A 班的 12 名学生参加了一次数学测试。下表显示了这些学生的测试成绩。然而，其中两名学生的成绩被墨水遮住了。已知…

A 班的 12 名学生参加了一次数学测试。下表显示了这些学生的测试成绩。然而，其中两名学生的成绩被墨水遮住了。已知下方的箱线图显示了测试成绩的分布情况，请解释为什么该信息可能不足以求出这两名学生缺失的成绩。进一步已知众数成绩为 63，求这两个缺失的成绩。答案 ... 和 ... (iii) B 班的 12 名学生也参加了相同的数学测试。他们的测试成绩分布显示在累积频数图像上。写一条比较两个班级数学测试成绩平均水平的评语，以及一条比较其成绩分布情况的评语。使用数据支持你的回答。答案 ... (b) Alice 掷了一个六面骰子 X 100 次。Bala 掷了另一个六面骰子 Y 80 次。他们各自得到“6”的次数记录在下表中。 (i) 分别求 Alice 和 Bala 掷出“6”的概率。答案 P(Alice 得到“6”) = ...

   P(Bala 得到“6”) = ...

(ii) 其中一个骰子是不均匀的。利用你在 (b)(i) 中的答案，判断骰子 X 或 Y 中哪一个更可能是均匀的六面骰子。解释你的答案。答案 ...

统计图表分析与解读集中趋势可能性度量统计图表简单推断离散程度 English

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完整解法

4(a)(i)

1. Identify the limitation of the box-and-whisker plot

A box-and-whisker plot summarises the data using only key values such as the minimum, lower quartile, median, upper quartile and maximum. It does not show the exact individual marks or how often each mark occurs.

答案

The box-and-whisker plot only shows the five-number summary: the minimum, lower quartile, median, upper quartile and maximum. It does not show all the individual marks, so different pairs of missing marks could give the same box-and-whisker plot.

4(a)(iii)

1. Read and compare the average marks using the medians

From the box-and-whisker plot for Class A,
$\text{median}_A \approx 65.$
From the cumulative frequency graph for Class B, at cumulative frequency $6$ ,
$\text{median}_B \approx 60\text{ to }65.$
So Class A has the higher average mark, or at least slightly higher depending on the exact graph reading.

2. Read quartiles and calculate the spread for Class A

For Class A,
$Q_1 \approx 55, \qquad Q_3 \approx 75.$
Hence
$\text{IQR}_A = Q_3-Q_1 \approx 75-55=20.$

3. Read quartiles and compare the distributions using interquartile ranges

For Class B,
$Q_1 \approx 50, \qquad Q_3 \approx 75\text{ to }80.$
Hence
$\text{IQR}_B \approx 75-50=25 \quad \text{to} \quad 80-50=30.$
Since $\text{IQR}_B$ is greater than $\text{IQR}_A$ , Class B's marks are more spread out. Class A's marks are more consistent.

答案

Class A has a higher average mark since its median is about $65$ , while Class B's median is about $60$ to $65$ .

Class B's marks are more spread out since its interquartile range is about $75-50=25$ to $80-50=30$ , compared with Class A's interquartile range of about $75-55=20$ .

4(b)((i))

1. Find Alice's experimental probability of obtaining a 6

Alice obtained a '6' $16$ times out of $100$ rolls.

$P(\text{obtaining a '6' by Alice})=\dfrac{16}{100}=\dfrac{4}{25}$

2. Find Bala's experimental probability of obtaining a 6

Bala obtained a '6' $18$ times out of $80$ rolls.

$P(\text{obtaining a '6' by Bala})=\dfrac{18}{80}=\dfrac{9}{40}$

答案

$P(\text{obtaining a '6' by Alice})=\dfrac{16}{100}=\dfrac{4}{25}$

$P(\text{obtaining a '6' by Bala})=\dfrac{18}{80}=\dfrac{9}{40}$

4(b)((ii))

1. Compare the experimental probabilities with the theoretical probability for an unbiased die

For an unbiased six-sided die,

$P(\text{obtaining a 6})=\frac{1}{6}\approx 0.167.$

From part (b)(i),

$P_X=\frac{4}{25}=0.16, \qquad P_Y=\frac{9}{40}=0.225.$

Since $0.16$ is closer to $0.167$ than $0.225$ is, die $X$ is likely to be the unbiased die.

答案

Die $X$ is likely to be unbiased, since Alice's probability $\frac{4}{25}=0.16$ is closer to $\frac{1}{6}\approx0.167$ than Bala's probability $\frac{9}{40}=0.225$ .

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完整解法

4(a)(i)

4(a)(iii)

4(b)((i))

4(b)((ii))

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