题目

Problem

The back-to-back stem and leaf diagram on page 3 shows information about the running times, in minutes, of $31$ Action films and $31$ Comedy films.

Back-to-back stem and leaf diagram

Key: $0\mid 9\mid 2$ means $90$ minutes for an Action film and $92$ minutes for a Comedy film.

(a) Write down the modal running time for the Action films.

(1)

Some of the quartiles for these two distributions are shown in the table below.

	Lower quartile	Median	Upper quartile
Action films	$121$	$b$	$138$
Comedy films	$a$	$117$	$c$

(b) Find the value of $a$ , the value of $b$ and the value of $c$ .

(3)

For the Action films,

\sum x=4016,\qquad \sum x^2=525056.

(3)

A measure of skewness is given by

\frac{\text{mean}-\text{mode}}{\text{standard deviation}}.

(d) Evaluate this measure of skewness for the Action films and describe the skewness.

(2)

(e) Comment on one difference between the distributions of the running times of the Action films and the Comedy films. State the statistics you have used to support your answer.

(1)

解答

(a)

解法一

思路

展开

众数就是出现次数最多的数据。Action films 的叶子在 $13$ stem 里有三个 $7$ ，所以 $137$ 出现最多。

答题过程

展开

The modal running time for the Action films is

\begin{align*} 137\text{ minutes}. \end{align*}

(b)

解法一

思路

展开

每组有 $31$ 个数据。中位数是第 $16$ 个数，下四分位数是前 $15$ 个数的中位数，也就是第 $8$ 个数；上四分位数是后 $15$ 个数的中位数，也就是第 $24$ 个数。

答题过程

展开

For $31$ values,

Q_1\text{ is the }8\text{th value},\qquad Q_2\text{ is the }16\text{th value},\qquad Q_3\text{ is the }24\text{th value}.

Reading from the stem and leaf diagram:

\begin{align*} a=106,\qquad b=129,\qquad c=126. \end{align*}

(c)

解法一

思路

展开

平均数直接用 $\sum x/n$ 。标准差用

\begin{align*} \sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}. \end{align*}

这里若使用样本标准差，结果会是 $12.6$ ；使用总体标准差，结果是 $12.4$ 。本解答用常见的 S1 数据标准差公式。

答题过程

展开

The mean is

\begin{align*} \bar{x} =&\,\frac{4016}{31}\\[3mm] =&\,129.548\ldots. \end{align*}

So the mean is

\begin{align*} 129.5\text{ minutes} \end{align*}

to $1$ decimal place.

The standard deviation is

\begin{align*} s =&\,\sqrt{\frac{525056}{31}-\left(\frac{4016}{31}\right)^2}\\[3mm] =&\,12.368\ldots. \end{align*}

So the standard deviation is

\begin{align*} 12.4\text{ minutes} \end{align*}

to $1$ decimal place.

(d)

解法一

思路

展开

把上一小题的平均数和标准差，以及 (a) 的众数代入公式。结果为负，说明是负偏态。

答题过程

展开

Using the measure of skewness,

\begin{align*} \frac{\text{mean}-\text{mode}}{\text{standard deviation}} =&\,\frac{129.5-137}{12.4}\\[3mm] =&\,-0.604\ldots. \end{align*}

The measure of skewness is approximately

\begin{align*} -0.6. \end{align*}

Therefore the distribution of the Action film running times is negatively skewed.

(e)

解法一

思路

展开

这题只要评论一个差异，并且用明确统计量支持。最直接比较中位数：Action 的中位数比 Comedy 大，所以 Action films 一般更长。

答题过程

展开

On average, Action films have longer running times than Comedy films, since the median running time for Action films is

\begin{align*} 129\text{ minutes} \end{align*}

whereas the median running time for Comedy films is

\begin{align*} 117\text{ minutes}. \end{align*}

IAL 2024 Oct Q1

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

(d)

解法一

思路

答题过程

(e)

解法一

思路

答题过程