Introductory Mathematics
& Statistics for Business
Chapter 16
The Normal Distribution
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Learning Objectives
• Identify the properties of the normal distribution and normal
curve
• Identify the characteristics of the standard normal curve
• Understand examples of normally distributed data
• Read z-score tables and find areas under the normal curve
• Find the z-score, given the area under the normal curve
• Compute proportions
• Check whether data follow a normal distribution
• Understand and apply the Central Limit Theorem
• Solve business problems that can be represented by a
normal distribution
• Calculate estimates and their standard errors
• Calculate confidence intervals for the population mean
• Calculate confidence intervals for the population proportion
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16.1 Introduction
• When the frequencies of observations for a large
population result in a frequency polygon that follows the
pattern of a smooth bell-shaped curve that population is
said to have a normal distribution.
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Introduction (cont…)
• The normal curve
– If the frequency polygon for a given set of observations that
have a normal distribution is made into a smooth curve, the
resulting curve is referred to as a normal curve.
– A specific normal curve is characterised by its mean (μ) and
standard deviation (σ).
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Introduction (cont…)
• The main features of a normal distribution are:
–
–
–
–
It is bell-shaped.
It is symmetric about the mean.
It is asymptotic to the horizontal axis.
Approximately 68% of the distribution lies within 1 standard
deviation of the mean;
– about 95% lies within 2 standard deviations of the mean;
– about 99.7% lies within 3 standard deviations of the mean.
– The location and dispersion will depend on the values of μ
and σ (see Figures 16.2 and 16.3).
– The total area under any normal curve is 1.
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16.2 An Example of Data that have an
Approximate Normal Distribution
• As part of a study by the author, random samples of
200 males and 200 females were selected from the
students enrolled in a large introductory statistics
course at Macquarie University, Sydney.
• Each student was asked to record his or her height, to
the nearest centimetre.
• The male heights have a mean of 178.6 cm and a
standard deviation of 7.0 cm, and the female heights
have a mean of 164.5 cm and a standard deviation of
7.7 cm.
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An Example of Data that have an
Approximate Normal Distribution (cont…)
• The histograms displaying the grouped frequency
• distributions for the male and female heights
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16.3 Areas Under The Normal Curve
• The proportions of observations that take on certain
values are represented by areas under the distribution
curve
• The proportion of observations that take on a value
between a and b is the area under the curve between
two vertical lines erected at a and b.
• We could calculate these areas and thus obtain values
for the proportions.
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Areas Under The Normal Curve (cont…)
• Standard Scores (z – scores)
– The z-score of a measurement is defined as the number of
standard deviations the measurement is away from the mean
– If the measurement is above the mean, the corresponding zscore is positive; but if the measurement is below the mean,
the corresponding z-score is negative
Standard score  z 
observed value - mean
standard deviation
– Thus, if a distribution has a mean of μ and a standard deviation
of σ, the corresponding z-score of an observation x is:
x 
z

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Areas Under The Normal Curve (cont…)
• Standard Scores (z – scores)
– The mean itself has a z-score of zero and that a value exactly
1 standard deviation from the mean has z = + 1 or – 1.
– The mean itself has a z-score of zero and that a value exactly
1 standard deviation from the mean has z = + 1 or – 1. The
larger a positive z-value, the further the given x is above the
mean, while large negative z-values correspond to extreme
values below the mean
– It is found in practice with most data sets that the vast
majority of observations have a z-score in the range roughly
–2 to + 2.
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16.4 Using the Table of the Standard
Normal Curve
• Table 6 gives the area under the standard normal curve
between the mean (zero) and any positive number z.
– Example
Using Table 6, determine the area under the standard
normal curve:
(a) between z = 0 and z = 1.50
(b) between z = –2.10 and z = 0
(c) between z = 0.60 and z = 1.80
(d) between z = –0.30 and z = 2.25
(e) to the right of z = 1.95
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Using the Table of the Standard Normal
Curve (cont…)
• Solution
(a) From Table 6, the area between z = 0 and z = 1.50 is
0.4332
(b) In order to find the area to the left of z = 0, the symmetry of
the normal curve can be used. In this case, the area
between z = –2.10 and z = 0 is the same as the area
between z = 0 and z = + 2.10. From Table 6, this area is
0.4821.
(c) The area between z = 0.60 and z = 1.80 may be found by
subtracting the area between z = 0 and z = 0.60 from the
area between z = 0 and z = 1.80. From Table 6:
The area between z = 0 and z = 1.80 is 0.4641.
The area between z = 0 and z = 0.60 is 0.2257.
Therefore, the required area is 0.4641 – 0.2257 = 0.2384.
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Using the Table of the Standard Normal
Curve (cont…)
• Solution (cont…)
(d) The area between z = –0.30 and z = 2.25 may be found by
adding the area between z = – 0.30 and z = 0 to the area
between z = 0 and z = 2.25. The area between z = – 0.30
and z = 0 equals the area between z = 0 and z = + 0.30 (by
symmetry). From Table 6:
The area between z = 0 and z = 0.30 is 0.1179.
The area between z = 0 and z = 2.25 is 0.4878.
Therefore, the required area is 0.1179 + 0.4878 = 0.6057.
(e) From the facts that the normal curve is symmetrical about
the mean and the total area under the curve is 1, it follows
that the area to the right of z = 0 is 0.5 and that the area to
the left of z = 0 is also 0.5. Thus, to find the area to the right
of z = 1.95, we subtract the area between z = 0 and z = 1.95
from 0.5. From Table 6:
The area between z = 0 and z = 1.95 is 0.4744.
Therefore, the required area is 0.5000 – 0.4744 = 0.0256.
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Using the Table of the Standard Normal
Curve (cont…)
• Conversion to raw scores
– In order to determine appropriate areas under any
normal curve, the z-score (or standard score) may be
calculated.
– The z-scores express the given problem in ‘standard
form’ so that the standard normal curve can be used.
 To convert a raw score of x (from a distribution with
mean μ and standard deviation σ) to a z-score,
subtract the mean from x and divide by the standard
deviation.
 To convert a z-score to a raw score x, multiply the zscore by the standard deviation and add this product
to the mean.
– In equation form:
x    Ζ
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16.5 Computation of Proportions
•
A proportion within a given interval
– If a set of data has a normal distribution, the proportion of
observations that lie in a particular interval can be found
using the following procedure:
1. Determine the z-score for each end point of the interval.
2. Find the area (from Table 6) for each z-value. (If the zvalue is negative, ignore the sign.)
3. lf the end points of the interval lie on opposite sides of
the mean, add the two areas found in Step 2. If the two
end points lie on the same side of the mean, subtract
the smaller area from the larger one.
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Computation of Proportions (cont…)
• A proportion greater or less than a given value
– Table 6 can also be used to determine what proportion of a
normal population is greater than a certain value or less
than a certain value
 To determine the proportion of a normal distribution
greater than a value of x, calculate the z-score
corresponding to x and find the area to the right of this
score.
 To determine the proportion of a normal distribution less
than a value of x, calculate the z-score corresponding to
x and find the area to the left of this score.
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Computation of Proportions (cont…)
• A value corresponding to a given proportion
– For some problems it is necessary to find the z-score for a given area
rather than the other way around.
 Example : Suppose that a particular MBA entrance exam is
designed so that the marks will be normally distributed with a
mean of 300 and a standard deviation of 60. The policy is that
the top 35% gain admission to the program. What is the lowest
mark that would gain admission?
 Solution
 From Table 6, z = 0.38 gives an area of 0.1480 while z = 0.39
gives an area of 0.1517. Thus, the z-score required is
somewhere between 0.38 and 0.39, say 0.385. this z-score
can be converted to a raw score:
x    z
 300  60  0.385
 323.1
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16.6 Checking Whether Data are
Normally Distributed
• Even if the distribution is believed to be normal, it is rare for
the mean μ and standard deviation s to be known precisely
much of statistical testing depends on whether or not the
data are normally distributed
• A wrong assumption can lead to an incorrect statistical test
being used on the data, which in turn can lead to a doubtful
conclusion
• The construction of a histogram and corresponding
frequency polygon of the data can give a rough idea of its
shape
• If the frequency polygon for a large sample clearly does not
have a normal shape, it is unlikely that the population does
either.
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16.7 The Central Limit Theorem
• Definition
– If random samples of size n are selected from a population
with mean μ and standard deviation σ, the means of the
samples are approximately normally distributed with mean μ

and standard deviation
even if the population itself is not
n
normally distributed, provided that n is not too small. The
approximation becomes more and more accurate as the
sample size n is increased.
– The Central Limit Theorem also says that any unusual
shapes of populations are suppressed. That is, even if
individual values come from a very non-normal population
(such as one that is skewed or bimodal), the sample means
tend to have a normal distribution.
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The Central Limit Theorem (cont…)
• Conversion of a sample mean to a z-score
– The formula for converting a mean of to a z-score is given by:
x 
z

n
– Where
  thepopulationmean
  the population stantard deviation
n  the size of the sample
x  the mean of the sample
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16.8 Confidence Intervals for an
Unknown Population Mean
• Point estimates
– A single estimate of an unknown population mean can be
obtained from a random sample
– Different random samples give different values of the mean
– A single estimate is referred to as a point estimate
– Accuracy depends on:
 variability of data in the population
 size of the random sample
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Confidence Intervals for an Unknown
Population Mean (cont…)
• The Standard error of the mean
Standard error of the mean provides the precise measure of
accuracy of a point estimate of the mean

Standard error of the mean 
n
where:
σ = population standard deviation
n = size of random sample
the value of σ can be replaced by the sample standard
deviation, s.
Standard error of the mean 
s
n
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Confidence Intervals for an Unknown
Population Mean (cont…)
• The meaning of confidence intervals for μ
– Instead of providing a single point estimate for μ, we
consider a range of values or an interval within which the
value of μ may lie.
– Be able to provide a probability or level of confidence that
this interval does indeed contain the true value of μ.
– Common probabilities to use are 0.90, 0.95 and 0.99
– Confidence intervals for μ are based on the values from
random samples
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Confidence Intervals for an Unknown
Population Mean (cont…)
• Construction of confidence intervals for μ
– Suppose that we have the following information:
 The population has a normal distribution.
 The population standard deviation (σ) is known.
 The sample size (n) can be of any size.
– Then a 95% confidence interval for μ is:

 

x

1
.
96
,
x

1
.
96


n
n

x = the mean of the sample
σ = the population standard deviation
n = the size of the sample
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16.9 Confidence Intervals for an
Unknown Population Proportion
• Point estimates
– To give a single estimate of an unknown population
proportion (π), use the value of the proportion (p) obtained
from a random sample taken from that population.
– A single estimate of π is referred to as a point estimate.
– Point estimates are particularly important in survey work, for
example, to get an idea of how the population sampled feels
about a certain issue.
– How accurate a point estimate is depends on two factors:
 how variable the data are in the population
 the size of the random sample used to make the
estimate
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Confidence Intervals for an Unknown
Population Proportion (cont…)
• The standard error of the proportion
– The precise measure of accuracy of a point estimate (p) of a
population proportion is provided by the standard error of the
proportion
– The formal definition is
standard error of the proportion 
1  
n
– Where
π = the population proportion
n = the size of the random sample
the value of π can be replaced by the sample standard proportion, p
standard error of the proportion 
p1  p
n
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Confidence Intervals for an Unknown
Population Proportion (cont…)
• The meaning of confidence intervals for π
– Instead of providing a single point estimate for π, we
consider a range of values or an interval within which the
value of π may lie.
– Be able to provide a probability or level of confidence that
this interval does indeed contain the true value of π.
– Common probabilities to use are 0.90, 0.95 and 0.99,
although any probability could be used.
– Confidence intervals for π are based on the values from
random samples
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Confidence Intervals for an Unknown
Population Proportion (cont…)
• Construction of confidence intervals for π
– We are able to construct an interval estimate for the
true value of an unknown population proportion π.
– The confidence interval for π is of the form






p
1

p
p
1

p
z
z
p 

,p 


n
n


– where the value of z is chosen as follows:
 z = 1.645 for a 90% confidence interval
 z = 1.96 for a 95% confidence interval
 z = 2.58 for a 99% confidence interval
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Conclusion
• Look at identifying the properties of the normal distribution and
normal curve
• Also identified the characteristics of the standard normal curve
• Understood examples of normally distributed data
• Read z-score tables and find areas under the normal curve
• Found the z-score, given the area under the normal curve
• Computed proportions
• Checked whether data follow a normal distribution
• Understood and applied the Central Limit Theorem
• Solved business problems that can be represented by a normal
distribution
• Calculated estimates and their standard errors
• Calculated confidence intervals for the population mean
• Calculated confidence intervals for the population proportion
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