一位眼科医生正在检测患者是否患有特定眼疾。

根据过去的记录，已知 5% 的患者患有此眼疾。

(a) 求在随机抽取的 30 名患者中

(i) 恰好有一名患者患有此眼疾的概率，

(ii) 不超过 2 名患者患有此眼疾的概率。

眼科医生声称患有此眼疾的患者比例已经改变。

随机抽取了 150 名患者样本，其中 13 人患有此眼疾。

(b) 使用适当的近似方法，进行适当的检验来调查眼科医生的声明。使用 5% 的显著性水平并清楚地陈述你的假设。

（第 4 题共 8 分）

令 $X$ 表示 30 名患者中患有该眼疾的人数，则 $X\sim B(30,0.05)$。把 $X=1$ 代入二项分布的单点概率公式。

Let $X$ be the number of patients, out of 30, who have the eye condition. Then

Therefore,

“不超过 2 人”对应 $X\leq2$，因此把 $X=0,1,2$ 的二项概率相加，或直接使用二项分布的累积概率。

Using the same binomial distribution, let $p_x=P(X=x)$, where

“比例已经改变”表示双尾检验。原假设下样本人数服从 $B(150,0.05)$；由于 $n$ 大而 $p$ 小，用均值 $np=7.5$ 的泊松分布近似。观察值 $13$ 在均值上方，因此计算上尾概率，并与双尾检验单侧的 $0.025$ 比较。

Let $p$ be the proportion of patients who have the eye condition. The hypotheses are

Under $H_0$, the number $Y$ of affected patients in the sample has distribution $B(150,0.05)$. Since $n$ is large and $p$ is small,

where $150\times0.05=7.5$.

For the observed value $Y=13$,

This probability is given to 3 significant figures. This is a two-tailed test, and

Therefore, $H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level to suggest that the proportion of patients with the eye condition has changed.

也可直接找出泊松近似下的双尾临界区域。每一尾的概率不能超过 $0.025$：下尾边界为 $Y\leq2$，上尾边界为 $Y\geq14$。观察值 $13$ 不在临界区域，因此结论相同。

Using $Y\mathrel{\dot\sim}\operatorname{Po}(7.5)$,

and

The adjacent outcomes would make the corresponding tail probability exceed $0.025$, so the critical region is

The observed value $Y=13$ is not in the critical region. Therefore, $H_0$ is not rejected, and there is insufficient evidence that the proportion of patients with the eye condition has changed.