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Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees a refund or a replacement for any calculator that malfunctions within 2 years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by the company malfunction within a 2 -year period. The company mailed a package of 10 randomly selected calculators to a store. a. Let \(x\) denote the number of calculators in this package of 10 that will be returned for refund or replacement within a 2-year period. Using the binomial probabilities table, obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Determine the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly 2 of the 10 calculators will be returned for refund or replacement within a 2-year period.

Step by step solution

01

Define the problem in terms of binomial distribution

Since the question involves multiple trials resulting in two outcomes: the calculator will malfunction or it won't, a binomial distribution model is ideal. The characteristics of a binomial experiment are satisfied: fixed number of trials, independence, the task can only end in two ways (success or failure) where failure is a malfunction here, and the probability of success is the same for each trial. Let the number of trials (n) be 10, the probability of success (p) be 0.05, and denote the number of successful trials (malfunctions) as \(x\). The probability mass function for a binomial distribution is \(P(x) = \binom{n}{x} (p^x) ((1-p)^{n-x})\). This can be used to calculate the probability of \(x\) calculators malfunctioning.

02

Calculate the probability distribution, mean and standard deviation

Use the binomial probability formula to calculate each possible outcome from \(x = 0\) to \(x = 10\). Plot the probabilities for each outcome on a graph to visualize the distribution. The mean and standard deviation for a binomial distribution can be calculated using the formulas \(\mu = np\) and \(\sigma = \sqrt {np(1-p)}\).

03

Calculate the probability of exactly 2 malfunctions

Use the binomial probability formula to calculate the probability of exactly 2 calculators malfunctioning, i.e., \(x = 2\). This represents the probability that exactly 2 of the 10 calculators will be returned for refund or replacement within a 2-year period.

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Key Concepts

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

Probability Mass Function

In the study of statistics, the probability mass function (PMF) is essential for understanding how probabilities distribute across discrete random variables. Here, we apply it to binomial distribution scenarios, like the calculator malfunctioning problem.
The PMF for a binomial distribution can be described using the formula:

• \[P(x) = \binom{n}{x} p^x (1-p)^{n-x}\]

where \(n\) is the number of trials, \(x\) is the number of successful outcomes (malfunctions in this case), \(p\) is the probability of a single success, and \(\binom{n}{x}\) is the binomial coefficient calculated as \(\frac{n!}{x!(n-x)!}\).
This formula helps elucidate the probability of each possible number of malfunctions occurring in a set number of trials.

Mean and Standard Deviation

Understanding the mean and standard deviation is crucial to gaining insights into a binomial distribution. These concepts give information about the distribution's center and spread.
The mean, denoted by \(\mu\), represents the average or expected number of successes. For a binomial distribution, it is calculated as:

• \[ \mu = np \]

where \(n\) is the number of trials and \(p\) is the probability of success in a single trial. For our example with calculator malfunctions, \(n = 10\) and \(p = 0.05\), giving a mean of 0.5.
The standard deviation, represented by \(\sigma\), indicates the variability or dispersion of the distribution. It is calculated by the formula:

• \[ \sigma = \sqrt{np(1-p)} \]

This calculation provides the standard deviation, which in turn helps assess how much the number of malfunctions can vary from the mean.

Binomial Probability Table

A binomial probability table is a helpful tool that displays the probabilities of different numbers of successes in a simple tabulated manner. It's especially useful when dealing with multiple trials, as seen in the distribution of calculator malfunctions.
To utilize a binomial probability table, calculate the probability for each possible outcome of \(x\) using the PMF and record these values. For instance, if you have 10 trials, as in our scenario, you'll get a probability value for each \(x\) from 0 to 10.
The table visually summarizes all the probabilities, making it easy to find specific probabilities, such as the probability of exactly 2 calculators malfunctioning.

Consumer Satisfaction

Consumer satisfaction is a key focus in any business, impacting customer loyalty and brand reputation. In the context of Johnson Electronics, reducing the likelihood of calculator malfunctions directly corresponds to increased consumer satisfaction.
By understanding the statistics around malfunctions, such as the mean number of defects expected and how they are distributed, the company can adopt proactive measures to reduce issues. Statistics like these inform decision-making processes, helping the company improve product quality and customer experience.
For example, knowing that 5% of calculators malfunction allows management to target quality checks and address potential issues before reaching consumers. This direct link between statistical insight and practical action steps significantly enhances overall consumer satisfaction.