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Chapter 6: Problem 85

Aircraft engines Engineers define reliability as the probability that an item will perform its function under specific conditions for a specific period of time. A certain model of aircraft engine is designed so that each engine has probability 0.999 of performing properly for an hour of flight. Company engineers test an \(\mathrm{SRS}\) of 350 engines of this model. Let X = the number that operate for an hour without failure. (a) Explain why \(\mathrm{X}\) is a binomial random variable. (b) Find the mean and standard deviation of \(\mathrm{X}\) . Interpret each value in context. (c) Two engines failed the test. Are you convinced that this model of engine is less reliable than it's supposed to be? Compute \(P(\mathrm{X} \leq 348)\) and use the result to justify your answer.

Short Answer

Expert verified
X is binomial. Mean = 349.65, SD = 0.59. P(X 鈮� 348) = 0.067, not significant.

Step by step solution

01

Define a Binomial Random Variable

A binomial random variable arises in a scenario where there are a fixed number of independent trials, each with only two outcomes: success or failure. In this exercise, the random variable \(X\) represents the number of engines that operate without failure among the 350 engines tested. Each engine either performs properly (success) or fails (failure), with the success probability for each being 0.999. Hence, \(X\) is a binomial random variable with \(n = 350\) trials and success probability \(p = 0.999\).
02

Calculate Mean and Standard Deviation

The mean of a binomial distribution is given by \(\mu = np\) and the standard deviation is given by \(\sigma = \sqrt{np(1-p)}\). Substituting the known values, we have:\[\mu = 350 \times 0.999 = 349.65\] and \[\sigma = \sqrt{350 \times 0.999 \times 0.001} = \sqrt{0.34965} \approx 0.59\].This means on average, 349.65 engines operate without failure, and there is a standard deviation of about 0.59 engines from this mean value.
03

Calculate Probability for Two Failures

To determine if the engine model is less reliable than supposed, we calculate the probability that 348 or fewer engines operate without failure, equivalent to two or more failures. Using the binomial distribution, we compute \(P(X \leq 348)\). With a binomial distribution table or calculator, \[P(X \leq 348) \approx 0.067\].This probability indicates the chance that 348 or fewer engines out of 350 operate properly. Since this probability is greater than 0.05, the event of 2 or more failures is not statistically significant, suggesting that the model is as reliable as expected.

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics that quantifies the likelihood of certain events happening. In the context of a **binomial random variable** like in our aircraft engine example, probability helps us to understand how often a specific outcome occurs out of multiple trials.

For instance, when assessing whether the engines perform properly, probability is used to predict how many out of 350 engines will succeed in operation. Here, each engine's performance for one hour is a trial, and the engine either succeeds or fails. Given the reliability is 0.999, this high probability means that there is only a very small chance (0.001) of any single engine failing during that hour.
  • When dealing with binomial variables:
    - The probability of success remains constant for each trial.
  • Outcomes are independent 鈥� one trial's outcome does not affect another.
By calculating probabilities in this manner, we can predict performance over a vast number of trials accurately.
Mean and Standard Deviation
The mean and standard deviation are statistical measures that describe the behavior of a binomial distribution. Understanding these can give us insights into the expected performance of the engines over many trials.

- **Mean (\(\mu\))**: This is the expected value or average outcome, calculated by multiplying the number of trial (\(n\)) by the probability of success (\(p\)). In our case, \(\mu = 350 \times 0.999 = 349.65\). This means we expect that on average, approximately 349.65 engines out of 350 will perform correctly within one hour.

- **Standard Deviation (\(\sigma\))**: It quantifies the variability of the random variable around the mean. Given by \(\sigma = \sqrt{np(1-p)}\), it shows how spread out the outcomes are. Here, \(\sigma = \sqrt{350 \times 0.999 \times 0.001} \approx 0.59\). This suggests that the number of operating engines is typically close to the mean, but there may be some slight variations around this average.

Understanding these measures allows us to better evaluate any surprising outcomes, like the two failures recorded.
Reliability
Reliability in engineering context is crucial as it measures the probability that a system functions without failure under stated conditions for a period of time. With this aircraft engine model, reliability is 0.999, indicating a high expectation of performance.

When assessing reliability, engineers might look at scenarios where an engine operates or fails, and compare it with predicted outcomes. In this case, even though two engines failed, calculating \(P(X \leq 348) \approx 0.067\) helps determine whether such failure rates exceed expectations.
  • A significance level often used is 0.05.
  • If the probability of observing such a failure or more is above this, it's not considered unusual.
Since the calculated probability (0.067) is higher than 0.05, the occurrence of two failures is not significant enough to say the model's reliability has decreased. This suggests the engine still meets its high reliability standards.

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