题目

Problem

At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.

Time taken, in minutes	$1-30$	$31-45$	$46-65$	$66-75$	$76-100$
Frequency	21	30	68	86	45

(a) Draw a histogram to represent this information.

(4)

(b) State which class interval contains the median.

(1)

(c) Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.

(1)

题目中文翻译

在一个夏令营中，250 名儿童参加了一次数学运算测试。完成测试所用的时间按最接近的整数分钟记录，结果汇总如下表。

所用时间，单位为分钟	$1-30$	$31-45$	$46-65$	$66-75$	$76-100$
频数	21	30	68	86	45

(a) 绘制直方图来表示这些信息。

(4)

(b) 指出中位数位于哪个组区间。

(1)

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

(a)

解法一

思路

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时间按最近分钟记录，组界需各伸0.5；组距不等，柱高用频数密度。

答题过程

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Use continuous class boundaries because times are recorded to the nearest minute:

Class boundaries (minutes)	Width	Frequency density
$0.5$ to $30.5$	$30$	$21/30=0.7$
$30.5$ to $45.5$	$15$	$30/15=2$
$45.5$ to $65.5$	$20$	$68/20=3.4$
$65.5$ to $75.5$	$10$	$86/10=8.6$
$75.5$ to $100.5$	$25$	$45/25=1.8$

Draw adjoining bars with these boundaries and heights. Label the horizontal axis “Time (minutes)” and the vertical axis “Frequency density”, using linear scales.

(b)

解法一

思路

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累计频数119之后才到第125、126项，均位于第四组。

答题过程

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The cumulative frequencies are $21,51,119,205,250$ . Both middle observations, the 125th and 126th, lie in the fourth class. Thus the median class is

\boxed{66\text{–}75\text{ minutes}}.

(c)

解法一

思路

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平均数与中位数不同，归因于分布不对称。

答题过程

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The distribution of completion times is not symmetrical; it is skewed.

CIE 9709 2022 March Paper 52 Q3

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程