过去，某地区有许多规模相近的农场，每个农场的小麦年产量的均值为 $24.0$ 吨，标准差为 $5.2$ 吨。去年夏天，所有这些农场都使用了一种新肥料，预计每个农场的小麦平均产量将超过 $24.0$ 吨。为了检验这一预期是否成立，一名科学家随机抽取 $50$ 个农场，记录它们去年夏天的小麦产量，得到样本均值为 $25.8$ 吨。

写出一个必要的假设，并在 $1\%$ 的显著性水平下进行检验。

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。

明确假设新肥料使用后的总体标准差仍为5.2，以总体平均产量作右尾大样本正态检验。

Assume that the population standard deviation remains 5.2 tonnes after the new fertiliser is used.

Let $\mu$ be the population mean wheat production per farm last summer, in tonnes.

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

The upper-tail 1% critical value is $2.326$. Since $2.44768\ldots>2.326$, we reject $H_0$.

There is sufficient evidence at the 1% significance level that the population mean wheat production per farm increased above 24.0 tonnes.