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ELEMENTARY STATISTICS 3E
William Navidi and Barry Monk
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
The Standard Normal Curve
Section 7.1
©McGraw-Hill Education.
Objectives
1.
2.
3.
4.
Use a probability density curve to describe a population
Use a normal curve to describe a normal population
Find areas under the standard normal curve
Find 𝑧-scores corresponding to areas under the normal curve
©McGraw-Hill Education.
Objective 1
Use a probability density curve to describe a
population
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Probability Density Curves
The following figure presents a relative
frequency histogram for the particulate
emissions of a sample of 65 vehicles.
If we had information on the entire population,
containing millions of vehicles, we could make
the rectangles extremely narrow.
The histogram would then look smooth and
could be approximated by a curve. The curve
used to describe the distribution of this
variable is called the probability density curve
of the random variable. The probability density
curve tells what proportion of the population
falls within a given interval.
©McGraw-Hill Education.
Area and Probability Density Curves
The area under a probability density curve between any two values 𝑎
and 𝑏 has two interpretations:
• It represents the proportion of the population whose values are
between 𝑎 and 𝑏.
• It represents the probability that a randomly selected value from
the population will be between 𝑎 and 𝑏.
©McGraw-Hill Education.
Properties of Probability Density Curves
The region above a single point has
no width, thus no area. Therefore, if
𝑋 is a continuous random variable,
𝑃(𝑋 = 𝑎) = 0 for any number 𝑎. This
means that 𝑷(𝒂 < 𝑿 < 𝒃) =
𝑷(𝒂 ≤ 𝑿 ≤ 𝒃) for any numbers 𝑎
and 𝑏.
For any probability density curve, the
area under the entire curve is 1,
because this area represents the
entire population.
©McGraw-Hill Education.
Objective 2
Use a normal curve to describe a normal population
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Normal Curves
Probability density curves comes in many varieties, depending on the
characteristics of the populations they represent. Many important
statistical procedures can be carried out using only one type of
probability density curve, called a normal curve.
A population that is represented by a normal curve is said to be
normally distributed, or to have a normal distribution.
©McGraw-Hill Education.
Properties of Normal Curves
The population mean determines the location of the peak. The
population standard deviation measures the spread of the
population. Therefore, the normal curve is wide and flat when the
population standard deviation is large, and tall and narrow when the
population standard deviation is small. The mean and median of a
normal distribution are both equal to the mode.
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Empirical Rule
The normal distribution follows the Empirical Rule.
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Objective 3
Find areas under the standard normal curve
*(TI-84 PLUS)
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Standard Normal Curve
A normal distribution can have any mean and any positive standard
deviation. However, the normal distribution with a mean of 0 and
standard deviation of 1 is known as the standard normal
distribution.
©McGraw-Hill Education.
𝑍-Scores
When finding an area under the standard normal curve, we use the
letter 𝑧 to indicate a value on the horizontal axis beneath the curve.
We refer to such a value as a 𝒛-score.
Since the mean of the standard normal distribution is 0:
• The mean has a 𝒛-score of 0.
• Points on the horizontal axis to the left of the mean have negative 𝒛scores.
• Points to the right of the mean have positive 𝒛-scores.
©McGraw-Hill Education.
Finding Areas with the TI-84 PLUS
On the TI-84 PLUS calculator, the normalcdf command is used to find
areas under a normal curve. Four numbers must be used as the
input. The first entry is the lower bound of the area. The second
entry is the upper bound of the area. The last two entries are the
mean and standard deviation. This command is accessed by pressing
2nd, Vars.
©McGraw-Hill Education.
Ex 1: Area Under Standard Normal (TI-84)
Find the area to the left of 𝑧 = 1.26.
Note the there is no lower endpoint,
therefore we use -1E99 (or -10^99)
which represents negative 1 followed by
99 zeroes. We select the normalcdf
command and enter -1E99 as the lower
endpoint, 1.26 as the upper endpoint, 0
as the mean and 1 as the standard
deviation.
The area to the left of 𝑧 = 1.26 is 0.8962.
©McGraw-Hill Education.
Ex 2: Area Under Standard Normal (TI-84)
Find the area to the right of 𝑧 = –0.58.
Note the there is no upper endpoint, therefore we use 1E99
(or 10^99) which represents the large number 1 followed
by 99 zeroes. We select the normalcdf command and enter
–0.58 as the lower endpoint, 1E99 as the upper endpoint, 0
as the mean and 1 as the standard deviation. The area to
the right of 𝑧 = –0.58 is 0.7190.
©McGraw-Hill Education.
Ex 3: Area Under Standard Normal (TI-84)
Find the area between 𝑧 = –1.45 and 𝑧 = 0.42.
We select the normalcdf command and enter –1.45 as the
lower endpoint, 0.42 as the upper endpoint, 0 as the mean
and 1 as the standard deviation.
The area between 𝑧 = –1.45 and 𝑧 = 0.42 is 0.5892.
©McGraw-Hill Education.
Objective 4
Find 𝑧-scores corresponding to areas under the normal
curve
*(TI-84 PLUS)
©McGraw-Hill Education.
𝑍-scores (Normal Values) From Areas
We have been finding areas under the normal curve from given
𝑧-scores.
Many problems require us to go in the reverse direction. That is, if
we are given an area, we need to find the 𝑧-score (value from the
population ) that corresponds to that area under the standard
normal curve.
©McGraw-Hill Education.
Normal Values From Areas on the TI-84 PLUS
The invNorm command on the TI-84 PLUS calculator returns the
value from the normal population with a given area to its left. This
command takes three values as its input. The first value is the area
to the left, the second and third values are the mean and standard
deviation, respectively. This command is accessed by pressing
2nd, Vars.
©McGraw-Hill Education.
Example 1: Finding 𝑧-Scores (TI-84 PLUS)
Find the 𝑧-score that has an area of 0.26 to its left.
We select the invNorm command and enter 0.26 as the area to the
left, 0 as the mean, and 1 as the standard deviation.
The z-score with an area of 0.26 to its left is approximately –0.64.
©McGraw-Hill Education.
Example 2: Finding 𝑧-Scores (TI-84 PLUS)
Find the 𝑧-score that has an area of 0.68 to its right.
Since the area to the right is 0.68, the area to
the left is 1 – 0.68 = 0.32.
We use the invNorm command with 0.32 as the
area to the left, 0 as the mean, and 1 as the
standard deviation. The z-score with an area of
0.68 to its right is approximately –0.47.
©McGraw-Hill Education.
Example 3: Finding 𝑧-Scores (TI-84 PLUS)
Find the 𝑧-scores that bound the middle 95% of the area under the
standard normal curve.
We first sketch a normal curve and shade in the
given area. Label the 𝑧-score on the left 𝑧1 and
the 𝑧-score on the right 𝑧2.
Now, since the area in the middle is 0.95, the
area in the two tails combined is 0.05. Half of
that area, which is 0.025, is to the left of 𝑧1.
We use the invNorm command with 0.025 as
the area to the left, 0 as the mean, and 1 as the
standard deviation. We find that 𝑧1 = –1.96. By
symmetry, 𝑧2 = 1.96.
©McGraw-Hill Education.
You Should Know . . .
• How to use a probability density curve to describe a
population
• How the shape of a normal curve is affected by the mean
and standard deviation
• How to find areas under a normal curve (Calculator)
• How to find 𝑧-scores corresponding to areas under a
normal curve (Calculator)
©McGraw-Hill Education.
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