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Chapter 6: Problem 8

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 point estimate of non-defective proportion is 0.85. b. Approximately 72.25% of the systems work properly.

Step by step solution

01

Calculate the Proportion of Defective Components

To find the proportion of defective components, divide the number of defective components by the total number of components sampled. So, the proportion of defective components \(\hat{p}_d\) is \(\hat{p}_d = \frac{12}{80} = 0.15\).
02

Calculate the Proportion of Non-Defective Components

The proportion of non-defective components \(\hat{p}_w\) is the complement of the proportion of defective components. Therefore, \(\hat{p}_w = 1 - \hat{p}_d = 1 - 0.15 = 0.85\).
03

Establish the Probability Model for the System

Given a system is constructed in series from two components, both components must be non-defective for the system to function. Let \(p\) be the probability that a single component is non-defective, \(p = \hat{p}_w\).
04

Calculate the Proportion of Functioning Systems

The system functions if both components are non-defective, which occurs with probability \(P = p^2\). Therefore, \(P = (0.85)^2\). When you calculate this, you get \(P = 0.7225\). So, approximately 72.25% of such systems work properly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Estimation
Point estimation is a statistical technique used to provide a single value or "point" that serves as a best guess of an unknown parameter of a population. In the context of the exercise, we want to estimate the proportion of components that are not defective. To do this, we utilize the sample information we have, specifically the point estimate formula:
  • For the non-defective components, it is calculated as the complement of the defective proportion found in the sample.
The proportion of non-defective components from the sample, \(\hat{p}_w = 1 - \hat{p}_d\),is essentially our point estimate for the population's non-defective rate. By using the observed data from our sample of 80 components, we concluded that approximately 85% of the components are non-defective. This point estimate helps us understand the broader characteristics of the entire set of components.
Probability
Probability is a measure of the likelihood of a particular event happening, and it's a fundamental concept often used to assess uncertainty in statistics. In this exercise, we use probability to determine the functioning likelihood of a series system.
  • For a component to work properly, it must not be defective, which we've estimated through our point estimation as a probability of 0.85.
The probability of a single component functioning properly plays a crucial role when predicting how two components linked in series will behave.
Series System Functionality
Understanding series system functionality involves recognizing how components work together to determine the system's overall performance. In a series system, all connected components must function for the system to be operational.
  • In this example, when two non-defective components are linked, and both need to work for the overall system to function, we can assess this by multiplying the probability of each independent component working.
Thus, the combined functionality probability is \(p^2\),where \(p\)is the probability a single component functions. If each component has a probability of 0.85 to function, then the system, comprised of two such components, has a probability of \((0.85)^2 = 0.7225\).This insight tells us about the system's reliability and helps in designing efficient systems.
Proportion Calculation
Proportion calculation involves determining the fraction of the total that possesses a particular attribute or condition. In statistical exercises like this, proportions provide vital insights into sample and population characteristics.
  • To calculate the proportion of defective components, you simply divide the number of defective units by the total sampled units.
  • Conversely, non-defective proportions are calculated by taking one minus the defective proportion, as \(1 - \hat{p}_d\).
This calculation is crucial for establishing accurate point estimates and informs decisions such as assessing system functionality or reliability. By understanding both defective and non-defective proportions, we gain clearer insight into the quality and dependability of the components we assess.

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Most popular questions from this chapter

a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each house. The resulting observations are \(103,156,118,89,125\), \(147,122,109,138,99\). Let \(\mu\) denote the average gas usage during January by all houses in this area. Compute a point estimate of \(\mu\). b. Suppose there are 10,000 houses in this area that use natural gas for heating. Let \(\tau\) denote the total amount of gas used by all of these houses during January. Estimate \(\tau\) using the data of part (a). What estimator did you use in computing your estimate? c. Use the data in part (a) to estimate \(p\), the proportion of all houses that used at least 100 therms. d. Give a point estimate of the population median usage (the middle value in the population of all houses) based on the sample of part (a). What estimator did you use?

At time \(t=0,20\) identical components are tested. The lifetime distribution of each is exponential with parameter \(\lambda\). The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that \(y=15\) of the 20 components are still in operation (so 5 have failed). Derive the mle of \(\lambda\). [Hint: Let \(Y=\) the number that survive 24 hours. Then \(Y \sim \operatorname{Bin}(n, p)\). What is the mle of \(p\) ? Now notice that \(p=P\left(X_{i} \geq 24\right)\), where \(X_{i}\) is exponentially distributed. This relates \(\lambda\) to \(p\), so the former can be estimated once the latter has been.]

Let \(X_{1}, \ldots, X_{n}\) be a random sample from a gamma distribution with parameters \(\alpha\) and \(\beta\). a. Derive the equations whose solutions yield the maximum likelihood estimators of \(\alpha\) and \(\beta\). Do you think they can be solved explicitly? b. Show that the mle of \(\mu=\alpha \beta\) is \(\hat{\mu}=\bar{X}\).

Let \(X\) denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of \(X\) is $$ f(x ; \theta)=\left\\{\begin{array}{cl} (\theta+1) x^{\theta} & 0 \leq x \leq 1 \\ 0 & \text { otherwise } \end{array}\right. $$ where \(-1<\theta\). A random sample of ten students yields data \(x_{1}=.92, x_{2}=.79, x_{3}=.90, x_{4}=.65, x_{5}=.86\), \(x_{6}=.47, x_{7}=.73, x_{9}=.97, x_{9}=.94, x_{10}=.77\). a. Use the method of moments to obtain an estimator of \(\theta\), and then compute the estimate for this data. b. Obtain the maximum likelihood estimator of \(\theta\), and then compute the estimate for the given data.

A sample of \(n\) captured Pandemonium jet fighters results in serial numbers \(x_{1}, x_{2}, x_{3}, \ldots, x_{\kappa}\). The CIA knows that the aircraft were numbered consecutively at the factory starting with \(\alpha\) and ending with \(\beta\), so that the total number of planes manufactured is \(\beta-\alpha+1\) (e.g., if \(\alpha=17\) and \(\beta=29\), then \(29-17+1=13\) planes having serial numbers \(17,18,19, \ldots, 28,29\) were manufactured). However, the CIA does not know the values of \(\alpha\) or \(\beta\). A CIA statistician suggests using the estimator \(\max \left(X_{i}\right)-\min \left(X_{i}\right)+1\) to estimate the total number of planes manufactured. a. If \(n=5, x_{1}=237, x_{2}=375, x_{3}=202, x_{4}=525\), and \(x_{5}=418\), what is the corresponding estimate? b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating \(\beta-\alpha+1\) ? Explain in one or two sentences.
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