金声玉亮2.0模型评测报告

问题

There is a single choice question about high school statistics. Answer the question by replying A, B, C or D.
Question: Researchers are conducting an experiment using a significance level of 0.05. The null hypothesis is, in fact, false. If they modify their experiment to use twice as many experimental units for each treatment, which of the following would be true?
A. The probability of a Type I error and the probability of a Type II error would both decrease.
B. The probability of a Type I error and the power would both increase.
C. The probability of a Type II error and the power would both increase.
D. The probability of a Type I error would stay the same and the power would increase.
Answer:

答案

模型	模型答案	正确答案
金声玉亮2.0	D. The probability of a Type I error would stay the same and the power would increase. Explanation: When the sample size is increased, the power of the test increases, meaning that the probability of correctly rejecting the null hypothesis when it is false increases. However, the probability of a Type I error, which is the probability of rejecting the null hypothesis when it is true, remains the same and is determined by the significance level (0.05 in this case). Therefore, the correct answer is D. Note: The other options are incorrect because: A. The probability of a Type I error remains the same, not decreases. B. The probability of a Type I error does not increase. C. The probability of a Type II error decreases, not increases, as the power increases.	D
deepseekr1	The null hypothesis is false, and increasing the sample size affects Type II error (β) and power (1−β). A larger sample size reduces β (lower Type II error probability) and increases power. The Type I error (α) remains fixed at the chosen significance level (0.05), as it is set by the researcher and independent of sample size. Answer: D. The probability of a Type I error would stay the same and the power would increase.	D
gpto1	D	D
GPT-4o	D. The probability of a Type I error would stay the same and the power would increase.	D
llama3.1-70B-Instruct	D	D
llama3.1-405B-Instruct	D. The probability of a Type I error would stay the same and the power would increase. Explanation: * The probability of a Type I error (α) is set by the researchers and is not affected by the sample size. Therefore, it would remain the same (0.05 in this case). * Increasing the sample size (using twice as many experimental units for each treatment) would increase the power of the test. Power is the probability of rejecting the null hypothesis when it is false, and a larger sample size provides more information and increases the likelihood of detecting a statistically significant effect. So, the correct answer is D. The probability of a Type I error would stay the same and the power would increase.	D
DeepSeek-Coder-V2	D. The probability of a Type I error would stay the same and the power would increase.	D
Qwen2-72B-Instruct	D. The probability of a Type I error would stay the same and the power would increase.	D
baichuan4	D	D
doubao	D Increasing the number of experimental units generally leads to a decrease in the probability of a Type II error and an increase in power. The significance level (0.05) determines the probability of a Type I error and remains the same if not changed explicitly. Power is the probability of correctly rejecting the null hypothesis when it is false, and it typically increases with more experimental units.	D
minimax	D. The probability of a Type I error would stay the same and the power would increase.	D
kimi	D. The probability of a Type I error would stay the same and the power would increase. Explanation: - The significance level (α) is set at 0.05, which represents the probability of a Type I error (rejecting the null hypothesis when it is actually true). This is a fixed value set before the experiment and does not change with the number of experimental units. - The power of a test is the probability of correctly rejecting a false null hypothesis (1 - β). It is affected by the sample size, among other factors. Increasing the number of experimental units (sample size) increases the power of the test because it reduces the standard error of the estimate, making the test more sensitive to detect a real effect if one exists. - Type II error (β) is the probability of failing to reject a false null hypothesis. Increasing the sample size will decrease the chance of making a Type II error, as the test becomes more powerful.	D
wenxinyiyan	D Explanation: * Type I error: Rejecting the null hypothesis when it is actually true. * Type II error: Failing to reject the null hypothesis when it is actually false. * Power: The probability of correctly rejecting the null hypothesis when it is false. In this scenario, the significance level (α) is 0.05, which means the probability of a Type I error is set at 0.05. Changing the number of experimental units does not affect the significance level, so the probability of a Type I error remains the same. However, increasing the sample size (in this case, by using twice as many experimental units for each treatment) typically increases the power of the test. With a larger sample size, it becomes easier to detect differences between groups, making it more likely to reject the null hypothesis when it is false. Therefore, the probability of a Type I error would stay the same (at 0.05), and the power would increase. This corresponds to option D.	D
zhipu	D. The probability of a Type I error would stay the same and the power would increase. Explanation: The significance level (α) of 0.05 is fixed and does not change with sample size. Therefore, the probability of a Type I error (rejecting a true null hypothesis) remains the same. However, increasing the sample size generally increases the power of the test, which is the probability of correctly rejecting a false null hypothesis. Hence, the power would increase. This makes option D the correct choice.	D
gpto1	D	D
deepseekv3	D. The probability of a Type I error would stay the same and the power would increase. Explanation: - Type I error (false positive) is the probability of rejecting the null hypothesis when it is true. It is determined by the significance level (0.05 in this case), which is chosen by the researcher and does not change with sample size. - Type II error (false negative) is the probability of failing to reject the null hypothesis when it is false. Increasing the sample size reduces the probability of a Type II error. - Power is the probability of correctly rejecting the null hypothesis when it is false. Since power = 1 - (probability of Type II error), increasing the sample size also increases power. Thus, when the sample size is doubled, the probability of a Type I error remains the same (as it is fixed by the significance level), while the power increases. Therefore, the correct answer is D.	D

模型

模型答案

正确答案

金声玉亮2.0

D. The probability of a Type I error would stay the same and the power would increase.

Explanation:

When the sample size is increased, the power of the test increases, meaning that the probability of correctly rejecting the null hypothesis when it is false increases. However, the probability of a Type I error, which is the probability of rejecting the null hypothesis when it is true, remains the same and is determined by the significance level (0.05 in this case). Therefore, the correct answer is D.

Note: The other options are incorrect because:

A. The probability of a Type I error remains the same, not decreases.

B. The probability of a Type I error does not increase.

C. The probability of a Type II error decreases, not increases, as the power increases.

deepseekr1

The null hypothesis is false, and increasing the sample size affects Type II error (β) and power (1−β). A larger sample size reduces β (lower Type II error probability) and increases power. The Type I error (α) remains fixed at the chosen significance level (0.05), as it is set by the researcher and independent of sample size.

**Answer:** D. The probability of a Type I error would stay the same and the power would increase.

gpto1

GPT-4o

D. The probability of a Type I error would stay the same and the power would increase.

llama3.1-70B-Instruct

llama3.1-405B-Instruct

D. The probability of a Type I error would stay the same and the power would increase.

Explanation:

* The probability of a Type I error (α) is set by the researchers and is not affected by the sample size. Therefore, it would remain the same (0.05 in this case).
* Increasing the sample size (using twice as many experimental units for each treatment) would increase the power of the test. Power is the probability of rejecting the null hypothesis when it is false, and a larger sample size provides more information and increases the likelihood of detecting a statistically significant effect.

So, the correct answer is D. The probability of a Type I error would stay the same and the power would increase.

DeepSeek-Coder-V2

D. The probability of a Type I error would stay the same and the power would increase.

Qwen2-72B-Instruct

D. The probability of a Type I error would stay the same and the power would increase.

baichuan4

doubao

D

Increasing the number of experimental units generally leads to a decrease in the probability of a Type II error and an increase in power. The significance level (0.05) determines the probability of a Type I error and remains the same if not changed explicitly. Power is the probability of correctly rejecting the null hypothesis when it is false, and it typically increases with more experimental units.

minimax

D. The probability of a Type I error would stay the same and the power would increase.

kimi

D. The probability of a Type I error would stay the same and the power would increase.

Explanation:
- The significance level (α) is set at 0.05, which represents the probability of a Type I error (rejecting the null hypothesis when it is actually true). This is a fixed value set before the experiment and does not change with the number of experimental units.
- The power of a test is the probability of correctly rejecting a false null hypothesis (1 - β). It is affected by the sample size, among other factors. Increasing the number of experimental units (sample size) increases the power of the test because it reduces the standard error of the estimate, making the test more sensitive to detect a real effect if one exists.
- Type II error (β) is the probability of failing to reject a false null hypothesis. Increasing the sample size will decrease the chance of making a Type II error, as the test becomes more powerful.

wenxinyiyan

D

Explanation:

* Type I error: Rejecting the null hypothesis when it is actually true.
* Type II error: Failing to reject the null hypothesis when it is actually false.
* Power: The probability of correctly rejecting the null hypothesis when it is false.

In this scenario, the significance level (α) is 0.05, which means the probability of a Type I error is set at 0.05. Changing the number of experimental units does not affect the significance level, so the probability of a Type I error remains the same.

However, increasing the sample size (in this case, by using twice as many experimental units for each treatment) typically increases the power of the test. With a larger sample size, it becomes easier to detect differences between groups, making it more likely to reject the null hypothesis when it is false.

Therefore, the probability of a Type I error would stay the same (at 0.05), and the power would increase. This corresponds to option D.

zhipu

D. The probability of a Type I error would stay the same and the power would increase.

Explanation: The significance level (α) of 0.05 is fixed and does not change with sample size. Therefore, the probability of a Type I error (rejecting a true null hypothesis) remains the same. However, increasing the sample size generally increases the power of the test, which is the probability of correctly rejecting a false null hypothesis. Hence, the power would increase. This makes option D the correct choice.

gpto1

deepseekv3

D. The probability of a Type I error would stay the same and the power would increase.

Explanation:
- **Type I error** (false positive) is the probability of rejecting the null hypothesis when it is true. It is determined by the significance level (0.05 in this case), which is chosen by the researcher and does not change with sample size.
- **Type II error** (false negative) is the probability of failing to reject the null hypothesis when it is false. Increasing the sample size reduces the probability of a Type II error.
- **Power** is the probability of correctly rejecting the null hypothesis when it is false. Since power = 1 - (probability of Type II error), increasing the sample size also increases power.

Thus, when the sample size is doubled, the probability of a Type I error remains the same (as it is fixed by the significance level), while the power increases. Therefore, the correct answer is D.