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Engineering Statistics ECIV 2305
5-2 Linear Combinations of Normal Random
Variables
The Distribution of Linear Combinations of
Normal Random Variables
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• Notice that if a = 1/σ and
b = −μ/σ, the resulting
linear function of X has a
standard normal
distribution,
Example 23: Piston Head Construction
Recall that the radius of a piston head X1 has a mean
value of 30.00 mm and a standard deviation of 0.05 mm,
and that the inside radius of a cylinder X2 has a mean
value of 30.25 mm and a standard deviation of 0.06 mm.
• What is the mean and the standard deviation of the gap
between the piston head and the cylinder?
• If the piston head radius and the cylinder radius are taken
to be normally distributed, what is the probability that a
piston head will not fit within a cylinder?
• Suppose that a piston performs optimally when the gap Y is
between 0.10mmand 0.35mm. What is the probability that
a piston performs optimally?
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Example 18 Tomato Plant Heights
Recall that three weeks after planting, the heights of tomato
plants have a mean of 29.4 cm and a standard deviation of 2.1
cm. Suppose that 20 tomato plants are planted.
• What is the distribution of the average tomato plant height after
three weeks of growth?
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Example 37 Concrete Block Weights
Suppose that a wall is constructed from 24 concrete blocks as illustrated in
Figure.
What is the distribution of the total weight of the wall??
What is the interval that it gives about a 99.7% chance that the wall has a weight
within three standard deviations of its mean value?