已知某种动物的质量服从正态分布，标准差为 $\sigma\,\mathrm{kg}$。一名研究人员从该物种随机抽取 $n$ 只动物，测得它们的质量，并用这些数据求出总体均值的两个置信区间，置信水平分别为 $\alpha\%$ 和 $90\%$。$\alpha\%$ 置信区间的宽度是 $90\%$ 置信区间宽度的 $1.414$ 倍。

(a) 求 $\alpha$。

(b) 已知 $\alpha\%$ 置信区间包含总体均值，求 $90\%$ 置信区间也包含总体均值的概率。

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。

同一份样本和同一总体标准差使宽度只随正态分位数变化，因此宽度比就是分位数之比。

For a confidence interval based on this sample, the width is $2z\sigma/\sqrt n$. Using $z_{0.95}\approx1.645$ for the $90\%$ interval, the given width ratio implies

Therefore

Hence $\boxed{\alpha=98.0}$ to $3$ significant figures.

两个区间来自同一样本且中心相同，较窄的 $90\%$ 区间包含于较宽区间；利用事件包含关系，而不是假设两个覆盖事件独立。

Both intervals have the same centre, $\overline x$, and the $90\%$ interval is contained in the wider $\alpha\%$ interval. Therefore the event that the $90\%$ interval covers the true mean is a subset of the corresponding event for the wider interval.

Let $A$ be the event that the $90\%$ interval covers $\mu$, and $B$ the event that the wider interval covers $\mu$. Since $A\subset B$, using part (a),