Chapter 6Analysis of Statistical Graphs 6.1

Open-ended Question Zone

Use a frequency distribution table to list a set of data which can be represented by the following cumulative frequency curve. Explain briefly.

4marks:2marksforcorrectanswer;2marksforclearexplanation.

Suggested solution

[Analysis:From the given cumulative frequency curve, we can see that the smaller the values of data, the higher the frequency is. We may then find the data suitable to this condition. ]

The following frequency distribution table shows the number of children in 200 families.

Number of children / Number of families
0 - 1 / 93
2 - 3 / 64
4 - 5 / 21
6 - 7 / 16
8 - 9 / 6

According to the above table, the smaller the number of children, the higher the frequency is. Thus this set of data can be represented by a cumulative frequency curve similar to the given one.

Solution of a student

Comments

This student can construct a suitable frequency distribution table, but no explanation has been provided on the relation between the table and the given graph.

The following frequency distribution table shows the monthly salaries of 75 employees in a restaurant.

Monthly salary ($) / Frequency
5 001 - 6 000 / 23
6 001 - 7 000 / x
7 001 - 8 000 / y
8 001 - 9 000 / 3
9 001 - 10 000 / 1
Total / 75

(a)Write down two sets of possible values of x and y.

(b)Do you think that the frequency of the first class could be the highest? Explain briefly.

6marks:(a)1markforcorrectanswer;1markforclearexplanation.
(b)1markforcorrectanswer;3marksforclearexplanation.

Suggested solution

[Analysis:For (b), we may first assume the frequency of the first class to be the highest and then see whether we can find a set of suitable values of x and y. ]

(a)Since the total number of data is 75,

When x10, y481038

When x20, y482028

x10, y38 and x20, y28 are two sets of possible values of x and y.

(b)Assume that the frequency of the first class, i.e. 23, is the highest.

Based on the assumption, the largest possible values of x and y are both 23, and the largest possible sum of x and y is 232346.

But from the result of (a), we know that xy48, i.e. the largest possible sum of x and y cannot be 46. Thus we know that the frequency of the first class could not be the highest.

Solution of a student

Comments

This student can write down two sets of possible values of x and y. He/She has tried to use a real-life experience to argue that the frequency of the first class can be the highest but this is not a reasonable argument because

1.no background information about the real-life situation has been provided in this question;

2.it is mathematically incorrect.

Exercise of Open-ended Questions 6

Level 1

1.Write down a daily example which can be represented by the following frequency curve.

4marks:2marksforcorrectanswer;2marksforclearexplanation.

2.The following figure shows the distribution of the heights of a group of soccer players. Describe the distribution and explain briefly.

4marks:2marksforcorrectanswer;2marksforclearexplanation.

3.The following frequency curve shows the heights of 1000 trees in a forest.

(a)Describe the distribution of the heights of the 1000 trees.

(b)Give a possible reason for such distribution.

4marks:(a)1markforcorrectanswer;1markforclearexplanation.
(b)1markforcorrectanswer;1markforclearexplanation.

Level 2

4.The following cumulative frequency table shows the maximum lifetime of 45 mobile phone batteries.

Time less than (hour) / Cumulative frequency
45.5 / 0
50.5 / 6
55.5 / 8
60.5 / x
65.5 / 20
70.5 / y
75.5 / 34
80.5 / 38
85.5 / 42
90.5 / 45

(a)Find two sets of possible values of x and y.

(b)Do you think it is possible for the class 61 hours - 65 hours to be the class with the highest frequency? Explain briefly.

6marks:(a)1markforcorrectanswer;1markforclearexplanation.
(b)1markforcorrectanswer;3marksforclearexplanation.