Problem Analysis #35

K-EXAONE-236B-A23B · Run 1

120.97s
13,465 toks
Problem Statement

A population is normally distributed with mean mm and standard deviation 55. From this population, a random sample of size 3636 is taken, and a 99%99\% confidence interval for the population mean mm is constructed using the sample mean. The resulting confidence interval is 1.2≤m≤a1.2 \leq m \leq a. Find the value of aa. (Here, ZZ denotes a standard normal random variable, and use P(∣Z∣≤2.58)=0.99\text{P}(|Z| \leq 2.58) = 0.99.) [3 points]Compute 10a10a.\

Ground Truth
55
Execution Trace
Reasoning Process

55\boxed{55}

Reasoning Process

[THOUGHT] [PYTHON] from fractions import Fraction

z = Fraction(258, 100) # 2.58 = 258/100 sigma = Fraction(5) n = 36 E = z * (sigma / 6) # since sqrt(36) = 6 a = 2 * E + Fraction(6, 5) # 1.2 = 6/5 result = 10 * a

print("a =", a) print("10a =", result) [/PYTHON] 55\boxed{55}

Tool Output
[PYTHON OUTPUT] a = 11/2 10a = 55 [/PYTHON OUTPUT]
Reasoning Process

To solve the problem, we begin by understanding how a confidence interval for the population mean $ from a normal distribution is constructed.

Given:

  • Population standard deviation: $$\sigma$$ = 5
  • Sample size: n=36n = 36
  • Confidence level: 99%
  • Z-score corresponding to 99% confidence: z∗=2.58z^* = 2.58
  • The confidence interval is given as 1.2 $$\leq$$ m $$\leq$$ a

This implies that the lower bound of the interval is 1.2 and the upper bound is aa. Since the confidence interval for the mean is symmetric about the sample mean $$\bar{x}$$, we can write:

Lower bound=xˉ−E=1.2Upper bound=xˉ+E=a\text{Lower bound} = \bar{x} - E = 1.2 \\ \text{Upper bound} = \bar{x} + E = a

Where EE is the margin of error, given by:

E=z∗⋅σn=2.58⋅536=2.58⋅56=2.58⋅0.8333≈2.15E = z^* \cdot \frac{\sigma}{\sqrt{n}} = 2.58 \cdot \frac{5}{\sqrt{36}} = 2.58 \cdot \frac{5}{6} = 2.58 \cdot 0.8333 \approx 2.15

Now, since the lower bound is 1.2:

xˉ=1.2+E=1.2+2.15=3.35\bar{x} = 1.2 + E = 1.2 + 2.15 = 3.35

Then the upper bound becomes:

a=xˉ+E=3.35+2.15=5.5a = \bar{x} + E = 3.35 + 2.15 = 5.5

Finally, compute:

10a=10×5.5=5510a = 10 \times 5.5 = 55

Final Answer:

55\boxed{55}
Final Model Output
55

Comparative Analysis (P35) (Average across runs)

Avg Time Taken

EXAONE-4.0.1-32B (high)
636.4
K-EXAONE-236B-A23B
109.6
Llama-VARCO-8B-Instruct
80.1
HCX-007(high)
49.6
Deepseek-V3.2
47.8
Kanana-2-30B-Thinking
46.1
Gemini-3-Pro-Preview
24.4
Kanana-2-30B-Thinking-2601
22.1
Solar-Pro-2 (31B)(high)
16.4
Solar-Open-100B
14.7
Claude-Opus-4.5
13.8
A.X-4.0 (72B)
13.8
Grok-4.1-fast
10.6
GPT-5.1 (high)
7.8
GPT-5.2 (high)
7.1
K-EXAONE-236B-A23B
6.3

Avg Token Usage

K-EXAONE-236B-A23B
12,259
Kanana-2-30B-Thinking
9,114
Kanana-2-30B-Thinking-2601
6,664
Llama-VARCO-8B-Instruct
6,041
HCX-007(high)
5,122
Gemini-3-Pro-Preview
4,633
Solar-Pro-2 (31B)(high)
4,521
Solar-Open-100B
4,423
Grok-4.1-fast
4,352
A.X-4.0 (72B)
3,481
Claude-Opus-4.5
3,452
GPT-5.2 (high)
3,014
K-EXAONE-236B-A23B
2,962
Deepseek-V3.2
2,931
GPT-5.1 (high)
2,481
EXAONE-4.0.1-32B (high)
1,752