Can you use a Normal model to approximate the distribution?

Question

BlackTom AI · Accepted Answer

The prompt presents a question and an answer, but the list of answer options is empty, which means there are no explicit alternatives to evaluate. We should address what is given and then connect it to the general concept.

First, the question asks: 'Can you use a Normal model to approximate the distribution?' In statistical practice, this typically refers to using a normal approximation for a binomial distribution, especially in the context of a hypothesis test or confidence interval.

Next, the provided answer states: 'Yes, because np and nq are both at least 10.' This reflects a common rule of thumb: when approximating a Binomial(n, p) with a Normal distribution, the counts np and n(1-p) (often denoted nq) should both be sufficiently large to justify the approximation. The typical thresholds are np ≥ 10 and n(1-p) ≥ 10, though some texts use slightly different cutoffs such as 5 or 10 depending on the desired accuracy.

Analyzing the reasoning behind this criterion:
- The normal approximation relies on the Central Limit Theorem, which says that as the sample size grows, the binomial distribution becomes more bell-shaped and closer to normal by nature of summing many independent Bernoulli trials.
- The conditions np ≥ 10 and n(1-p) ≥ 10 help ensure that the distribution is not too skewed and that the standard normal approximation provides a reasonable estimate for probabilities around the mean and in the tails.
- If either np < 10 or n(1-p) < 10, the binomial distribution can be skewed, and the normal approximation may be inaccurate; in those cases, one might use a different method (e.g., exact binomial calculations or a continuity correction) or a different distributional approximation.

Since no alternative options are provided, we cannot compare the proposed justification against other statements. If there were options, we would assess:
- Option that supports the normal approximation criterion (np and nq at least 10) as correct.
- Options that say the approximation is always valid regardless of np and nq, which would be incorrect because small np or nq leads to poor approximation.
- Options that mention different thresholds (like 5 or 15) and evaluate their reasonableness based on context and required accuracy.

In summary, the core justification behind the given answer aligns with the standard rule of thumb for using a Normal model to approximate a Binomial: ensure both np and nq are sufficiently large (commonly at least 10) to achieve a reasonable approximation, while being mindful that stricter or looser thresholds may be used depending on the specific tolerance for error.

BUSI_V 291 202 2024W2 Ch 07 Quiz Level 3

Can you use a Normal model to approximate the distribution?

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