题目

Problem

At a shop, past figures show that 35% of customers pay by credit card. Following the shop’s decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35%

(a) Test Hadi’s belief at the 5% level of significance. State your hypotheses clearly.

(5)

For a random sample of 20 customers,

(b) show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.

You may assume that 35% is the true proportion of customers who pay by credit card.

(4)

(Total for Question 4 is 9 marks)

题目中文翻译

一家商店过去的数据显示，35% 的顾客使用信用卡付款。商店决定不再收取信用卡使用手续费后，随机抽取了 20 名顾客，其中 11 人使用信用卡付款。

Hadi 认为，现在使用信用卡付款的顾客比例高于 35%。

(a) 在 5% 显著性水平下检验 Hadi 的看法，并清楚写出假设。

对于随机抽取的 20 名顾客，

(b) 证明 11 比使用信用卡付款顾客人数的均值高出不到 2 个标准差。

可以假设使用信用卡付款顾客的真实比例为 35%。

（第 4 题共 9 分）

解答

(a)

解法一

思路

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Hadi 认为比例增加，因此使用上尾单侧二项检验。在原假设下计算 20 人中至少 11 人使用信用卡的概率，并与 5% 显著性水平比较。结论必须结合“使用信用卡付款的顾客比例”这一情境表述。

答题过程

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Let $p$ be the proportion of customers who now pay by credit card. The hypotheses are

H_0:p=0.35, \qquad H_1:p>0.35.

Under $H_0$ , let

X\sim\operatorname{B}(20,0.35).

The p-value is

\begin{align*} P(X\geqslant11) =&\,1-P(X\leqslant10) \\[4mm] =&\,1-0.9468 \\[4mm] =&\,0.0532. \end{align*}

Since $0.0532>0.05$ , do not reject $H_0$ . There is insufficient evidence at the 5% significance level to support Hadi’s belief that the proportion of customers paying by credit card is now greater than 35%.

解法二

思路

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也可以使用官方评分资料给出的临界域路线。检查相邻上尾概率，找出使概率不超过 0.05 的最小人数，再判断观测值 11 是否落入临界域。

答题过程

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Use the same hypotheses, with $X\sim\operatorname{B}(20,0.35)$ under $H_0$ . Now,

P(X\geqslant12)=0.0196<0.05,

whereas

P(X\geqslant11)=0.0532>0.05.

Therefore, the critical region is

X\geqslant12.

The observed value $11$ does not lie in the critical region, so do not reject $H_0$ . There is insufficient evidence that the proportion of customers paying by credit card has increased.

(b)

解法一

思路

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在真实比例为 0.35 的假设下，使用二项分布公式求人数的均值与标准差。再把 11 与均值之差除以标准差；结果小于 2，即完成题目要求的证明。

答题过程

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For $X\sim\operatorname{B}(20,0.35)$ ,

E(X)=20(0.35)=7

and

\begin{align*} \operatorname{sd}(X) =&\,\sqrt{20(0.35)(0.65)} \\[4mm] =&\,\sqrt{4.55} \\[4mm] =&\,2.13307\ldots \end{align*}

The number of standard deviations by which 11 exceeds the mean is

\frac{11-7}{2.13307\ldots} =1.87522\ldots<2.

Therefore, 11 lies less than 2 standard deviations above the mean, as required.

IAL 2019 Oct S2 Q4

题目

解答

(a)

解法一

思路

答题过程

解法二

思路

答题过程

(b)

解法一

思路

答题过程