题目

Problem

A manufacturer produces plates. The proportion of plates that are flawed is 45%, with flawed plates occurring independently.

A random sample of 10 of these plates is selected.

(a) Find the probability that the sample contains

(i) fewer than 2 flawed plates,

(ii) at least 6 flawed plates.

(4)

George believes that the proportion of flawed plates is not 45%. To assess his belief George takes a random sample of 120 plates. The random variable $F$ represents the number of flawed plates found in the sample.

(b) Using a normal approximation, find the maximum number of plates, $c$ , and the minimum number of plates, $d$ , such that

P(F\leqslant c)\leqslant0.05

and

P(F\geqslant d)\leqslant0.05,

where $F\sim B(120,0.45)$

(7)

The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.

(c) Use a suitable hypothesis test, at the 5% level of significance, to assess the manufacturer’s claim. State your hypotheses clearly.

(4)

(Total for Question 3 is 15 marks)

题目中文翻译

一家制造商生产盘子。盘子有缺陷的比例为 45%，且缺陷盘子彼此独立出现。

随机抽取 10 个盘子。

(a) 求样本中：

(i) 少于 2 个有缺陷盘子的概率；

(ii) 至少 6 个有缺陷盘子的概率。

(4)

George 认为有缺陷盘子的比例不是 45%。为了检验他的看法，他抽取了 120 个盘子。随机变量 $F$ 表示样本中有缺陷盘子的数量。

(b) 使用正态近似，求最大的盘子数 $c$ 和最小的盘子数 $d$ ，使得

P(F\leqslant c)\leqslant0.05

且

P(F\geqslant d)\leqslant0.05

其中 $F \sim B(120,0.45)$ 。

(7)

制造商声称，在生产工艺改变后，有缺陷盘子的比例下降了。改变工艺后随机抽取 30 个盘子，其中 8 个有缺陷。

(4)

（第 3 题共 15 分）

解答

(a)(i)

解法一

思路

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每个盘子有缺陷的概率为 0.45，且各盘子彼此独立，所以缺陷盘子数服从二项分布。“少于 2 个”包括 0 个和 1 个，可直接计算累积概率。

答题过程

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Let $X$ be the number of flawed plates in the sample. Then

X\sim B(10,0.45).

Therefore,

P(X<2)

\begin{align*} =&\,P(X\leqslant1) \\[4mm] =&\,(0.55)^{10}\\ &\,\hspace{2pt}+10(0.45)(0.55)^9 \\[4mm] =&\,0.0232571\ldots \\[4mm] =&\,\boxed{0.0233}. \end{align*}

(a)(ii)

解法一

思路

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“至少 6 个”是“不超过 5 个”的补集。沿用 (i) 的二项分布，计算 $1-P(X\leqslant5)$ 。

答题过程

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P(X\geqslant6)

\begin{align*} =&\,1-P(X\leqslant5) \\[4mm] =&\,1-0.738437\ldots \\[4mm] =&\,0.261562\ldots \\[4mm] =&\,\boxed{0.262}. \end{align*}

(b)

解法一

思路

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用正态分布近似 $B(120,0.45)$ ，其均值为 54、方差为 29.7。左右两尾各对应标准正态的 5% 分位点 $\pm1.64485$ 。计算 $P(F\leqslant c)$ 时用边界 $c+0.5$ ，计算 $P(F\geqslant d)$ 时用边界 $d-0.5$ ，最后根据最大或最小整数要求取值。

答题过程

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The mean and variance are

\mu=120(0.45)=54

and

\begin{align*} \sigma^2=&\,120(0.45)(0.55)\\[4mm] =&\,29.7. \end{align*}

Hence

F\mathrel{\dot\sim}N(54,29.7).

For the lower tail, using a continuity correction,

\frac{c+0.5-54}{\sqrt{29.7}} \leqslant-1.64485.

Therefore,

\begin{align*} &\,54-1.64485\sqrt{29.7}\\[4mm] =&\,45.035944\ldots \end{align*}

c+0.5\leqslant45.035944\ldots

so $c\leqslant44.535944\ldots$ . The maximum integer value is

\boxed{c=44}.

For the upper tail,

\frac{d-0.5-54}{\sqrt{29.7}} \geqslant1.64485.

Therefore,

\begin{align*} &\,54+1.64485\sqrt{29.7}\\[4mm] =&\,62.964055\ldots \end{align*}

d-0.5\geqslant62.964055\ldots

so $d\geqslant63.464055\ldots$ . The minimum integer value is

\boxed{d=64}.

(c)

解法一

思路

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制造商声称缺陷率下降，因此进行下尾单侧检验。以原比例 0.45 为原假设，计算在 $B(30,0.45)$ 下出现 8 个或更少缺陷盘子的概率，并与 5% 比较。

答题过程

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Let $p$ be the proportion of flawed plates. The hypotheses are

\begin{align*} H_0:&\,p=0.45,\\[4mm] H_1:&\,p<0.45 \end{align*}.

Under $H_0$ , let

Y\sim B(30,0.45).

The p-value is

\begin{align*} P(Y\leqslant8)=&\,0.0312056\ldots \end{align*}

Since $0.0312<0.05$ , $H_0$ is rejected. There is sufficient evidence at the 5% significance level to support the manufacturer’s claim that the proportion of flawed plates has decreased.

解法二

思路

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也可以按临界域路线作答。检查相邻累积概率可知， $P(Y\leqslant8)<0.05$ ，但 $P(Y\leqslant9)>0.05$ ，所以 5% 下尾临界域是 $Y\leqslant8$ 。

答题过程

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Use the hypotheses

\begin{align*} H_0:&\,p=0.45,\\[4mm] H_1:&\,p<0.45 \end{align*},

with $Y\sim B(30,0.45)$ under $H_0$ . Now,

\begin{align*} P(Y\leqslant8)=&\,0.0312056\ldots\\[4mm] <&\,0.05, \end{align*}

whereas

\begin{align*} P(Y\leqslant9)=&\,0.0694073\ldots\\[4mm] >&\,0.05. \end{align*}

Hence the critical region is $Y\leqslant8$ . The observed value $8$ lies in the critical region, so $H_0$ is rejected. The data support the manufacturer’s claim that the proportion of flawed plates has decreased.

S2 2020 Oct Q3

题目

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(a)(i)

解法一

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(a)(ii)

解法一

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(b)

解法一

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(c)

解法一

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解法二

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