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Chapter 3: Problem 2

For each of the following exercises, determine the range (possible values) of the random variable. A batch of 500 machined parts contains 10 that do not conform to customer requirements. The random variable is the number of parts in a sample of five parts that do not conform to customer requirements.

Short Answer

Expert verified
The range of the random variable is 0 to 5.

Step by step solution

01

Identify the Nature of the Random Variable

The random variable in this problem represents the number of non-conforming parts in a sample of size five, drawn from a batch of 500 parts, where 10 parts are non-conforming. This random variable follows a hypergeometric distribution because we are selecting without replacement from a finite population.
02

Determine the Possible Values of the Random Variable

The number of non-conforming parts in a sample of five can range from 0 to 5. However, not all of these values are possible because there are only 10 non-conforming parts in total. Thus, the values must also consider the constraints imposed by the total number of non-conforming and conforming parts.
03

Calculate the Maximum Possible Non-Conforming Parts

In a sample of five parts, the most non-conforming parts we can have is equal to the smaller of 5 (the sample size) or 10 (the total number of non-conforming parts). Thus, the maximum possible number of non-conforming parts in the sample is 5.
04

Determine the Minimum Possible Non-Conforming Parts

The minimum number of non-conforming parts in the sample is 0, which would occur if all five parts drawn are conforming.
05

Define the Range of the Random Variable

Given the calculations, the range of the random variable (number of non-conforming parts in the sample of five) is from 0 to 5, because we can have from 0 to 5 non-conforming parts out of 10 available non-conforming parts in a batch of 500.

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Key Concepts

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

Random Variable
In probability and statistics, a random variable is a fundamental concept representing an outcome from a random phenomenon. Specifically, in the context of this exercise, the random variable is the number of non-conforming parts in a sample of five from a batch of 500.

This random variable is particularly interesting because it does not describe individual parts but instead captures the number that fails to meet customer requirements. The key aspect here is that we are considering a scenario where selection happens without replacement, indicating that each selected part is distinctly different from the others in the batch.
  • The random variable can take discrete values since we count the number of specific occurrences (non-conforming parts).
  • Because of the nature of the selection process, this specific problem falls under the category of a hypergeometric distribution, which occurs frequently in quality control scenarios.
Non-Conforming Parts
Non-conforming parts refer to those components within a batch that do not meet predetermined specifications or customer requirements. In our context, out of 500 parts, only 10 are non-conforming.

Understanding non-conformance is crucial for identifying flaws in manufacturing or production processes. Such analysis helps businesses manage quality control, reduce waste, and improve customer satisfaction.
  • The number of non-conforming parts impacts the quality assessment of the batch as a whole.
  • In statistical terms, identifying the number of these parts in differing sample sizes helps in estimating the overall batch quality.
By considering non-conforming parts, companies can take preventative measures to avoid defects and enhance product reliability.
Sample Size
Sample size, in this context, is the subset of the whole batch of 500 parts that is tested to estimate the number of non-conforming parts. For this exercise, this size is set to five.

The significance of sample size cannot be overstated, as it directly influences the accuracy and reliability of statistical inferences made about the entire batch. A carefully chosen sample allows evaluators to accurately estimate the proportion of faulty parts without examining every item.
  • The larger the sample size, the more accurate the representation of the population typically is.
  • However, in the hypergeometric distribution, the sample is determined without replacement, which means the population size has a direct impact on the sample outcome.
This subtlety means that, even with a small sample, quality inspectors can draw significant conclusions about the overall batch quality.
Finite Population
A finite population refers to a population of a limited size from which we draw samples for analysis. In this exercise, the population consists of 500 machined parts.

Understanding the concept of a finite population is critical when calculating probabilities such as finding defective units because it's important to account for the exhaustive nature of the selection process.
  • Unlike an infinite population, each selection affects the remaining pool of items in a finite population.
  • The hypergeometric distribution is particularly suited for scenarios involving finite populations and without replacement.
With finite populations, each draw alters the probability distribution, thereby necessitating thoughtful consideration in statistical calculations.

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Most popular questions from this chapter

The demand for water use in Phoenix in 2003 hit a high of about 442 million gallons per day on June 27 (http://phoenix.gov/WATER/wtrfacts.html). Water use in the summer is normally distributed with a mean of 310 million gallons per day and a standard deviation of 45 million gallons per day. City reservoirs have a combined storage capacity of nearly 350 million gallons. a. What is the probability that a day requires more water than is stored in city reservoirs? b. What reservoir capacity is needed so that the probability that it is exceeded is \(1 \% ?\) c. What amount of water use is exceeded with \(95 \%\) probability? d. Water is provided to approximately 1.4 million people. What is the mean daily consumption per person at which the probability that the demand exceeds the current reservoir capacity is \(1 \%\) ? Assume that the standard deviation of demand remains the same.

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