| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B. If Ha is true, the probability of failing to reject H0 is 0.2. Explanation: The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true. In this case, the power of the test is 0.8, which means that if Ha: p = 0.7 is true, the probability of rejecting H0: p = 0.6 is 0.8. Therefore, the probability of failing to reject H0 when Ha is true is 1 - 0.8 = 0.2, which is the probability of a Type II error. The other options are not valid conclusions: A. The probability of committing a Type I error is not given in the problem. The power of the test only tells us about the probability of rejecting H0 when Ha is true, not about the probability of rejecting H0 when H0 is true. C. The probability of committing a Type II error is indeed 0.2, not 0.3. D. Since options A and C are not valid conclusions, option D is also incorrect. |
B |
| deepseekr1 | Answer: B The power of a test (0.8) represents the probability of correctly rejecting \( H_0 \) when \( H_a \) is true. Since power = \( 1 - \beta \), the Type II error probability (\( \beta \)) is \( 1 - 0.8 = 0.2 \). This directly supports **Option B**: If \( H_a \) is true, the probability of failing to reject \( H_0 \) (Type II error) is 0.2. - **Option A** is invalid because the Type I error probability (\( \alpha \)) is not provided or calculable from the given information. - **Option C** incorrectly states the Type II error probability as 0.3 instead of 0.2. - **Option D** is incorrect since only **B** is valid. Thus, the correct answer is **B**. |
B |
| gpto1 | B | B |
| GPT-4o | B. If Ha is true, the probability of failing to reject H0 is 0.2. Explanation: The power of a test, which is 0.8 in this case, is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. Therefore, if the alternative hypothesis (Ha: p = 0.7) is true, the probability of failing to reject the null hypothesis (H0: p = 0.6) is 1 - 0.8 = 0.2. This is known as the probability of a Type II error (β). |
B |
| llama3.1-70B-Instruct | B |
B |
| llama3.1-405B-Instruct | B. If Ha is true, the probability of failing to reject H0 is 0.2. Explanation: The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true. In this case, the power of the test for Ha: p = 0.7 is 0.8, which means that if the true population proportion is 0.7, there is an 80% chance of rejecting the null hypothesis H0: p = 0.6. The probability of failing to reject H0 when Ha is true is equal to 1 minus the power of the test, which is 1 - 0.8 = 0.2. Therefore, if Ha is true, the probability of failing to reject H0 is 0.2, which is option B. Option A is incorrect because the probability of committing a Type I error (rejecting a true null hypothesis) is not provided in the question. Option C is also incorrect because the probability of committing a Type II error (failing to reject a false null hypothesis) is 0.2, not 0.3. Option D is incorrect because only option B is a valid conclusion. |
B |
| DeepSeek-Coder-V2 | B | B |
| Qwen2-72B-Instruct | B. If Ha is true, the probability of failing to reject H0 is 0.2. | B |
| baichuan4 | D | B |
| doubao | B The power of the test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. If the power is 0.8, then the probability of failing to reject H0 when Ha is true is 1 - 0.8 = 0.2. We don't have enough information to determine the probability of a Type I error or the probability of a Type II error from the given information. |
B |
| minimax | B. If Ha is true, the probability of failing to reject H0 is 0.2. Explanation: The power of the test is the probability of correctly rejecting the null hypothesis (H0) when the alternative hypothesis (Ha) is true. In this case, the power is given as 0.8, which means that there is an 80% chance of correctly rejecting H0 when p = 0.7 (Ha is true). This implies that there is a 1 - 0.8 = 0.2 (or 20%) chance of failing to reject H0 when Ha is true, which is the probability of committing a Type II error (β). Options A and C are not correct because the power of the test does not directly tell us the probability of committing a Type I error (α), which is the probability of rejecting H0 when it is true. The probability of committing a Type I error is typically set by the significance level (α) of the test, which is not provided in the question. Similarly, the power of the test does not directly tell us the probability of committing a Type II error; rather, it is the complement of the power. |
B |
| kimi | B. If Ha is true, the probability of failing to reject H0 is 0.2. Explanation: - The power of a test is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. In this case, the power is given as 0.8, which means there is an 80% chance of correctly rejecting H0 when Ha is true. - The probability of failing to reject H0 when Ha is true is the complement of the power, which is 1 - power = 1 - 0.8 = 0.2. - The probability of committing a Type I error (rejecting a true null hypothesis) is given by the significance level of the test, which is not provided in the question. - The probability of committing a Type II error (failing to reject a false null hypothesis) is not directly given by the power of the test, but it is the complement of the power when Ha is true, which is 0.2, not 0.3 as stated in option C. Therefore, option B is the correct answer. |
B |
| wenxinyiyan | B Explanation: The power of a test is defined as the probability of rejecting the null hypothesis (H0) when the alternative hypothesis (Ha) is true. It is given as 0.8 in this question. A Type II error occurs when we fail to reject H0 when Ha is actually true. The probability of a Type II error is therefore the complement of the power of the test, i.e., 1 - 0.8 = 0.2. Option B directly states this: "If Ha is true, the probability of failing to reject H0 is 0.2." This is a valid conclusion based on the given information. Options A and C are incorrect because the question does not provide any information about the significance level of the test (which would be related to the probability of a Type I error) or the specific probability of a Type II error other than through the power of the test. |
B |
| zhipu | B Explanation: - The power of a test is defined as the probability of correctly rejecting the null hypothesis (H0) when the alternative hypothesis (Ha) is true. In this case, the power is given as 0.8. - The probability of failing to reject H0 when Ha is true (which is the probability of committing a Type II error, denoted as β) is 1 - power. Therefore, β = 1 - 0.8 = 0.2. Let's evaluate each option: A. The probability of committing a Type I error (α) is not provided in the question. The given information does not allow us to determine α. B. If Ha is true, the probability of failing to reject H0 is 0.2. This is correct based on the calculation above. C. The probability of committing a Type II error is 0.3. This is incorrect because the correct value is 0.2. D. All of the above are valid conclusions. This is incorrect because only option B is valid. Therefore, the correct answer is B. |
B |
| gpto1 | B | B |
| deepseekv3 | The correct answer is **B**. ### Explanation: - **Power of the test** is the probability of correctly rejecting \( H_0 \) when \( H_a \) is true. In this case, the power is 0.8 for \( H_a: p = 0.7 \). - **Type II error** occurs when we fail to reject \( H_0 \) when \( H_a \) is true. The probability of a Type II error is \( 1 - \text{power} \), which is \( 1 - 0.8 = 0.2 \). - **Type I error** occurs when we reject \( H_0 \) when \( H_0 \) is true. The probability of a Type I error is not provided in the question and cannot be determined from the given information. Now, let's evaluate the options: - **A:** The probability of committing a Type I error is not given, so this is not a valid conclusion. - **B:** If \( H_a \) is true, the probability of failing to reject \( H_0 \) is the probability of a Type II error, which is 0.2. This is a valid conclusion. - **C:** The probability of committing a Type II error is 0.2, not 0.3, so this is incorrect. - **D:** Since only option B is correct, this is not a valid conclusion. Thus, the correct answer is **B**. |
B |