过去，某品种蠕虫的平均体长为 $10.3\,\mathrm{cm}$，标准差为 $2.6\,\mathrm{cm}$。气候发生变化后，有人认为该品种蠕虫的平均体长有所下降。随机抽取 $100$ 条该品种的蠕虫，测得它们的体长，样本均值为 $9.8\,\mathrm{cm}$。

假设标准差仍为 $2.6\,\mathrm{cm}$，在 $2\%$ 的显著性水平下检验平均体长是否下降。

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。

检验总体均长是否下降，使用已知不变标准差及100的样本量作左尾正态检验。

Let $\mu$ be the population mean worm length after the climate change, in centimetres.

By the Central Limit Theorem, the sample mean is approximately normal. The test statistic is

The lower-tail 2% critical value is $-2.054$. Since $-1.92308\ldots>-2.054$, we do not reject $H_0$.

There is insufficient evidence at the 2% significance level that the population mean length has decreased.