{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"frontierscience","formal_name":"FrontierScience","introduction":"専門的な科学課題を解く能力を評価するベンチマークです。公開データはolympiadとresearchに分かれ、競技問題と研究課題を区別して扱います。\n\nFrontierScience evaluates the ability to solve expert-level scientific tasks. Its public data separates olympiad and research problems so that competition and research tasks can be examined independently.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/openai/frontierscience","indexing_mode":"noindex"},"task_id":"b16968ff-0f9e-5b31-b264-458fe7fc45e4","task_key":"olympiad--test--1dba0086~2dbca8~2d4b25~2d8843~2de6e428f9284d","task_revision_id":"1","upstream_id":"1dba0086-bca8-4b25-8843-e6e428f9284d","short_description":"Assuming we conducted an experiment to determine a physical quantity \\\\(X\\\\). We…","config":"olympiad","split":"test","body":"{\"problem\":\"Assuming we conducted an experiment to determine a physical quantity \\\\\\\\(X\\\\\\\\). We measure the physical quantity \\\\\\\\(X\\\\\\\\), `\\\\(N\\\\)` times independently, with the results being `\\\\(x_{1}, x_{2}, ..., x_{N}\\\\)`, each random variable \\\\\\\\(X_i\\\\\\\\) follows a Gaussian distribution.\\n\\nAn experiment is a sampling process, and our goal is to use `\\\\(X_{1},X_{2},...,X_{N}\\\\)` to find the best estimates for the parameters `\\\\(μ\\\\)` and `\\\\(σ\\\\)` in the Gaussian distribution, where \\\\\\\\(\\\\mu\\\\\\\\) and \\\\\\\\(\\\\sigma\\\\\\\\) are the mean and standard deviation of the quantity \\\\\\\\(X\\\\\\\\). Here, we define `\\\\(\\\\hat{\\\\mu}\\\\)` as the unbiased estimate of \\\\\\\\(\\\\mu\\\\\\\\) and \\\\\\\\(S\\\\\\\\) as the unbiased estimate of \\\\\\\\(\\\\sigma\\\\\\\\). It is given that an unbiased estimator \\\\\\\\(\\\\hat{x}\\\\\\\\) of a quantity \\\\\\\\(x\\\\\\\\) satisfies the property \\\\\\\\(E(\\\\hat{x}) = x\\\\\\\\).\\n\\nIt can be shown that `\\\\(\\\\hat{\\\\mu}\\\\)` follows a Gaussian distribution with mean `\\\\(μ\\\\)` and standard deviation `\\\\(\\\\frac{s}{\\\\sqrt{N}}\\\\)`. `\\\\(\\\\frac{s}{\\\\sqrt{N}}\\\\)` is referred to as the \\\"uncertainty of the mean\\\" or \\\"Type `\\\\(A\\\\)` uncertainty.\\\"\\n\\nUncertainty is a quantity defined by the measurement results, representing the reliability of our measurement. We usually express the experimental results as `\\\\(\\\\hat\\\\mu\\\\pm\\\\frac{s}{\\\\sqrt{N}}\\\\)`. If systematic errors are well excluded, then the true value has approximately a 68% chance of falling within this interval.\\n\\nSince there is Type `\\\\(A\\\\)` uncertainty, there is also Type `\\\\(B\\\\)` uncertainty. Type `\\\\(B\\\\)` uncertainty takes into account the minimum scale of the measuring instrument. For example, when measuring length with a ruler, if the smallest scale on the ruler is `\\\\(a\\\\)`, then the region from `\\\\(a/2\\\\)` to the left of a scale line to `\\\\(a/2\\\\)` to the right will all be read as the same value.\\n\\nReturning to the previous situation, but assuming that all our experimental results `\\\\(X_{1}, X_{2}, ..., X_{N}\\\\)` are increased by `\\\\(X_{B}\\\\)`, where `\\\\(X_{B}\\\\)` is the error caused by aligning with the scale line.\\n\\nConsider `\\\\(X_{B}\\\\)` as a random variable, assuming `\\\\(X_{B}\\\\)` follows a uniform distribution between `\\\\(−a/2\\\\)` and `\\\\(a/2\\\\)`.  Find the uncertainty of `\\\\(\\\\hat\\\\mu\\\\)`, in terms of \\\\\\\\(s, a, N\\\\\\\\).\\n\\nThink step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.\",\"subject\":\"physics\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/openai/frontierscience","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}