Chapter 8: Q. 8.6 (page 393)

8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?

Short Answer

Expert verified

The components is $n = 56$ .

Step by step solution

Given information

The components need to have on hand, approximately $90$ percent. The stock would last at least $35$ days.

Explanation

Let $X_{i}$ denote the lifetime of $i$ th component. Understand that $X_{1}, X_{2}, \dots$ is a sequence of independent and identically distributed random variables, each having a mean.
$μ = E [X_{i}] = \int_{0}^{1} x f (x) d x = \int_{0}^{1} 2 x^{2} d x = {2 \frac{x^{3}}{3}|}_{0}^{1} = \frac{2}{3}$
Then the variance is,
$σ^{2} = Var (X_{i}) = \int_{0}^{1} x^{2} f (x) d x - {(\frac{2}{3})}^{2} = \int_{0}^{1} 2 x^{3} d x - \frac{4}{9} = {2 \frac{x^{4}}{4}|}_{0}^{1} - \frac{4}{9} = \frac{1}{18}$
Let $S_{n} = \sum_{i = 1}^{n} X_{i}$ indicates the time of $n$ th failure.

Verify that the value of $n$ is satisfied.
$P \{S_{n} \geq 35\} \approx . 90 ?$

Explanation

Using the central limit theorem:

$.90 \approx P \{S_{n} \geq 35\} = 1 - P \{S_{n} \leq 35\} = 1 - P \{\frac{S_{n} - n μ}{σ \sqrt{n}} \leq \frac{35 - n μ}{σ \sqrt{n}}\} \approx \approx 1 - Φ (\frac{35 - n μ}{σ \sqrt{n}}) \Rightarrow Φ (\frac{35 - n μ}{σ \sqrt{n}}) \approx . 10 \Rightarrow \frac{35 - n μ}{σ \sqrt{n}} \approx Φ^{- 1} (. 10) = - 1.28$

Then,

$\begin{matrix} \frac{\sqrt{18} (105 - 2 n)}{3 \sqrt{n}} \approx - 1.28 \end{matrix} n = 56$

Hence, the components $n = 56$

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8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?

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8.6 . In Self-Test Problem 8.5, how many components would one need to have on hand to be approximately 90percent certain that the stock would last at least 35days?

Short Answer

Step by step solution

Given information

Explanation

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

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8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?