A manufacturer tests components for failure during a short reliability test. For component , define where . Assume that are IID. An engineer tests components and argues that the sampling distribution of should be approximately Gaussian because . Which assessment is most accurate?

Question

A manufacturer tests components for failure during a short reliability test. For component   , define    where   . Assume that    are IID. An engineer tests    components and argues that the sampling distribution of    should be approximately Gaussian because   . Which assessment is most accurate?

Accepted Answer

The engineer is incorrect because the probability of observing no failures is   , and so the sampling distribution of    is heavily concentrated at zero and is poorly approximated by a Gaussian distribution.

Answer

The engineer is correct because    is sufficiently large for the Central Limit Theorem whenever the population variance is finite.

Answer

The engineer is correct because    is small, and therefore the sampling distribution of    must be approximately Gaussian.

Answer

The engineer is incorrect because the Central Limit Theorem cannot be applied to Bernoulli random variables, since they are discrete.

Answer

The engineer is incorrect because a Gaussian approximation to the sampling distribution of a sample mean of Bernoulli random variables is valid only when   .

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A manufacturer tests components for failure during a short reliability test. For...

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