An engineer has IID Gaussian observations where both and are unknown. The engineer makes the following argument:

"The standardised sample mean is standard normal. Therefore, I can replace the unknown by the sample standard deviation and obtain the exact confidence interval Afterall, since the chi-squared confidence interval for is exact for a Gaussian population, this substitution must also be exact."

Which assessment is correct?

Question

An engineer has IID Gaussian observations where both and are unknown. The engineer makes the following argument:

"The standardised sample mean is standard normal. Therefore, I can replace the unknown by the sample standard deviation and obtain the exact confidence interval Afterall, since the chi-squared confidence interval for is exact for a Gaussian population, this substitution must also be exact."

Which assessment is correct?

Accepted Answer

The argument is incorrect because replacing    by the random variable    changes the sampling distribution of the standardised quantity; the resulting standardised quantity follows a Student's    distribution, not a standard normal distribution.

Answer

The argument is correct because Gaussian observations ensure that every standardised estimator has a standard normal distribution.

Answer

The argument is incorrect because    and    are biased estimators of    and   , respectively.

Answer

The argument is correct because replacing any unknown population parameter by an unbiased estimator preserves the sampling distribution of the standardised quantity.

Answer

The argument is incorrect because the chi-squared confidence interval for    is itself only a large-sample approximation.

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An engineer has IID Gaussian observations where both and are unknown. T...

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