Transcript Z-scores

Normal Distribution
MATH 102
Contemporary Math
S. Rook
Overview
• Section 15.4 in the textbook:
– Normal distribution
– Z-scores
– Converting raw scores to z-scores
– Applications
Normal Distribution
Normal Distribution
• When a sufficient number of data are collected, the
resulting histogram becomes nearly symmetric
– i.e. split into two equal halves
• By connecting the tops of the bars of the histogram,
of the data we obtain a bellshaped curve more commonly
known as the Normal Curve
• If a set of data can be modeled
by a normal curve, we can
calculate the proportion of the
data in ANY interval
Properties of the Normal
Distribution
• Given that a set of data follows a normal distribution, the
following properties apply to the resulting normal curve:
–
–
–
–
–
It is bell-shaped
Its highest point is the mean
It is symmetric with respect to the mean
The total area under it is 1
Approximately:
• 68% of the data lies within 1
standard deviation of the mean
• 95% of the data lies within 2
standard deviations of the mean
• 99.7% of the data lies within 3
standard deviations of the mean
Normal Distribution (Example)
Ex 1: Assume we have a normal distribution
with a mean of 10 and a standard deviation of
2. Use the 68-95-99.7 rule to find the
percentage of values in the desired interval:
a) Between 10 and 12
b) Above 14
c) Below 10
Normal Distribution (Example)
Ex 2: Assume we have a normal distribution
with a mean of 12 and a standard deviation of
3. Use the 68-95-99.7 rule to find the
percentage of values in the desired interval:
a) Above 6
b) Below 9
c) Between 15 and 18
Z-scores
Z-scores
• Examples 1 and 2 allowed us to determine the
percentage of data values that lay within 1, 2, or 3
standard deviations of the mean
• Z-scores allow us to determine the percentage of
data that lie within ANY number of standard
deviations (sds) of the mean of a standard normal
distribution
– e.g. z = 2.3 refers to 2.3 sds to the RIGHT of the mean
and z = -1.7 refers to 1.7 sds to the LEFT of the mean
• The standard normal distribution has a mean of 0
and a standard deviation of 1
Calculating Percentage/Proportion
of Area
• To calculate the percentage or proportion of
area under a standard normal curve, we use
the standard normal table
– A z-score such as z = -1.92 is looked up by finding
the first two digits (-1.9) in the rows and then the
value of the hundredths (0.02) in the columns
• e.g. What is the value associated by looking up
z = -1.92?
– The area obtained represents the proportion of
data values that lie to the LEFT of (below) the
given z-score
Calculating Percentage/Proportion
of Area (Continued)
• To find the percentage or proportion of area:
– Below a z-score – simply look up the z-score in the
table
– Above a z-score – look up the z-score and
subtract it from 1
• Recall that the area underneath the entire normal
curve is 1
– Between two z-scores – look up both z-scores and
subtract the smaller from the larger
• It may help to draw the normal curve
Z-scores (Example)
Ex 3: Assume a data set follows the normal
distribution and use the standard normal table
to find the specified percentage of data values
under the standard normal curve:
a) Below z = -1.35
b) Above z = 2.00
c) Between z = -0.98
and z = 1.72
d) Above z = -2.73
e) Below z = 0.08
f) Between z = 1.01
and z = 1.99
Converting Raw Scores to
Z-scores
Converting Raw Scores to Z-scores
• Finding the proportion of data values using z-scores
works ONLY with data adhering to a standard
normal distribution
• However, given that the data comes from a normal
distribution and the distribution’s mean & sd, we
can convert a value into a z-score using the
formula z  x   where:

μ is the mean of the data
σ is the sd of the data
x is any value
Converting Raw Scores to Z-scores
(Example)
Ex 4: Given a mean, a standard deviation, and a
raw score from a data set that comes from a
normal distribution, find the corresponding zscore:
a) μ = 80, σ = 5, x = 87
b) μ = 21, σ = 4, x = 14
c) μ = 38, σ = 10.3, x = 48
Applications
Applications (Example)
Ex 5: A machine fills bags of candy, but due to slight
irregularities in the operation of the machine, not
every bag gets exactly the same number of pieces.
Assume that the number of pieces per bag has a
mean of 200 and a standard deviation of 2.
Assuming a normal distribution, what proportion of
the bags will have:
a) Less than 197 pieces of candy?
b) More than 204 pieces of candy?
c) Between 199 and 201 pieces of candy?
Applications (Example)
Ex 6: A supervisor observes and records the amount
of time his employees take for lunch. Suppose the
length of the average lunch is 42.5 minutes with a
standard deviation of 5.8 minutes. Assuming a
normal distribution, what proportion of the
supervisor’s employees have a lunch break:
a) Less than 50 minutes?
b) More than 40 minutes?
c) Between 43 to 45 minutes?
Summary
• After studying these slides, you should know how to do
the following:
– Use the 68-95-99.7 rule to find the percentage of values in a
desired interval of a normal distribution
– Compute the proportion of the standard normal curve using zscores
– Compute a z-score given the mean, standard deviation, and a
raw score from a normal distribution
– Solve application problems involving normal distributions
• Additional Practice:
– See problems in Section 15.4
• Next Lesson:
– Study for the Final Exam!