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: Suppose you did 10 independent tests of the form H0: μ = 25 versus Ha: μ < 25, each at the α = 0.05 significance level. What is the probability of committing a Type I error and incorrectly rejecting a true H0 with at least one of the 10 tests?
A. 0.05
B. 0.40
C. 0.50
D. 0.60
Answer:

答案

模型	模型答案	正确答案
金声玉亮2.0	The correct answer is B. 0.40. When you perform multiple independent tests, the probability of committing a Type I error (incorrectly rejecting a true null hypothesis) increases. In this case, you are performing 10 independent tests, each at a significance level of α = 0.05. The probability of not committing a Type I error on a single test is 1 - α = 1 - 0.05 = 0.95. Since the tests are independent, the probability of not committing a Type I error on all 10 tests is (0.95)^10 ≈ 0.60. Therefore, the probability of committing a Type I error on at least one of the 10 tests is 1 - 0.60 = 0.40, which is option B.	B
deepseekr1	The problem involves calculating the probability of committing at least one Type I error in 10 independent hypothesis tests, each conducted at a 5% significance level (α = 0.05). A Type I error occurs when a true null hypothesis (H₀: μ = 25) is incorrectly rejected. For a single test, the probability of not committing a Type I error is \(1 - \alpha = 0.95\). Since the tests are independent, the probability of no Type I errors in all 10 tests is \(0.95^{10}\). The probability of at least one Type I error is then: \[ 1 - 0.95^{10}. \] Calculating \(0.95^{10}\) step-by-step: \[ 0.95^{10} \approx 0.5987. \] Thus: \[ 1 - 0.5987 = 0.4013 \approx 0.40. \] This matches option B (0.40), which represents the probability of at least one Type I error across the 10 tests. Answer: B	B
gpto1	B	B
GPT-4o	To answer this question, we need to understand the probability of committing a Type I error in at least one of the 10 independent tests. A Type I error occurs when we incorrectly reject a true null hypothesis (H0). The probability of a Type I error for a single test is given as α = 0.05. The probability of not committing a Type I error in a single test is therefore 1 - 0.05 = 0.95. For 10 independent tests, the probability of not committing a Type I error in all 10 tests is 0.95^10. Thus, the probability of committing at least one Type I error in 10 tests is: 1 - (0.95^10). Calculating this: 0.95^10 ≈ 0.5987. 1 - 0.5987 ≈ 0.4013. So, the probability of committing a Type I error in at least one of the 10 tests is approximately 0.40. Answer: B. 0.40	B
llama3.1-70B-Instruct	B	B
llama3.1-405B-Instruct	The correct answer is B. 0.40. When you perform multiple independent tests, the probability of committing a Type I error (incorrectly rejecting a true null hypothesis) increases. In this case, you are performing 10 independent tests, each at a significance level of α = 0.05. The probability of not committing a Type I error on a single test is 1 - α = 1 - 0.05 = 0.95. Since the tests are independent, the probability of not committing a Type I error on all 10 tests is (0.95)^10 ≈ 0.60. Therefore, the probability of committing a Type I error on at least one of the 10 tests is 1 - 0.60 = 0.40, which is option B.	B
DeepSeek-Coder-V2	A	B
Qwen2-72B-Instruct	B. 0.40	B
baichuan4	B. 0.40	B
doubao	C The probability of not committing a Type I error in a single test is 1 - 0.05 = 0.95. The probability of not committing a Type I error in all 10 tests is 0.95^10 ≈ 0.60. So the probability of committing a Type I error with at least one of the 10 tests is 1 - 0.60 = 0.40.	B
minimax	The probability of committing a Type I error in a single test is α, which is given as 0.05. If the tests are independent, the probability of not committing a Type I error in a single test is (1 - α), which is (1 - 0.05) = 0.95. The probability of not committing a Type I error in all 10 tests is (1 - α)^10, which is 0.95^10. To find the probability of committing at least one Type I error in any of the 10 tests, we subtract the probability of not committing a Type I error in all tests from 1: 1 - (1 - α)^10 = 1 - 0.95^10 ≈ 1 - 0.5987 ≈ 0.4013 This value is closest to 0.40, so the answer is: B. 0.40	B
kimi	B. 0.40	B
wenxinyiyan	B Explanation: In each individual test, the probability of committing a Type I error (i.e., rejecting a true H0) is α = 0.05. Since the 10 tests are independent, the probability of not committing a Type I error in any of the 10 tests is (1 - 0.05)^10. Therefore, the probability of committing a Type I error in at least one of the 10 tests is 1 - (1 - 0.05)^10 ≈ 1 - 0.5987 ≈ 0.4013, which is closest to 0.40 among the given options. Note: The exact calculation would give a value slightly different from 0.40, but since the options are rounded to two decimal places, we choose the closest option.	B
zhipu	To determine the probability of committing a Type I error (incorrectly rejecting a true null hypothesis, \( H_0 \)) at least once in 10 independent tests, we need to consider the complement of not committing a Type I error in any of the tests. The probability of not committing a Type I error in a single test is \( 1 - \alpha \). Given \( \alpha = 0.05 \), the probability of not committing a Type I error in one test is: \[ 1 - 0.05 = 0.95 \] Since the tests are independent, the probability of not committing a Type I error in all 10 tests is: \[ (0.95)^{10} \] Now, the probability of committing a Type I error at least once in the 10 tests is the complement of not committing a Type I error in any of the tests: \[ 1 - (0.95)^{10} \] Let's calculate \( (0.95)^{10} \): \[ (0.95)^{10} \approx 0.5987 \] Therefore, the probability of committing a Type I error at least once in 10 tests is: \[ 1 - 0.5987 \approx 0.4013 \] The closest option to 0.4013 is: B. 0.40 Answer: B	B
gpto1	B	B
deepseekv3	To solve this problem, we need to calculate the probability of committing at least one Type I error when performing 10 independent tests, each at the α = 0.05 significance level. ### Step 1: Understand the probability of not committing a Type I error in a single test The probability of not committing a Type I error in a single test is: \[ 1 - \alpha = 1 - 0.05 = 0.95 \] ### Step 2: Calculate the probability of not committing a Type I error in all 10 tests Since the tests are independent, the probability of not committing a Type I error in all 10 tests is: \[ (1 - \alpha)^{10} = (0.95)^{10} \] ### Step 3: Calculate the probability of committing at least one Type I error The probability of committing at least one Type I error is the complement of the probability of not committing any Type I error in all 10 tests. Thus: \[ P(\text{at least one Type I error}) = 1 - (0.95)^{10} \] ### Step 4: Compute the numerical value Using a calculator: \[ (0.95)^{10} \approx 0.5987 \] So: \[ P(\text{at least one Type I error}) = 1 - 0.5987 = 0.4013 \] ### Step 5: Match the result to the options The closest option to 0.4013 is B. 0.40. Answer: B	B

模型

模型答案

正确答案

金声玉亮2.0

The correct answer is B. 0.40.

When you perform multiple independent tests, the probability of committing a Type I error (incorrectly rejecting a true null hypothesis) increases. In this case, you are performing 10 independent tests, each at a significance level of α = 0.05.

The probability of not committing a Type I error on a single test is 1 - α = 1 - 0.05 = 0.95.

Since the tests are independent, the probability of not committing a Type I error on all 10 tests is (0.95)^10 ≈ 0.60.

Therefore, the probability of committing a Type I error on at least one of the 10 tests is 1 - 0.60 = 0.40, which is option B.

deepseekr1

The problem involves calculating the probability of committing at least one Type I error in 10 independent hypothesis tests, each conducted at a 5% significance level (α = 0.05). A Type I error occurs when a true null hypothesis (H₀: μ = 25) is incorrectly rejected.

For a single test, the probability of **not** committing a Type I error is \(1 - \alpha = 0.95\). Since the tests are independent, the probability of **no** Type I errors in all 10 tests is \(0.95^{10}\). The probability of **at least one** Type I error is then:
\[
1 - 0.95^{10}.
\]

Calculating \(0.95^{10}\) step-by-step:
\[
0.95^{10} \approx 0.5987.
\]
Thus:
\[
1 - 0.5987 = 0.4013 \approx 0.40.
\]

This matches option **B** (0.40), which represents the probability of at least one Type I error across the 10 tests.

**Answer: B**

gpto1

GPT-4o

To answer this question, we need to understand the probability of committing a Type I error in at least one of the 10 independent tests.

A Type I error occurs when we incorrectly reject a true null hypothesis (H0). The probability of a Type I error for a single test is given as α = 0.05.

The probability of not committing a Type I error in a single test is therefore 1 - 0.05 = 0.95.

For 10 independent tests, the probability of not committing a Type I error in all 10 tests is 0.95^10.

Thus, the probability of committing at least one Type I error in 10 tests is:

1 - (0.95^10).

Calculating this:

0.95^10 ≈ 0.5987.

1 - 0.5987 ≈ 0.4013.

So, the probability of committing a Type I error in at least one of the 10 tests is approximately 0.40.

Answer: B. 0.40

llama3.1-70B-Instruct

llama3.1-405B-Instruct

DeepSeek-Coder-V2

Qwen2-72B-Instruct

B. 0.40

baichuan4

B. 0.40

doubao

C

The probability of not committing a Type I error in a single test is 1 - 0.05 = 0.95. The probability of not committing a Type I error in all 10 tests is 0.95^10 ≈ 0.60. So the probability of committing a Type I error with at least one of the 10 tests is 1 - 0.60 = 0.40.

minimax

The probability of committing a Type I error in a single test is α, which is given as 0.05. If the tests are independent, the probability of not committing a Type I error in a single test is (1 - α), which is (1 - 0.05) = 0.95.

The probability of not committing a Type I error in all 10 tests is (1 - α)^10, which is 0.95^10.

To find the probability of committing at least one Type I error in any of the 10 tests, we subtract the probability of not committing a Type I error in all tests from 1:

1 - (1 - α)^10 = 1 - 0.95^10 ≈ 1 - 0.5987 ≈ 0.4013

This value is closest to 0.40, so the answer is:

B. 0.40

kimi

B. 0.40

wenxinyiyan

B

Explanation:
In each individual test, the probability of committing a Type I error (i.e., rejecting a true H0) is α = 0.05. Since the 10 tests are independent, the probability of not committing a Type I error in any of the 10 tests is (1 - 0.05)^10.

Therefore, the probability of committing a Type I error in at least one of the 10 tests is 1 - (1 - 0.05)^10 ≈ 1 - 0.5987 ≈ 0.4013, which is closest to 0.40 among the given options.

Note: The exact calculation would give a value slightly different from 0.40, but since the options are rounded to two decimal places, we choose the closest option.

zhipu

To determine the probability of committing a Type I error (incorrectly rejecting a true null hypothesis, \( H_0 \)) at least once in 10 independent tests, we need to consider the complement of not committing a Type I error in any of the tests.

The probability of not committing a Type I error in a single test is \( 1 - \alpha \). Given \( \alpha = 0.05 \), the probability of not committing a Type I error in one test is:

\[ 1 - 0.05 = 0.95 \]

Since the tests are independent, the probability of not committing a Type I error in all 10 tests is:

\[ (0.95)^{10} \]

Now, the probability of committing a Type I error at least once in the 10 tests is the complement of not committing a Type I error in any of the tests:

\[ 1 - (0.95)^{10} \]

Let's calculate \( (0.95)^{10} \):

\[ (0.95)^{10} \approx 0.5987 \]

Therefore, the probability of committing a Type I error at least once in 10 tests is:

\[ 1 - 0.5987 \approx 0.4013 \]

The closest option to 0.4013 is:

B. 0.40

Answer: B

gpto1

deepseekv3

To solve this problem, we need to calculate the probability of committing at least one Type I error when performing 10 independent tests, each at the α = 0.05 significance level.

### Step 1: Understand the probability of not committing a Type I error in a single test
The probability of **not** committing a Type I error in a single test is:
\[
1 - \alpha = 1 - 0.05 = 0.95
\]

### Step 2: Calculate the probability of not committing a Type I error in all 10 tests
Since the tests are independent, the probability of **not** committing a Type I error in all 10 tests is:
\[
(1 - \alpha)^{10} = (0.95)^{10}
\]

### Step 3: Calculate the probability of committing at least one Type I error
The probability of committing **at least one** Type I error is the complement of the probability of not committing any Type I error in all 10 tests. Thus:
\[
P(\text{at least one Type I error}) = 1 - (0.95)^{10}
\]

### Step 4: Compute the numerical value
Using a calculator:
\[
(0.95)^{10} \approx 0.5987
\]
So:
\[
P(\text{at least one Type I error}) = 1 - 0.5987 = 0.4013
\]

### Step 5: Match the result to the options
The closest option to 0.4013 is **B. 0.40**.

Answer: **B**