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IAL 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.

Back-to-back stem and leaf diagram for Birch and Maple trees

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.

Grid with Maple box plot
(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)

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 Maple 数据后，box plot 不变，表示下四分位数、上四分位数、端点等不会被这些新数据改变。根据排序给出的新数据是

27, a, 48, 2a.\begin{align*} 27,\ a,\ 48,\ 2a. \end{align*}

要保持 27,a,48,2a27,a,48,2a 的升序，且不改变 Maple box plot，最终范围会由 aa 和 2a2a 的位置限制出来。

答题过程

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Since the four values are in ascending order,

a⩽48\begin{align*} a\leqslant48 \end{align*}

and

48⩽2a.\begin{align*} 48\leqslant2a. \end{align*}

So

a⩾24.\begin{align*} a\geqslant24. \end{align*}

For the Maple box plot to remain unchanged, the new value aa must not move the lower part of the box plot, giving

a⩾36.\begin{align*} a\geqslant36. \end{align*}

Also the new value 2a2a must not exceed the upper whisker value 8080, so

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

Hence

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

Therefore the possible range is

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