Question

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 SRS of 350 engines of this model. Let X = the number that operate for an hour without failure. Explain why X is a binomial random variable.

Answer 1

X is a binomial random variable because the testing of aircraft engines involves a fixed number of independent trials with a consistent probability of success for each trial and only two possible outcomes.

Step-by-step explanation:

The student's question involves understanding why the variable X is a binomial random variable. X represents the number of aircraft engines that will operate for an hour without failure. We know this is a binomial situation because:

• There are a fixed number of trials (tests on 350 engines).
• Each trial has only two possible outcomes: an engine either functions properly for an hour or it does not.
• The probability of an engine functioning properly for an hour is consistent across all trials at 0.999.
• Each engine's performance is independent of the others'.

X therefore follows a binomial distribution with parameters n = 350 (the number of trials) and p = 0.999 (the probability of success on each trial).

Answer 2

X is a binomial random variable because the testing of each engine is an independent trial with two outcomes - success or failure - and each trial has the same probability of success.

Step-by-step explanation:

X is considered a binomial random variable because it fulfills the criteria for a binomial distribution. This includes the scenario where there are a fixed number of trials (350 engine tests), each trial is independent, there are only two possible outcomes (an engine performs properly for an hour, or it doesn't), and the probability of success (engine performing without failure) is the same for each trial (0.999).

Therefore, in the context of the student's question, we can utilize the binomial distribution to model the random variable X, which represents the number of engines operating for an hour without failure during the test.