Download Section 12.5

Chapter 12
Statistics
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Chapter 12: Statistics
12.1
12.2
12.3
12.4
12.5
Visual Displays of Data
Measures of Central Tendency
Measures of Dispersion
Measures of Position
The Normal Distribution
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Section 12-5
The Normal Distribution
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The Normal Distribution
• Understand the distinction between discrete and
continuous random variables.
• Understand the properties of a normal curve.
• Use a table of standard normal curve areas.
• Interpret normal curve areas in several ways.
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Discrete and Continuous Random
Variables
A random variable that can take on only certain
fixed values is called a discrete random
variable. A variable whose values are not
restricted in this way is a continuous random
variable.
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Definition and Properties of a Normal
Curve
A normal curve is a symmetric, bell-shaped
curve. Any random variable whose graph has this
characteristic shape is said to have a normal
distribution.
On a normal curve, if the quantity shown on the
horizontal axis is the number of standard
deviations from the mean, rather than values of
the random variable itself, then we call the curve
the standard normal curve.
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Normal Curves
B
A
S
C
0
S is standard, with mean = 0, standard deviation = 1
A has mean < 0, standard deviation = 1
B has mean = 0, standard deviation < 1
C has mean > 0, standard deviation > 1
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Properties of Normal Curves
The graph of a normal curve is bell-shaped and
symmetric about a vertical line through its center.
The mean, median, and mode of a normal curve are all
equal and occur at the center of the distribution.
Empirical Rule About 68% of all data values of a
normal curve lie within 1 standard deviation of the
mean (in both directions), and about 95% within
2 standard deviations, and about 99.7% within
3 standard deviations.
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Empirical Rule
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Example: Applying the Empirical Rule
Suppose that 280 sociology students take an exam
and that the distribution of their scores can be
treated as normal. Find the number of scores
falling within 2 standard deviations of the mean.
Solution
A total of 95% of all scores lie within 2 standard
deviations of the mean.
(.95)(280)= 266 scores
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A Table of Standard Normal Curve
Areas
To answer questions that involve regions other than
those within 1, 2, or 3 standard deviations of the
mean, we can refer to the table on page 690 or other
tools such as a computer or calculator.
The table gives the fraction of all scores in a normal
distribution that lie between the mean and z standard
deviations from the mean. Because of symmetry, the
table can be used for values above the mean or
below the mean.
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Example: Applying the Normal Curve
Table
Use the table to find the percent of all scores that
lie between the mean and the following values.
a) 1.5 standard deviations above the mean
b) 2.62 standard deviations below the mean
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Example: Applying the Normal Curve
Table
Solution
a) Here z = 1.50. Find 1.50 in the z column. The table
entry is 0.433, so 43.3% of all values lie between
the mean and 1.5 standard deviations above the
mean.
x
z = 1.5
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Example: Applying the Normal Curve
Table
Solution (continued)
b) Even though it is below the mean, the table still
works because of the symmetry. Find 2.62 in the
z column. The table entry is 0.496, so 49.6% of all
values lie between the mean and 2.62 standard
deviations below the mean.
z = –2.62
x
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Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
Solution
z = –1.7
x
z = 2.55
z = 2.55 leads to an area of 0.495 and z = 1.7
leads to an area of 0.455. Add these areas to
get 0.495 + 0.455 = 0.950.
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Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
z = 0.61 z = 2.63
Solution
z = 2.63 leads to an area of 0.496 and z = 0.61
leads to an area of 0.229. Subtract these areas
to get 0.496 – 0.229 = 0.267.
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Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
z = 2.14
Solution
z = 2.14 leads to an area of 0.484. The total
area to the right of the mean is 0.500. Subtract
these values to get 0.500 – 0.484 = 0.016.
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Interpreting Normal Curve Areas
In a standard normal curve, the following three
quantities are equivalent.
1. Percentage (of total items that lie in an
interval)
2. Probability (of a randomly chosen item lying
in an interval)
3. Area (under the normal curve along an
interval)
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Example: Probability with Volume
The volumes of soda in bottles from a small company
are distributed normally with mean 12 ounces and
standard deviation 0.15 ounce. If 1 bottle is randomly
selected, what is the probability that it will have more
than 12.33 ounces?
Solution
12.33 corresponds to a z-value of 2.2. The probability
is equal to the area above z = 2.2, which is
0.500 – 0.486 = 0.014 (or 1.4%).
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Example: Finding z-Values for Given Areas
under the Normal Curve
Assuming a normal distribution, find the z-value
meeting the condition that 39% of the area is to
the right of z.
11%
Solution
39%
Because 50% of the area lies to the right of the mean,
there must be 11% of the area between the mean and z.
From the table, A = 0.110 corresponds to z = 0.28.
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Example: Finding z-Values for Given Areas
under the Normal Curve
Assuming a normal distribution, find the z-value
meeting the condition that 76% of the area is to
the left of z.
26%
Solution
50%
24%
There must be 26% of the area between the mean
and z. From the table, A = 0.260 corresponds
closely to z = 0.71.
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