题目

Problem

The company Seafield requires contractors to record the number of hours they work each week. A random sample of $38$ weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.

Stem and leaf diagram

The quartiles for this distribution are summarised in the table below.

\begin{array}{c|ccc} &Q_1&Q_2&Q_3\\ \hline &x&26&y \end{array}

(a) Find the values of $w$ , $x$ and $y$

(3)

Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than

Q_3+1.0\times(Q_3-Q_1)

(b) Showing your working clearly, identify any outliers that Kiana finds.

(2)

Box plot grid

(3)

(d) Use the formula

\text{skewness}=\frac{(Q_3-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}

to find the skewness of these data. Give your answer to $2$ significant figures.

(2)

Kiana’s new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.

(e) Explain why Landacre might prefer to know Kiana’s mean, rather than median, number of hours worked per week.

(1)

解答

(a)

解法一

思路

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stem-and-leaf 每行右侧给了该 stem 的总数。第二行一共有 $10$ 个 leaf，所以缺的 $w$ 要使这一行仍按大小顺序排列。共有 $38$ 个数据， $Q_1$ 是第 $10$ 个， $Q_3$ 是第 $29$ 个。

答题过程

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The missing leaf is

\begin{align*} w=8. \end{align*}

The lower quartile is the $10$ th value, so

\begin{align*} x=19. \end{align*}

The upper quartile is the $29$ th value, so

\begin{align*} y=37. \end{align*}

(b)

解法一

思路

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按题目定义，只找大于上界的 outliers。先算 $Q_3+1.0(Q_3-Q_1)$ 。

答题过程

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\begin{align*} Q_3+1.0(Q_3-Q_1) =&\,37+(37-19)\\[3mm] =&\,55. \end{align*}

The observations greater than $55$ are

\begin{align*} 59\quad\text{and}\quad64. \end{align*}

Therefore Kiana finds $59$ and $64$ as outliers.

(c)

解法一

思路

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箱体用 $Q_1=19$ 、median $=26$ 、 $Q_3=37$ 。由于 $59$ 和 $64$ 是 outliers，上须单独标出；upper whisker 到最大非 outlier，也就是 $51$ 。lower whisker 到最小值 $14$ 。

答题过程

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Draw the box plot using

\begin{gathered} \text{lower whisker}=14,\quad Q_1=19,\quad Q_2=26,\\[3mm] Q_3=37,\quad \text{upper whisker}=51,\\[3mm] \text{outliers}=59,\ 64. \end{gathered}

(d)

解法一

思路

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直接代入公式。注意这里公式等价于 $\dfrac{Q_3-2Q_2+Q_1}{Q_3-Q_1}$ 。

答题过程

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\begin{align*} \text{skewness} =&\,\frac{(37-26)-(26-19)}{37-19}\\[3mm] =&\,\frac{11-7}{18}\\[3mm] =&\,0.222\ldots \end{align*}

So the skewness is

\begin{align*} 0.22 \end{align*}

to $2$ significant figures.

(e)

解法一

思路

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Landacre 是为了估算雇佣成本，因此总工作时长更相关。mean 会使用所有数据，包括较大的工作时长；median 只反映中间位置。

答题过程

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Landacre may prefer the mean because it uses all the data, including the unusually large numbers of hours, so it is more useful for estimating employment cost.

IAL 2022 June Q1

题目

解答

(a)

解法一

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(b)

解法一

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(c)

解法一

思路

答题过程

(d)

解法一

思路

答题过程

(e)

解法一

思路

答题过程