题目

Problem

The lengths of 250 leaves of a certain type of plant are measured, correct to the nearest centimetre. The results are summarised in the table below.

Length (cm)	5–9	10–14	15–19	20–24	25–29	30–39
Frequency	18	28	60	72	48	24

(a) On the grid below, draw a cumulative frequency graph to illustrate this information.

(4)

(b) $38\%$ of these leaves are of length $k\,\mathrm{cm}$ or more.

Use your graph to find an estimate for $k$ .

(2)

(3)

题目中文翻译

测量某种植物的 250 片叶子的长度，精确到最接近的厘米。结果汇总如下表。

长度（cm）	5–9	10–14	15–19	20–24	25–29	30–39
频数	18	28	60	72	48	24

(a) 在下方的网格上画出累积频数图，表示这些信息。

(4)

(b) 这些叶子中有 $38\%$ 的长度至少为 $k\,\mathrm{cm}$ 。

利用你所画的图求 $k$ 的估计值。

(2)

(3)

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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记录到最接近的厘米，应以连续组界描点，先累加频数，再画平滑递增曲线。

答题过程

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The cumulative frequencies are

18,\ 46,\ 106,\ 178,\ 226,\ 250.

Because lengths are recorded to the nearest centimetre, plot

(4.5,0),\ (9.5,18),\ (14.5,46),\ (19.5,106),

(24.5,178),\ (29.5,226),\ (39.5,250).

Join the points with a smooth increasing curve. Label the axes “Length (cm)” and “Cumulative frequency”.

(b)

解法一

思路

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有38%在k以上，所以k对应的累计比例为62%，先求人数155再读图。

答题过程

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The required cumulative frequency is

250(1-0.38)=155.

Draw a horizontal line from cumulative frequency 155 to the curve, then read down to the length axis. This gives an estimate of $k=23\,\mathrm{cm}$ .

(c)

解法一

思路

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各组使用中点，乘频数后求加权平均。结果是分组估计值，不是实际逐片测量均值。

答题过程

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The class midpoints are $7,12,17,22,27,34.5$ cm. Hence

\begin{align*} \sum fm=&\,18(7)+28(12)+60(17)\\[4mm] &\,\hspace{2pt}+72(22)+48(27)+24(34.5)\\[4mm] =&\,5190. \end{align*}

The estimated mean length is

\frac{5190}{250}=\boxed{20.76\,\mathrm{cm}}.

CIE 9709 2025 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