Notes for math 141
Section 8.5-8.6
Finite Mathematics
8.5-8.6: The Normal Distribution
Up until now, we have been dealing with finite discrete random variables. In finding
the probability distribution, we could list the possible values in a table and represent
it with a histogram.
Definition: For a continuous random variable, a probability density function
is defined to represent the probability distribution.
Example 1:
Note that the for a continues random variable X, P (X ≤ x) = P (X < x)
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Definition:Definition: We concentrate on a special class of continuous probability
distributions known as normal distributions. Each normal distribution is defined
by µ and σ. Each normal distribution has the following characteristics:
1.
2.
3.
4.
The
The
The
The
area under the curve is always 1.
curve never crosses the x axis.
peak occurs directly above µ
curve is symmetric about a vertical line passing through the mean.
Example 2:
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Definition: The standard normal variable usually denoted by Z has a normal probability distribution with µ = 0 and σ = 1.
To find P (a ≤ X ≤ b) where X is a random variable with mean µ and standard
deviation σ
1. Type 2ND and then VARS to get to the distribution menu.
2. Select option 2 or scroll down to normcdf and hit ENTER.
3. Type a, comma, b, comma, µ, comma, σ.
4. Close the parentheses and hit ENTER.
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Example 3: Find and sketch the following:
a) P (Z ≤ 1.79)
b) P (Z ≥ 3.49)
c) P (−2 ≤ Z ≤ 1.79)
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Example 4: According to the data released by the Chamber of Commerce of a certain city, the weekly wages of factory workers are normally distributed with a mean of
$600 and a standard deviation of $50. What is the probability that a worker selected
at random from the city makes a weekly wage
a) of less than $600?
b) of more than $760?
c) between $575 and $650?
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Suppose X is a random variable with normal distribution with mean µ and standard
deviation σ. Find a such that P (X ≤ a) = p
1. Type 2ND and then VARS to get to the distribution menu.
2. Selection option 3 or scroll down to invNorm and hit ENTER.
3. Type p, a comma, µ, a comma, and σ.
4. Close the parentheses and hit ENTER.
Example 5: Let Z be the standard normal variable. Find the values of a if a satisfies:
a) P (Z ≤ a) = 0.8907
b)P (Z ≥ a) = 0.2460
c) P (−a ≤ Z ≤ a) = 0.7820
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Notes for math 141
Section 8.5-8.6
Finite Mathematics
Example 6: The scores on an Econ exam were normally distributed with a mean of
72 and a standard deviation of 16. If the instructor assigns a grade of A to 15% of
the class what is the lowest score a student may have to obtain a A?
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