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Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by this company malfunction within a 2 -year period. The company recently mailed 500 such calculators to its customers. a. Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2-year period. b. What is the probability that 27 or more of the 500 calculators will be returned for refund or replacement within a 2 -year period? c. What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2 -year period?

Step by step solution

01

Define the Constants

The total number of trials \(n\) is 500 (number of calculators) and the probability of success \(p\) in each trial is \(0.05\) (probability of one calculator malfunctioning).

02

Calculate Probability for Exactly 29 Malfunctions

Using the formula for binomial distribution, calculate the probability of exactly 29 calculators malfunctioning. \( P(X=29) = C(500, 29) \times (0.05^{29}) \times ((1-0.05)^{500-29}) \)

03

Calculate Probability for 27 or More Malfunctions

Calculate the probability that 27 or more calculators malfunction. Here we need to use the CDF of binomial distribution and subtract it from 1 as it gives probability of less than 27 outcomes. \( P(X \geq 27) = 1 - P(X < 27) = 1 - P(X<=26) = 1 - CDF(26) \)

04

Calculate Probability for 15 to 22 Malfunctions

Finally, compute the probability for number of malfunctions being between 15 and 22 inclusive. This involves CDF of binomial distribution as well. \( P(15 \leq X \leq 22) = P(X <= 22) - P(X < 15) = CDF(22) - CDF(14) \)

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Key Concepts

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

Probability Theory

Probability theory is a branch of mathematics that deals with analyzing random phenomena. It provides the tools necessary to quantify uncertainty and make predictions about the likelihood of various outcomes. In the context of our exercise, we are dealing with the probability that a certain number of calculators malfunction.The binomial distribution is a key concept here, as it models the number of successes in a fixed number of independent trials. Each trial has two possible outcomes: success (a malfunction) or failure (no malfunction). The probability of success in each trial is constant, denoted as \( p \), which is 0.05 in our case.

• Number of trials \( n \) - 500 calculators
• Probability of success \( p \) - 0.05
• Number of successes \( k \) - Varies per calculation (e.g., exactly 29, 27 or more, etc.)

Understanding probability theory helps us calculate these probabilities accurately, enabling businesses like Johnson Electronics to make informed decisions about their product guarantees.

Consumer Satisfaction

Consumer satisfaction is crucial for businesses that want to retain customers and maintain a positive reputation. It involves ensuring that a product or service meets or exceeds customer expectations. For Johnson Electronics, consumer satisfaction involves guaranteeing replacements or refunds for calculators that malfunction within a two-year period. Focusing on consumer satisfaction not only builds trust with customers but also helps in understanding consumer needs and improving product quality over time. By analyzing probability calculations, the company can predict potential issues and proactively address them to heighten consumer satisfaction.

• Build trust with strong guarantees
• Analyze customer feedback and product performance
• Continuously improve product quality based on data

Effective consumer satisfaction strategies can transform potential complaints into opportunities for customer loyalty and brand advocacy.

Quality Control

Quality control is an essential aspect of manufacturing, ensuring that products meet certain standards of quality and function before reaching the customer. It involves systematic processes and checks to maintain product quality, minimizing defects and ensuring reliability. At Johnson Electronics, quality control targets reducing the occurrence of malfunctions. Despite all efforts, a small percentage (5%) of calculators still malfunction within two years. Understanding and managing such variability is where statistical tools like binomial distribution play a role. Here are some key strategies used in quality control:

• Regular inspections and testing of products
• Implementing quality assurance frameworks
• Using statistical methods to identify defect trends

Applying these strategies helps isolate and mitigate problems effectively, resulting in fewer customer returns and higher quality products.

Statistical Calculation

Statistical calculation involves using mathematical methods to analyze data and draw inferences about populations. In our context, it's crucial for computing the likelihood of certain events, like the number of calculator malfunctions.Key statistical tools used in the exercise include:

• Binomial distribution for calculating precise probabilities
• Cumulative Distribution Function (CDF) for understanding probabilities over a range of outcomes
• Combinatorial methods for determining the number of possible ways an event can occur

For instance, to find the probability of exactly 29 malfunctions out of 500 calculators, we use the binomial probability formula:\[ P(X=29) = C(500, 29) \times (0.05^{29}) \times (0.95^{471}) \]Utilizing these calculations helps the company assess their risk and manage their quality assurance processes more effectively.