given X,  and  [ z = X –  ]

Use the z-score tables on pages 580-581 to answer each of the following
1. Suppose the average height of a high school student is 60 inches with a standard
deviation of 1.5 inches. What is the z-score for a freshman who has a height of :
a)
b)
c)
d)
58 inches
60.15 inches
56.25 inches
64 inches
2. Based on the information in #1 determine the percent of a population that will have
heights in the following ranges:
a)
b)
c)
less than 58 inches
more than 64 inches
between 56.25 inches and 60.15 inches
3. Based on the results in #2, determine the number of students in a population of 10000
that will have heights in the ranges:
a)
b)
c)
less than 58 inches
more than 64 inches
between 56.25 inches and 60.15 inches
4. Suppose the average teenage romantic relationship is normally distributed with a mean
number of 100 days with a standard deviation of 30 days.
a)
b)
c)
d)
What is the likelihood that a randomly selected couple’s relationship has lasted
for 90 days or fewer?
What is the probability that a randomly selected teenage bliss goes on less than
14 days?
How likely is it to randomly select a couple to find their admiration (relationship)
has lasted 180 days or more?
What is the likelihood that a randomly selected couple’s relationship survived
between 90 and 150 days?
Calculating X given Z,  and 
1.
Given a normally distributed data set whose mean is 50 and whose standard deviation
is 10, what value of x would each of the following z-scores have?
a) z = 1.00
d) z = -0.50
2.
b) z = -1.00
e) z = -1.81
c) z = 2.5
f) z = -2.24
Determine the corresponding z-score for each of the given percentages or
probabilities:
a) 5.94%
b) 29.12%
c) 12%
d) 0.9693
e) 0.6640
f) 0.5340
Page 2 of 3
Chapter 5: Statistics
Math 2201
Z-Scores
3.
Each 450 g box of cereal is routinely returned if its mass has a z-score of -2.7 or less.
Research has shown that the standard deviation of masses is 8 g and is normally
distributed. What is the minimum sized cereal box that is not returned? What percent
of boxes is returned?
4.
Jacob wrote 4 midterm exams. His marks, the class average and standard deviation
are shown in the table.
Calculate his z-score for each course and rank his courses in order of performance
from best to worst.
5.
The average playing time of a defense player in a certain hockey league is 17.2
minutes per game with a standard deviation of 2.2 minutes. In a league that has 300
defense players, how many will play between 15.5 and 20 minutes?
6.
It is known that the mean value of repairs, due to an accident, for both a Corvette and
a Civic is $3500. However, the standard deviation for the Corvette is $1200, while the
standard deviation for the Civic is only $800. If the cost of repairs is normally
distributed, determine the probability that the repairs costs will be over $5000 for
both cars.
7.
Cars are undercoated as a protection against rust. A car dealer determines the mean
life of protection is 65 months and the standard deviation is 4.5 months. If a company
undercoats 500 cars, how many will be rusty before 5 years are up?
8.
The number of red blood cells is normally distributed, with a mean of 4.8 and a
standard deviation of 0.3. To be in the top 5% (i.e. higher than 95% of people), what
count would someone need to have? (Hint: 95% is the value from the z-table. First
determine the corresponding z-score)
9.
The ABC Company produces bungee cords. When the manufacturing process is
running well, the lengths of the bungee cords produced is normally distributed, with a
mean of 45.2cm and a standard deviation of 1.3cm. Bungee cords that are shorter than
42.0cm or longer than 48.0cm are rejected by the quality control workers. a) If 20000
bungee cords are manufactured each day, how many bungee cords would you expect
would be ejected?
10. A bakery sells an average of 1200 donuts per day with a standard deviation of 200
donuts. If the data is normally distributed, determine the probability, to the nearest
hundredth, that the bakery would sell: a) more than 1300 donuts b) between 1300 &
1500 donuts c) less than 900
Questions from Textbook
Pages 292 – 293 #1 - 16
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