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S1 2024 May Q1

A Level / Edexcel / S1

IAL 2024 May Paper · Question 1

题目

Problem

A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 3535 Birch trees and 3232 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 5∣6∣35\mid6\mid3 means 6565 cm for a Birch tree and 6363 cm for a Maple tree.

The median height that these Maple trees grow in one year is 4545 cm.

(a) Find the value of kk, used in the stem and leaf diagram.

(1)

(b) Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees.

(2)

The researcher defines an outlier as an observation that is

greater than Q3+1.5(Q3−Q1)\text{greater than }Q_3+1.5(Q_3-Q_1)

or

less than Q1−1.5(Q3−Q1).\text{less than }Q_1-1.5(Q_3-Q_1).

(c) Show that there is only one outlier amongst the Birch trees.

(2)

The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.

(d) On the same grid draw a box plot for the heights that the Birch trees grow in one year.

(4)

(e) Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees. State the values of any statistics you have used to support your comment.

(1)

The researcher realises he has missed out 44 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 44 Maple trees are 2727 cm, aa cm, 4848 cm, 2a2a cm.

Given that there is no change to the box plot for the Maple trees given on page 3

(f) find the range of possible values for aa. Show your working clearly.

(3)
题目中文翻译

一名研究人员调查 Birch 和 Maple 两种树的生长情况。记录 3535 株 Birch 和 3232 株 Maple 幼苗一年内长高的高度，精确到最接近的厘米；结果汇总在上方的背靠背茎叶图中。图中 5∣6∣35\mid6\mid3 表示 Birch 为 6565 cm、Maple 为 6363 cm。

这些 Maple 树一年长高高度的中位数为 4545 cm。

(a) 求茎叶图中 kk 的值。

(b) 求 Birch 树一年长高高度的下四分位数和上四分位数。

研究人员把大于 Q3+1.5(Q3−Q1)Q_3+1.5(Q_3-Q_1) 或小于 Q1−1.5(Q3−Q1)Q_1-1.5(Q_3-Q_1) 的观测值定义为离群值。

(c) 证明 Birch 树中只有一个离群值。

上方网格显示了 Maple 树一年长高高度的箱线图。

(d) 在同一网格上画出 Birch 树一年长高高度的箱线图。

(e) 评论 Birch 与 Maple 两种树生长分布的任一差异，并写出用来支持评论的统计量数值。

研究人员发现漏记了 44 个 Maple 数据，按升序为 2727 cm、aa cm、4848 cm、2a2a cm。已知加入这些数据后 Maple 的箱线图不变。

(f) 求 aa 的可能取值范围，清楚写出过程。

解答

(a)

解法一

思路

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Maple 有 3232 个数据，所以中位数是第 1616 个和第 1717 个的平均数。题目说中位数是 4545，因此第 1616 和第 1717 个都应围绕 4545。

答题过程

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For the Maple trees, the median is 4545 cm. Reading the ordered Maple data from the stem and leaf diagram, this gives

k=3.\begin{align*} k=3. \end{align*}

(b)

解法一

思路

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Birch 有 3535 个数据。中位数是第 1818 个；下四分位数是前 1717 个的中位数，也就是第 99 个；上四分位数是后 1717 个的中位数，也就是第 2727 个。

答题过程

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For 3535 Birch trees,

Q1 is the 9th value\begin{align*} Q_1\text{ is the }9\text{th value} \end{align*}

and

Q3 is the 27th value.\begin{align*} Q_3\text{ is the }27\text{th value}. \end{align*}

Reading from the stem and leaf diagram,

Q1=39,Q3=57.\begin{align*} Q_1=39,\qquad Q_3=57. \end{align*}

(c)

解法一

思路

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先求 IQR，再求上下 outlier boundary。Birch 的最小值是 2828，最大值是 8585。只要 8585 超过上界，而 2828 没有低于下界，就只有一个 outlier。

答题过程

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The interquartile range is

Q3−Q1=57−39=18.\begin{align*} Q_3-Q_1=57-39=18. \end{align*}

The lower outlier boundary is

39−1.5(18)=12.\begin{align*} 39-1.5(18)=12. \end{align*}

The upper outlier boundary is

57+1.5(18)=84.\begin{align*} 57+1.5(18)=84. \end{align*}

The minimum Birch value is 2828, which is not less than 1212. The maximum Birch value is 8585, which is greater than 8484.

Therefore there is only one outlier, namely

85.\begin{align*} 85. \end{align*}

(d)

解法一

思路

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画 Birch box plot 需要五数概括，并且要把 outlier 单独标出。由于 8585 是 outlier，右端 whisker 不能画到 8585，而应画到最大非 outlier 值 7676。

答题过程

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For the Birch trees,

min⁡=28,Q1=39,Q2=48,Q3=57.\begin{align*} \min=28,\quad Q_1=39,\quad Q_2=48,\quad Q_3=57. \end{align*}

The largest value that is not an outlier is

76.\begin{align*} 76. \end{align*}

The value

85\begin{align*} 85 \end{align*}

is plotted separately as an outlier.

So the Birch box plot should have whiskers at 2828 and 7676, a box from 3939 to 5757, a median line at 4848, and a separate outlier at 8585.

(e)

解法一

思路

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只需说一个合理差异，并给统计量。最直接是比较中位数：Birch 的中位数是 4848，Maple 的中位数是 4545。

答题过程

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On average, Birch trees grow slightly taller than Maple trees in one year, since the median for Birch trees is

48 cm\begin{align*} 48\text{ cm} \end{align*}

whereas the median for Maple trees is

45 cm.\begin{align*} 45\text{ cm}. \end{align*}

(f)

解法一

思路

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加入 44 个数据后箱线图不变，因此 aa 与 2a2a 必须落在不会改变原有四分位数和端点的位置。由有序数据可得到 aa 与 2a2a 各自允许的范围，再取交集。

答题过程

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For the lower quartile of the Maple data to remain unchanged, aa must satisfy

36⩽a⩽43.\begin{align*} 36\leqslant a\leqslant43. \end{align*}

For the upper part of the box plot to remain unchanged, 2a2a must satisfy

54⩽2a⩽80.\begin{align*} 54\leqslant2a\leqslant80. \end{align*}

Therefore

27⩽a⩽40.\begin{align*} 27\leqslant a\leqslant40. \end{align*}

Combining the two ranges gives

36⩽a⩽40.\begin{align*} 36\leqslant a\leqslant40. \end{align*}