题目

Problem

A large number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows.

Student	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
Practical examination	66	63	24	52	59	76	88	51	48	36	91	72	68	67	60
Written examination	63	57	39	50	47	71	87	65	56	39	78	70	61	62	70

Use a sign test, at the 10% significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.

[5]

题目中文翻译

大量学生正在学习一门物理课程。他们通过一次实践考试和一次笔试进行评估。随机抽取 15 名学生，他们在这两项考试中获得的百分制成绩如下。

学生	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
实践考试	66	63	24	52	59	76	88	51	48	36	91	72	68	67	60
笔试	63	57	39	50	47	71	87	65	56	39	78	70	61	62	70

在 10% 显著性水平下，使用符号检验检验实践考试分数平均而言是否高于笔试分数。

(5)

原卷通用作答要求

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。

解答

解法一

思路

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每位学生两门考试的成绩是一对。数出实践成绩高于笔试的学生人数，计算单侧二项概率。

答题过程

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Define the paired difference as practical mark minus written mark. Test $H_0$ that its population median is zero against $H_1$ that it is positive. The practical mark is higher for $10$ students and lower for $5$ , with no ties. Under $H_0$ , the number $S$ of positive differences has distribution $\operatorname{Binomial}(15,1/2)$ . Hence

\begin{align*} &\,\Pr(S\ge10)=\sum_{j=10}^{15}\binom{15}{j}(1/2)^{15}\\ =&\,0.150878\ldots. \end{align*}

Since this exceeds $0.10$ , we fail to reject $H_0$ . There is insufficient evidence at the $10\%$ level that practical examination marks tend to be higher than written marks.

CIE 9231 2024 June Paper 42 Q2

题目

解答

解法一

思路

答题过程