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Chapter 3: Problem 173

From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that \(25 \%\) of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful data? (a) Assume that the company samples five customers. (b) Assume that the company samples 10 customers.

Short Answer

Expert verified
Both (a) and (b) use the hypergeometric distribution to calculate the probability mass function for samples of sizes 5 and 10, respectively.

Step by step solution

01

Understanding the Problem

We are given a population of 500 customers and want to find the probability distribution of useful feedback from a sample of customers. The key here is that the selection is without replacement, indicating a hypergeometric distribution.
02

Setting up for Hypergeometric Distribution

For the hypergeometric distribution, we have: Total population \( N = 500 \), number of successes in population \( K = 0.25 \times 500 = 125 \), sample size \( n \), and the number of observed successes \( k \).
03

Calculating for (a) five customers

For part (a), we have a sample size \( n = 5 \). The probability mass function (PMF) is given by \[ P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}} \] where \( K = 125 \) (customers providing useful data), \( N-K = 375 \) (other customers).
04

Calculating PMF for (a)

For each possible \( k \text{ (0 to min(n, K))} \), calculate: \[ P(X = k) = \frac{{\binom{125}{k} \binom{375}{5-k}}}{{\binom{500}{5}}} \] This results in a specific probability for each \( k \) from 0 to 5.
05

Calculating for (b) ten customers

For part (b), the sample size \( n = 10 \). We use the same formula: \[ P(X = k) = \frac{{\binom{125}{k} \binom{375}{10-k}}}{{\binom{500}{10}}} \] where \( k \) can range from 0 to 10.
06

Calculating PMF for (b)

For each \( k \text{ (0 to min(10, 125))} \), compute the probability \( P(X = k) \) using the formula above. This gives a complete distribution for the sample of 10 customers.

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Key Concepts

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

Probability Mass Function
A Probability Mass Function (PMF) describes the probability of specific outcomes in a discrete random variable. In the context of the hypergeometric distribution, it helps us understand the likelihood of observing different numbers of 'successes' in a sample.
For example, in the given exercise, we aim to know the probability of getting a certain number of customers providing useful data when sampling without replacement.
The PMF is expressed as:
  • \( P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}} \)
  • Where:
    • \( N \): Total population size
    • \( K \): Number of successes in the population
    • \( n \): Sample size
    • \( k \): Number of observed successes
This equation allows us to calculate the probabilities for different numbers of successes \( k \) in our sample. This function is crucial for understanding the distribution of useful data collected.
Sampling Without Replacement
Sampling without replacement is a method of selecting items from a population where each item is not returned to the population once it has been chosen. This technique drastically changes the probabilities of selecting subsequent items since the pool of possible selections becomes smaller and smaller.
In the exercise, this method of sampling is used when choosing customers to provide useful data. This affects the dynamics of how data is gathered, often making calculations more complex compared to sampling with replacement.
  • As items are not replaced, the characteristics and probabilities of the remaining population shift with each selection.
  • This approach is vital when the total number of items is finite and affects analytical scenarios where hypergeometric distributions are applied, emphasizing unique conditions each drawn sample faces.
Understanding this concept can clarify why we opt for a hypergeometric distribution to determine required probabilities.
Applied Statistics
Applied statistics is integral to solving real-world problems, utilizing data to perform analyses and draw inferences. In the given context, we apply statistical methods, such as the hypergeometric distribution, to analyze the sample data.
The probability calculations made help make informed decisions about the population based on sample data.
  • Using real sample data to make generalized conclusions about larger populations is a core element of applied statistics.
  • Understanding theoretical statistical models like the hypergeometric distribution and implementing them in practical scenarios where assumptions match the real-world conditions is crucial for reliable decision-making.
Thus, applied statistics plays a critical role in making sense of the data involving customers and their propensity to provide useful feedback.
Population and Sample
In statistical terms, a population refers to the complete set from which we draw data, while a sample is a subset of this population containing the actual observations used for analysis.
In our exercise, we have a population of 500 customers from which we draw samples of different sizes.
  • The population contains all potential cases of interest, here all customers eligible to give useful insights.
  • Sampling involves selecting specific individuals (e.g., 5 or 10 customers) to focus analyses on and derive conclusions for the whole population.
  • Effective sampling can provide a more practical and cost-effective way to gather data than surveying the full population, especially when dealing with large numbers like 500.
Understanding the relation between population and sample helps discern the validity and scope of statistical findings.

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