Use the following information to answer questions 1-4.

Just before a referendum on a harmonized sales tax, a local newspaper polls 381 voters in an attempt to predict whether the proposition will pass. Suppose that the proposition actually has the support of 54 % of the voters. What is the probability the newspaper's sample will lead it to predict defeat? Be sure to verify that the assumptions and conditions necessary for your analysis are met.

One of the assumptions is met assuming the sample is a simple random sample. . Choose the best answer to complete the sentence.

Question

Use the following information to answer questions 1-4.

Just before a referendum on a harmonized sales tax, a local newspaper polls 381 voters in an attempt to predict whether the proposition will pass. Suppose that the proposition actually has the support of 54 % of the voters. What is the probability the newspaper's sample will lead it to predict defeat? Be sure to verify that the assumptions and conditions necessary for your analysis are met.

One of the assumptions is met assuming the sample is a simple random sample. . Choose the best answer to complete the sentence.

BlackTom AI · Accepted Answer

Question restatement: Use the information provided to answer questions 1-4. One of the questions asks to fill in the blank: 'One of the assumptions is met assuming the sample is a simple random sample. Choose the best answer to complete the sentence.' The answer field shows a single option: 'the sample is a simple random sample.'

Option analysis:
- The intended completion is asserting that the assumption being referred to is that the sample is drawn in a way that justifies the statistical methods used. If the sentence says 'One of the assumptions is met assuming the sample is a simple random sample,' then selecting 'the sample is a simple random sample' would logically complete the idea that the sampling method is the key assumption enabling the use of standard sampling arguments (independence, representative sampling, and the applicability of the binomial/normal approximations for a proportion).

Why this option is conceptually appropriate: 
- A fundamental prerequisite for many inferential procedures about proportions is that the data come from a simple random sample (SRS). An SRS ensures that each individual in the population has an equal chance of being included and that the observations are approximately independent. This underpins the validity of the sampling distribution of the sample proportion and the use of np and n(1-p) checks for normal approximation.
- The sentence as written frames a condition for the analysis; identifying the sampling method as the key assumption aligns with standard practice in hypothesis testing and confidence interval construction for proportions.

What would happen if this assumption were not met (why this option remains the best fit):
- If the sample were not a simple random sample (e.g., convenience sampling, voluntary response, or cluster sampling without proper design effects accounted for), the theoretical justification for using the binomial distribution (or its normal approximation) to model the sample proportion would be weakened or invalid. This would cast doubt on the calculated probability of predicting defeat and on any associated confidence statements, because the independence and identical distribution properties would be compromised.
- Without SRS, other complexities (design effects, weighting, clustering) would need to be explicitly accounted for, and the simple np and n(1-p) criteria could fail, making normal approximation unreliable.

Notes on the numerical probability (contextual, not required for choosing the option):
- If you were asked to compute the probability that the sample leads to predicting defeat given the true support is 54%, you would model X ~ Bin(n=381, p=0.54) for the number of supporters in the sample, and assess P(p_hat < 0.5) or P(X ≤ 0.5n) (with continuity correction as appropriate). This typically results in a probability around a few percent (roughly 5–6%), but the exact value depends on whether you use a binomial calculation or a normal approximation with continuity correction.

In summary, the option 'the sample is a simple random sample' appropriately identifies the key design assumption needed to justify the sampling distribution and subsequent inferences about the proportion, making it the best completion for the sentence in the given context.

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