(a) 一名标枪运动员随机选取了自己的 $50$ 次投掷，记录投掷距离。样本均值为 $72.3\,\mathrm{m}$，总体方差的一个无偏估计值为 $64.3\,\mathrm{m}^2$。

求该运动员投掷距离总体均值的一个 $92\%$ 置信区间。

(b) 一名铁饼运动员想计算自己投掷距离总体均值的一个 $92\%$ 置信区间。他根据一周中最先进行的 $50$ 次投掷来计算。

评价这一方法。

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。

样本量50可用正态近似，92%区间临界值对应累计概率0.96。

The sample size is 50, so the central limit theorem gives an approximately normal sampling distribution for the sample mean. For a 92% confidence interval, $z=\Phi^{-1}(0.96)=1.750686\ldots$. The interval is

Thus the interval for the population mean throw length is

一周最早50次不是随机抽样，可能不能代表所有投掷。

The first 50 throws in one week are not a random sample and may not represent all his throws, so the resulting confidence interval may be unreliable.