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Statistics / Mathematical Statistics with Applications 7 / Chapter 4 / Problem 147E

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

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A machine used to fill cereal boxes dispenses, on the

Chapter 4, Problem 147E

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QUESTION:

Problem 147E

A machine used to fill cereal boxes dispenses, on the average, μ ounces per box. The manufacturer wants the actual ounces dispensed Y to be within 1 ounce of μ at least 75% of the time. What is the largest value of σ, the standard deviation of Y, that can be tolerated if the manufacturer’s objectives are to be met?

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QUESTION:

Problem 147E

A machine used to fill cereal boxes dispenses, on the average, μ ounces per box. The manufacturer wants the actual ounces dispensed Y to be within 1 ounce of μ at least 75% of the time. What is the largest value of σ, the standard deviation of Y, that can be tolerated if the manufacturer’s objectives are to be met?

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Answer:

Step 1 of 1:

A machine used to fill cereal boxes dispenses, on the average, $\mu{}$ ounces per box. The manufacturer wants the actual ounces dispensed $Y$ to be within 1 ounce of $\mu{}$ at least $75\%$ of the time.

What is largest value of $\sigma{},$ the standard deviation of $Y,$ that can be tolerated if the manufacturer’s objectives are to be met?

Tchebysheff’s Theorem Let $Y$ be a random variable with mean $\mu{}$ and finite variance ${\sigma{}}^{2}$ . Then, for any constant $k>0$ ,

$P(|Y\ -\mu{}|\leq{}k\sigma{})\geq{}1\ -\frac{1}{{k}^{2}}\ or\ P(|Y\ -\mu{}|\ \geq{}k\sigma{})\ \leq{}\frac{1}{{k}^{2}}.$ …….(1)

This result applies for any probability distribution, whether the

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