一家工厂生产一包包饼干。每包饼干中饼干的总质量服从均值为 $\mu$ 的正态分布。随机抽取 12 包，记录每包内容物的质量 $x$ g。结果总结如下。

(a) 求 $\mu$ 的 99% 置信区间。

对该样本进行原假设 $\mu = k$ 的检验，显著性水平为 5%。该检验不支持备择假设 $\mu < k$。

(b) 求 $k$ 的最大可能值。

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。

总体正态、方差未知。先从汇总值求无偏方差，再用自由度 $11$ 的双侧 $t$ 区间。

The sample mean is $\bar x=2390/12$, and the unbiased variance estimate is

The two-sided $99\%$ interval uses $t_{0.995,11}=3.106$, giving

To three significant figures the interval is $(196,202)$ g.

这是左尾检验。$k$ 越大，检验统计量越小，所以将统计量设在左尾临界点得到上界。

For a hypothesised mean $k$, the statistic is $t=(\bar x-k)/\sqrt{s^2/12}$. Its lower $5\%$ critical value with $11$ degrees of freedom is $-1.796$. The limiting value of $k$ for non-rejection is therefore

At this boundary the one-sided $p$-value is $0.05$. The greatest value to the accuracy of the calculation is $200.8$ g.