一项面向客运航空公司的飞行员培训项目会筛查申请者的色盲情况。以往记录显示，被识别为色盲的申请者比例为 0.045。

(a) 写出一个合适的模型，用于表示在总共 $n$ 名申请者中，被识别为色盲的人数分布。

(b) 说明使该分布适合此情境的一个必要假设。

(c) 使用适当的近似方法，求在 120 名申请者中恰好 5 人被识别为色盲的概率。

(d) 解释为什么 (c) 中使用的近似是适当的。

Jaymini 声称该培训项目的所有申请者中有 75% 最终会成为飞行员。

从该培训项目的申请者中随机抽取 96 人，其中 67 人最终成为飞行员。

(e) 使用适当的近似方法，在 5% 显著性水平下检验 Jaymini 的主张。

清楚陈述你的假设。

（第 5 题共 14 分）

每名申请者只有“被识别为色盲”或“未被识别为色盲”两种结果，概率固定为 $0.045$，因此人数适合用二项分布建模。

Let $X$ be the number of applicants identified as colour blind. A suitable model is

二项模型需要各次试验独立且成功概率不变。结合题目情境，可说明不同申请者的色盲识别结果相互独立，或该比例随时间保持不变。

The applicants’ colour-blindness outcomes are independent.

原分布为 $\operatorname{B}(120,0.045)$。由于 $n$ 大而 $p$ 很小，用参数 $\lambda=np=5.4$ 的泊松分布近似，再计算恰好 5 人的单点概率。

Let $X$ be the number identified as colour blind. Then

Using the Poisson approximation,

Therefore, to 3 significant figures,

泊松分布近似二项分布的适用条件是试验次数较大、单次成功概率较小。本题 $n=120$ 较大而 $p=0.045$ 很小，符合条件。

The approximation is appropriate because the binomial distribution has a large value of $n$ and a very small value of $p$:

令 $p$ 为申请者最终成为飞行员的概率。主张是 $p=0.75$，而检验是否“不同于”该比例，所以采用双侧检验。样本量足够大，可用正态分布近似二项分布，并对观测值 67 使用连续性修正 $67.5$。双侧 5% 检验的单尾临界概率是 $0.025$。

Let $p$ be the probability that an applicant goes on to become a pilot.

Under $H_0$, if $X$ is the number who become pilots, then

The mean and variance are

Use $Y\sim\operatorname{N}(72,18)$ to approximate $X$, and let $Z\sim\operatorname{N}(0,1)$. With a continuity correction,

The tail probability is $0.144$ to 3 significant figures. Since this is a two-tailed test,

Therefore, we do not reject $H_0$. There is insufficient evidence at the 5% significance level against Jaymini’s claim that 75% of applicants go on to become pilots.

也可以考察等价事件“未成为飞行员”。此时零假设比例为 $0.25$，样本中未成为飞行员的有 $96-67=29$ 人；使用连续性修正 $28.5$ 计算右尾概率，结论应与解法一完全一致。

Let $q$ be the probability that an applicant does not go on to become a pilot.

There are $96-67=29$ applicants in the sample who did not become pilots. Under $H_0$,

Use $V\sim\operatorname{N}(24,18)$ to approximate $W$, and let $Z\sim\operatorname{N}(0,1)$. Using a continuity correction,

Again, the tail probability rounds to $0.144$, and $0.144>0.025$, so we do not reject $H_0$. There is insufficient evidence at the 5% significance level against Jaymini’s claim.

不必查尾概率，也可以把标准化后的观测值与双侧检验的临界值比较。5% 双侧检验使用正态临界值 $\pm1.96$；若要把临界域写回成为飞行员的人数，也要在两端分别使用连续性修正。

Let $p$ be the probability that an applicant goes on to become a pilot. Test

Under $H_0$, $X\sim\operatorname{B}(96,0.75)$, approximated by $Y\sim\operatorname{N}(72,18)$. For a two-tailed 5% test, the standard normal critical values are $\pm1.96$.

For the observed value $X=67$, the continuity-corrected standardised value is

This lies between $-1.96$ and $1.96$, so it is not significant.

Equivalently, using continuity corrections, the lower rejection boundary satisfies

and the upper rejection boundary satisfies

Since $X$ is integer-valued, the approximate critical region is

The observed value $67$ is not in this region. We do not reject $H_0$: there is insufficient evidence at the 5% significance level against Jaymini’s claim that 75% of applicants go on to become pilots.