State-wide surveys indicates that 14.5% of toddlers in Ney York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Is this a binomial scenario?单项选择题
A
Binomial
B
Not binomial

Question

State-wide surveys indicates that 14.5% of toddlers in Ney York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Is this a binomial scenario?单项选择题

A

Binomial

B

Not binomial

BlackTom AI · Accepted Answer

This situation meets the four key requirements for a binomial distribution: a fixed number of trials, only two possible outcomes per trial, independent trials (or approximately independent), and a constant probability of success.

1. Relevant knowledge points and concepts
- Binomial distribution: arises when there are n independent trials, each with two outcomes (success/failure) and constant success probability p. The random variable X counts the number of successes.
- Independence: when sampling from a very large population without replacement, trials can be treated as approximately independent if the sample is less than 10% of the population (10% condition).
- Parameters and moments: E[X] = np, Var(X) = np(1 − p).

2. Step-by-step reasoning
- Identify trials: selecting 20 toddlers ⇒ fixed number n = 20.
- Two outcomes: each toddler is either obese (success) or not obese (failure) ⇒ binary outcome.
- Constant probability: statewide obesity proportion is p = 0.145 for each randomly chosen toddler.
- Independence check: sample of 20 is negligible compared to the total number of toddlers in New York (much larger than 10 × 20 = 200), so sampling without replacement yields approximately independent trials.

3. Calculations (intermediate steps)
- n = 20, p = 0.145.
- Expected value: E[X] = n p = 20 × 0.145 = 2.90.
- Variance: Var(X) = n p (1 − p) = 20 × 0.145 × 0.855 = 2.4795.
- Standard deviation: sqrt(2.4795) ≈ 1.575.

4. Why the correct answer is correct
All four binomial conditions are satisfied: fixed n, binary outcomes, approximately independent trials (10% rule holds), and constant success probability p = 0.145. Therefore X ~ Binomial(n = 20, p = 0.145).

5. Why the other option is incorrect
"Not binomial" would apply only if one of the binomial conditions failed—e.g., variable sample size, more than two outcomes, non-constant p across trials, or strong dependence between trials. None of these issues are present here.

6. Conclusion
Because the scenario satisfies the binomial criteria, the correct choice is Binomial, which matches the provided answer.

State-wide surveys indicates that 14.5% of toddlers in Ney York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Is this a binomial scenario?单项选择题

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