Download 6-2B Lecture

Section 6 – 2B:
Interpreting the Area under a Normal Probability Distribution
Probability Interpretation for the area under a normal curve
The area under a normal curve for any interval of the random variable x
is the probability of selecting a single number from the data set
and having it be within the stated interval.
Proportion Interpretation for the area under a normal curve
The area under a normal curve for any interval of the random variable x
is the proportion of the data set that is within the stated interval.
Example 1
Example 2
shaded
area
= .8367
shaded
area
= .3556
x= 4.3
x
x= 45.7
The interval for x is x > 4.3
The interval for x is x < 45.7
Probability Statement:
Probability Statement:
P(x > 4.3) = .3556
P(x < 45.7) = .8367
Probability Interpretation:
If I randomly select one value from the
data set, the probability that it will be a
number more than 4.3 is .3556
Probability Interpretation:
If I randomly select one value from the
data set, the probability that it will be a
number less than 45.7 is .8367
Proportion Interpretation:
The proportion of the data with a value
more than 4.3 is .3556
Proportion Interpretation:
The proportion of the data with a value
less than 45.7 is .8367
Section 6 – 2B
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x
© 2012 Eitel
Example 3
shaded
area
= .7926
x= 12.6
x= 16.5
x
The interval for x is 12.6 < x < 16.5
Probability Statement:
P(12.6 < x < 16.5) = .7926
Probability Interpretation:
If I randomly select one value from the
data set, the probability that it will be a
number between 12.6 and 16.5 is .7926
Proportion Interpretation:
The proportion of the data with a value
between 12.6 and 16.5 is .7926
Section 6 – 2B
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© 2012 Eitel
Displaying and Interpreting the Area under a Normal Probability Distribution
Label the given mean and standard deviation on each normal curve below. Label the x
value(s) on the horizontal axis. Draw and label the shaded area that would represent the answer to
the probability statement given.
Example 1
A data set is normally distributed with µ x = 3 , σ x = 1 and
P(x < 4.2) = .8849
shaded
area
= .8849
µx = 3
σx =1
x= 4.2
x
Step 1:
A data set that is normally distributed means that the density curve of the data set will be a continuous
normal curve. Draw a normal curve and label the axis as the x axis.
Step 2:
µ x = 3 says that the mean for the data is located on the x axis where the graph has itʼs highest point.
σ x = 1 says that the standard deviation for the data set is 1.
Write
µx = 3
on the x axis under the peak of the graph where the mean is located
σx =1
Step 3:
x < 4.2 says that the shaded area in question is less than 4.2
Draw and label a vertical line at x = 4.2 and shade the area to the left
Step 4:
P( x < 4.2) = .8849 says that the area under the graph less than (to the left of) 4.2 is .8849
Label the shaded area equal to .8849
Interpretation:
The interpretation of P( x < 4.2) = .8849 is, if I randomly select one value from the data set then
the probability that it will be a number less than 4.2 is .8849
A lay person would say in English that if one value from the data set is selected at random there is
about an 88% chance that the number selected will be a number less than 4.2
Section 6 – 2B
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© 2012 Eitel
Example 2
A data set is normally distributed with µ x = 4.5 , σ x = 2.6
and P( x > 5.7) = .3222
shaded
area
= .3222
µ x = 4.5 x= 5.7
σ x = 2.6
x
Step 1:
A data set that is normally distributed means that the density curve of the data set will be a continuous
normal curve. Draw a normal curve and label the axis as the x axis.
Step 2:
µ x = 4.5 says that the mean for the data is located on the x axis where the graph has itʼs highest point.
σ x = 2.6 says that the standard deviation for the data set is 2.6
Write
µ x = 4.5
on the x axis under the peak of the graph where the mean is located
σ x = 2.6
Step 3:
x > 5.7 says that the area in question is more than 5.7
Draw and label a vertical line at x = 5.7 and shade the area to the right
Step 4:
P( x > 5.7) = .3222 says that the area under the graph greater than (to the right of) 5.7 is .3222
Label the shaded area equal to .3222
Interpretation:
The interpretation of P( x > 5.7) = .3222 is that if I randomly select one value from the data set
above then the probability that it will be a number more than 5.7 is .3222
A lay person would say in English that if one value from the data set is selected at random there is
about a 32% chance that the number selected will be a number more than 5.7
Section 6 – 2B
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© 2012 Eitel
Example 3
A data set is normally distributed with µ = 5.7 , σ = 1.6
and
P(3.5 < x < 4.8) = .2023
shaded
area
= .2023
x= 3.5
x= 4.8 µ = 5.7
σ = 1.6
x
Step 1:
A data set that is normally distributed means that the density curve of the data set will be a continuous
bell shaped curve. Draw a normal curve and label the axis as the x axis.
Step 2:
µ = 5.7says that the mean for the data is located on the x axis where the graph has itʼs highest point.
σ = 1.6 says that the standard deviation for the data set is 1.6
Write
µ = 5.7
on the x axis under the peak of the graph where the mean is located
σ = 1.6
Step 3:
3.5 < x < 4.8 says that the area in question is between 3.5 and 4.8
NOTE: Both 3.5 and 4.8 are both to the left of the mean.
Draw and label a vertical line at x = 3.5 and x = 4.8 and shade the area between the lines.
Step 4:
P(3.5 < x < 4.8) = .2023 says that the area under the graph between 3.5 and 4.8 is .2023
Label the shaded area equal to .2023
Interpretation:
The interpretation of P(3.5 < x < 4.8) = .2023 is that if I randomly select one value from the data set
above then the probability that it will be a number between 3.5 and 4.8 is .2023
A lay person would say in English that if one value is selected from the data set there is about a
20% chance that the number will be between 3.5 and 4.8
Section 6 – 2B
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© 2012 Eitel