已知某国有 $20\%$ 的住户人数超过 $4$。Laxmi 认为她所在镇的这一比例低于 $20\%$。她在镇上随机抽取 $40$ 户，记录其中人数超过 $4$ 的户数。然后，她用二项分布在 $2.5\%$ 的显著性水平下进行检验。

(a) 求发生第一类错误的概率。

(b) 写出该检验的拒绝域。

Laxmi 发现样本中恰有 $2$ 户的人数超过 $4$。

(c) 解释为什么 Laxmi 不可能犯第二类错误。

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。

低于既有比例对应左尾检验。逐一核对相邻整数的累计概率，确定实际拒绝域后才能求第一类错误概率。

Let $K$ be the number of households containing more than four people in the sample. Test $H_0:p=0.2$ against $H_1:p<0.2$, where $p$ is the population proportion for the town.

Under $H_0$, $K\sim\mathrm B(40,0.2)$. We have

But

Thus the critical region at the $2.5\%$ level is $K\le2$. A Type I error occurs when $H_0$ is true and the observation falls in this region, so

直接承接(a)验证的边界，写出完整的整数拒绝域。

Using part (a), the rejection region is $\boxed{K\le2}$, where $K$ is the number of sampled households containing more than four people.

观测值二已经处于拒绝域；第二类错误却需要未拒绝原假设。

The observation $K=2$ lies in the rejection region, so Laxmi rejects $H_0$. A Type II error requires failing to reject a false null hypothesis; since she rejects it, her decision cannot be a Type II error.