Question

it is known that 64% of americans personally worry a great deal about federal spending and the budget deficit. which of the following needs to be satisfied in order to be able to use the central limit theorem to describe the sampling distribution of the proportion of americans who personally worry a great deal about federal spending and the budget deficit?

1. the sample needs to be random.
2. if sampling without replacement, the sample size needs to be less than 10% of the population size.
3. the sample size should not be less than 50.
4. the expected numbers of those who do and do not personally worry a great deal about federal spending and the budget deficit need to both be at least 10.

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Explanation:

To apply the Central Limit Theorem to a sampling distribution of a proportion, the following conditions must be met:

1. The sample must be randomly selected to avoid bias.
2. When sampling without replacement, the sample size should be less than 10% of the population to ensure independence of observations.
3. The success-failure condition: the expected number of successes ($np$) and failures ($n(1-p)$) must both be at least 10, where $p$ is the population proportion.

A minimum sample size of 50 is not a required condition for the CLT for proportions.

Answer:

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