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IAL 2024 Oct Q5

A Level / Edexcel / S1

IAL 2024 Oct Paper · Question 5

题目

Problem

Histogram showing time worked by freelance photographers

The histogram shows the number of hours worked in a given week by a group of 6464 freelance photographers.

(a) Give a reason to justify the use of a histogram to represent these data.

(1)

Given that 1616 of these freelance photographers spent between 1010 and 2020 hours working in this week,

(b) estimate the number that spent between 1212 and 2424 hours working in this week.

(3)

(c) Find an estimate for the median time spent working in this week by these 6464 freelance photographers.

(2)

Charlie decides to model these data using a normal distribution. Charlie calculates an estimate of the mean to be 23.923.9 hours to one decimal place.

(d) Comment on Charlie’s decision to use a normal distribution. Give a justification for your answer.

(2)

解答

(a)

解法一

思路

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Histogram 适合连续型数据。这里的工作时间是连续变量,所以可以用 histogram 表示。

答题过程

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Time worked is continuous, so a histogram is suitable.

(b)

解法一

思路

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先用 10102020 小时这一组的频数 1616 定出纵轴比例。然后从图上读出相应的频率密度,用面积表示频数。

题目要求 12122424,它横跨 10102020 以及 20202525 两组,所以要分两段算面积。

答题过程

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For the interval 1010 to 2020 hours,

frequency=16.\begin{align*} \text{frequency}=16. \end{align*}

Since the class width is 1010, the frequency density is

1610=1.6.\begin{align*} \frac{16}{10}=1.6. \end{align*}

From the histogram, the frequency in the interval 2020 to 2525 hours is 1414.

Therefore the estimated number between 1212 and 2424 hours is

estimated number=201210(16)+24205(14)=12.8+11.2=24.\begin{align*} \text{estimated number} =&\,\frac{20-12}{10}(16)\\[3mm] &\,\hspace{2pt}+\frac{24-20}{5}(14)\\[3mm] =&\,12.8+11.2\\[3mm] =&\,24. \end{align*}

(c)

解法一

思路

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一共有 6464 人,所以中位数位置取第 3232 个数据。由 histogram 的面积可知,到 2020 小时累计 2121 人,到 2525 小时累计 3535 人,所以第 3232 个在 20202525 这一组内。

在这一组内做线性插值。

答题过程

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The median is the 3232nd value.

From the histogram, the cumulative frequency up to 2020 hours is 2121, and the cumulative frequency up to 2525 hours is 3535.

So the median lies in the interval 2020 to 2525 hours.

Using linear interpolation,

median=20+32213521(2520)=20+1114(5)=23.928.\begin{align*} \text{median} =&\,20+\frac{32-21}{35-21}(25-20)\\[3mm] =&\,20+\frac{11}{14}(5)\\[3mm] =&\,23.928\ldots. \end{align*}

So the estimated median is

23.9 hours\begin{align*} 23.9\text{ hours} \end{align*}

to 11 decimal place.

(d)

解法一

思路

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正态分布是对称的,因此平均数和中位数应当接近。Charlie 的平均数是 23.923.9,上一小题估计的中位数也是约 23.923.9,所以这个决定是合理的。

答题过程

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Charlie’s decision is reasonable.

The estimated mean is 23.923.9 hours and the estimated median is approximately 23.923.9 hours. Since these are approximately equal, the data are consistent with a normal distribution.