| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B. The distribution of the sample proportion will be less spread out. Explanation: When the sample size increases, the standard deviation of the sampling distribution of the sample proportion decreases. This means that the distribution of the sample proportion will be less spread out, as the increased sample size provides more precise estimates of the population proportion. In this case, increasing the sample size from 200 to 2,000 will result in a more concentrated distribution of the sample proportion, making option B the correct answer. Option A is incorrect because a larger sample size will not make the distribution more spread out. Option C is also incorrect because the spread of the distribution will be affected by the increased sample size. Option D is incorrect because the distribution of the sample proportion will more closely resemble a normal distribution, not a binomial distribution, as the sample size increases. Therefore, the correct answer is B. The distribution of the sample proportion will be less spread out. |
B |
| deepseekr1 | Answer: B. The distribution of the sample proportion will be less spread out. **Explanation:** When the sample size increases from 200 to 2,000, the standard error of the sample proportion (calculated as \(\sqrt{\frac{p(1-p)}{n}}\)) decreases because it is inversely proportional to the square root of the sample size. A smaller standard error means the sampling distribution becomes narrower (less spread out). This aligns with option B. Options A and C are incorrect because the spread decreases, not increases or remains the same. Option D is incorrect because a larger sample size strengthens the normal approximation (Central Limit Theorem), making the distribution less like the discrete binomial distribution and more like a continuous normal distribution. |
B |
| gpto1 | B | B |
| GPT-4o | B. The distribution of the sample proportion will be less spread out. Increasing the sample size reduces the standard error of the sample proportion, making the distribution of the sample proportion less spread out. |
B |
| llama3.1-70B-Instruct | B |
B |
| llama3.1-405B-Instruct | B. The distribution of the sample proportion will be less spread out. Reasoning: When the sample size increases, the standard deviation of the sampling distribution of the sample proportion decreases. This is because the formula for the standard deviation of the sampling distribution of the sample proportion is: σ_p = sqrt(p(1-p)/n) where σ_p is the standard deviation, p is the population proportion, and n is the sample size. As the sample size (n) increases, the denominator of the formula increases, which results in a smaller standard deviation. A smaller standard deviation means that the distribution of the sample proportion will be less spread out. Therefore, increasing the sample size from 200 to 2,000 will result in a less spread out distribution of the sample proportion, making option B the correct answer. |
B |
| DeepSeek-Coder-V2 | B | B |
| Qwen2-72B-Instruct | B. The distribution of the sample proportion will be less spread out. | B |
| baichuan4 | B. The distribution of the sample proportion will be less spread out. | B |
| doubao | B. The distribution of the sample proportion will be less spread out. Increasing the sample size generally leads to a narrower and more concentrated distribution of the sample proportion, reducing the spread. | B |
| minimax | B. The distribution of the sample proportion will be less spread out. | B |
| kimi | B. The distribution of the sample proportion will be less spread out. Increasing the sample size from 200 to 2,000 will make the distribution of the sample proportion more precise and less variable. This is because a larger sample size reduces the standard error of the sample proportion, which is the measure of how much the sample proportion can be expected to vary from the true population proportion. The standard error of the sample proportion is given by the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the true proportion of the population and \( n \) is the sample size. As \( n \) increases, \( SE \) decreases, leading to a less spread out distribution of the sample proportion. |
B |
| wenxinyiyan | B Explanation: When increasing the sample size, the distribution of the sample proportion will tend to be less spread out. This is because as the sample size increases, the variability of the sample proportion decreases, making the distribution narrower and more tightly centered around the true population proportion. This concept is related to the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample mean (or in this case, the sample proportion) approaches a normal distribution with a mean equal to the population mean (or proportion) and a standard deviation that decreases as the sample size increases. Therefore, increasing the sample size from 200 to 2,000 will result in a distribution of the sample proportion that is less spread out. |
B |
| zhipu | B. The distribution of the sample proportion will be less spread out. Increasing the sample size generally reduces the standard error of the sample proportion, leading to a distribution that is less spread out. This is because a larger sample size provides more information and reduces the variability in the estimate of the population proportion. |
B |
| gpto1 | B | B |
| deepseekv3 | B. The distribution of the sample proportion will be less spread out. Explanation: Increasing the sample size from 200 to 2,000 will reduce the standard error of the sample proportion. The standard error is inversely proportional to the square root of the sample size, so a larger sample size leads to a smaller standard error, making the distribution of the sample proportion less spread out. |
B |