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Chapter 6: Q8E (page 262)

In a random sample of 80 components of a certain type, 12 are found to be defective.

a. Give a point estimate of the proportion of all such components that are not defective.

b. A system is to be constructed by randomly selecting two of these components and connecting them in series, as shown here.

The series connection implies that the system will function if and only if neither component is defective (i.e., both components work properly). Estimate the proportion of all such systems that work properly. (Hint: If p denotes the probability that a component works properly, how can P (system works) be expressed in terms of p ?)

Short Answer

Expert verified

(a) The required point estimate value is\({\rm{0}}{\rm{.85}}\).

(b) The required probability of interest value is\(0.723\).

Step by step solution

01

Concept introduction

In statistics, point estimation is the process of estimating an estimated value of a population's parameter鈥攕uch as the mean (average)鈥攆rom random samples of the population. The exact accuracy of any one approximation is unknown, but probabilistic claims about the accuracy of numbers determined over a large number of experiments can be built.

02

Calculating using a point estimate

(a)

There are 80 components in all, 12 of which are faulty. As a result, there are\({\rm{80 - 12 = 68}}\)components that are in good working order. The percentage of all such components that are not faulty is calculated using a point estimate.

\(\begin{array}{c}{\rm{\hat p = }}\frac{{{\rm{68}}}}{{{\rm{80}}}}\\{\rm{ = 0}}{\rm{.85}}\end{array}\)

Hence, the required point estimate value is \({\rm{0}}{\rm{.85}}\).

03

Calculating probability of interest

(b)

The probability of interest is,

\(\begin{array}{c}{\rm{P( system works )}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{p \times p}}\\{\rm{ = }}{{\rm{p}}^{\rm{2}}}\end{array}\)

1) Both components must function properly.

Therefore, to calculate this probability, replace \(p\)with \(\hat p\)and get,

\(P({\rm{ system works }}) \approx {0.85^2} = 0.723.\)

Hence, the required probability of interest value is \(0.723\).

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Most popular questions from this chapter

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