题目
The lengths of a random sample of roads in a certain region were measured. Using the results, a confidence interval for the mean length, in metres, of all roads in this region was found to be .
(a) Find the mean length of the roads in the sample.
(b) Calculate an estimate of the standard deviation of the lengths of roads in this region.
It is now given that the lengths of roads in this region are normally distributed.
(c) State, with a reason, whether this fact would make any difference to your calculation in part (b).
题目中文翻译
在某地区随机抽取 条道路，测量其长度。根据测量结果，得到该地区所有道路的平均长度（以米为单位）的 置信区间为 。
(a) 求样本中 条道路的平均长度。
(b) 估算该地区道路长度的标准差。
现已知该地区的道路长度服从正态分布。
(c) 说明这一条件是否会改变你在 (b) 中的计算，并给出理由。
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
(a)
解法一
思路
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对称置信区间的中心为样本均值。
答题过程
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The sample mean is the midpoint of the interval:
(b)
解法一
思路
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95%区间的半宽为9米，用半宽等于1.96倍标准误反求总体标准差估计值。
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Let be the estimated population standard deviation. The margin of error is
Hence
Thus
(c)
解法一
思路
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样本量50足够大，原计算已经通过中心极限定理使用样本均值的近似正态分布。
答题过程
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No. The sample size of 50 is large, so the Central Limit Theorem already permits a normal approximation for the sample mean, even without knowing that the population is normal.