91Ó°ÊÓ

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

Calculation For exactly 29 returns

To calculate the probability that exactly 29 out of the 500 calculators will be returned, we can use the binomial probability formula. It is defined as follows: \[P(k; n, p) = C(n, k) * p^k * (1-p)^{n-k}\]Where:- \(P(k; n, p)\) is the probability that exactly \(k\) out of \(n\) trials will be successful,- \(C(n, k)\) is the binomial coefficient (`n choose k`),- \(p\) is the probability of success on an individual trial, - \(1-p\) is the probability of failure on an individual trial.Substitute \(k = 29\), \(n = 500\) and \(p = 0.05\) into the formula to get the result.

02

Calculation for 27 or more returns

To calculate the probability that 27 or more out of the 500 calculators will be returned, we need to add up the probabilities for 27 through 500 returns. However, calculating the binomial probability for each of these and adding them up would be cumbersome, so we use the compliment rule. It states that the probability of an event happening is 1 minus the probability of it not happening.So, the probability that 27 or more calculators will be returned is the same as 1 minus the probability that 26 or less calculators will be returned. We calculate the cumulative probability for 26 or less returns and subtract it from 1.

03

Calculation for 15 to 22 returns

To calculate the probability that between 15 and 22 out of the 500 calculators will be returned, we once again use the binomial distribution's cumulative property. To get the answer, we first calculate the cumulative probability for up to 22 returns and then subtract the cumulative probability for up to 14 returns. This will give us the probability for exactly 15 to 22 calculators being returned.

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Key Concepts

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

Consumer Satisfaction

Consumer satisfaction is a crucial aspect of any business. At Johnson Electronics, ensuring happy customers is a top priority. To achieve this, they offer a guarantee that includes a refund or replacement for any malfunctioning calculator within two years from purchase. This approach builds trust and encourages customers to choose their products.

Happy customers are more likely to return and recommend products to others, enhancing the company's reputation.

• Offering refunds or replacements can significantly enhance customer loyalty.
• Reliable customer service often leads to positive word-of-mouth promotion.

By prioritizing consumer satisfaction, companies like Johnson Electronics can maintain a competitive edge in the market.

Probability Calculation

Calculating probability involves determining the likelihood of a specific event. In this scenario, we're interested in finding probabilities related to the number of calculator returns.

The binomial probability formula is a powerful tool for such calculations. It allows us to find the probability of a specific number of successes in a series of independent trials. The formula can be intimidating, but understanding each component simplifies the process:

• \(P(k; n, p) = C(n, k) \times p^k \times (1-p)^{n-k}\)
  
  Here, \(C(n, k)\) is the binomial coefficient, \(p\) is the probability of success, and \(1-p\) is the probability of failure.

Using these elements, we can effectively calculate the desired probability for the given scenario.

Binomial Distribution

A binomial distribution is a statistical method used to model the outcome of experiments with two possible outcomes, such as success or failure. In this case, a calculator either malfunctions (success) or works fine (failure).

For Johnson Electronics, the binomial distribution helps forecast calculator returns. Essential characteristics include:

• **Fixed Number of Trials:** There are 500 calculators.
• **Two Possible Outcomes:** Either a calculator malfunctions, or it doesn't.
• **Constant Probability:** Each calculator has a \(5\%\) chance of malfunctioning.

Recognizing these aspects helps us use binomial distribution effectively to solve consumer satisfaction-related problems.

Cumulative Probability

Cumulative probability helps identify the probability of multiple events occurring up to a certain point. To find how many calculators among 500 will be returned, this method is especially useful.

For example, calculating the probability that 27 or more calculators will be returned requires us to:

• First, calculate the cumulative probability for 26 or fewer returns.
• Subtract this result from 1 to find the probability for 27 or more returns.

Understanding cumulative probability enables us to handle more complex probability problems, focusing on ranges rather than single point events.