Engineering Statistics ECIV 2305

5-2 Linear Combinations of Normal Random Variables

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The Distribution of Linear Combinations of Normal Random

Variables

• 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 WeightsSuppose 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?