The basis for using a normal probability distribution to approximate the sampling distribution of and is _____.

Question

The basis for using a normal probability distribution to approximate the sampling distribution of  and  is _____.

BlackTom AI · Accepted Answer

The question asks about the basis for using a normal probability distribution to approximate the sampling distributions of x-bar and p-hat.
Option 1: the empirical rule. This describes approximate percentages in a normal distribution (68-95-99.7%), but it does not justify why we can use a normal model for sampling distributions. It’s a descriptive rule, not a foundational justification for normality of sampling distributions.
Option 2: Bayes’ theorem. Bayes’ theorem relates prior and posterior probabilities and is central to Bayesian inference, not to why sampling distributions of sample means or proportions are approximately normal. It does not provide the basis for the normal approximation in frequentist sampling.
Option 3: Chebyshev’s theorem. Chebyshev’s inequality gives bounds on the probability that a random variable deviates from its mean for any distribution with finite variance, but it does not guarantee or explain why the sampling distribution is normal. It’s a general bound, not a justification for normality.
Option 4: the central limit theorem. The CLT states that, under mild conditions, the sampling distribution of the sample mean (and, with certain caveats, the sample proportion) tends toward a normal distribution as the sample size grows, regardless of the population distribution. This provides the theoretical foundation for using a normal model to approximate these sampling distributions, especially for larger samples.

The basis for using a normal probability distribution to approximate the sampling distribution of and is _____.单项选择题

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