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Question 1

The table below summarizes the data for a histogram using class widths and frequency densities.

Class interval	Width	Frequency density
2.0–2.4	0.4	8.0
2.4–3.0	0.6	6.5
3.0–3.2	0.2	20.0
3.2–4.0	0.8	3.75

Calculate the frequency in each class, rounding to the nearest integer.

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Question 2

Data for a histogram is given in the table below.

Class interval	Frequency density
40–50	1.5
50–60	2.1
60–80	0.9
80–100	0.6

Calculate the frequency for each class interval.

[3]

Question 3

You will draw a histogram with vertical axis scaled so that $1$ grid unit represents $0.5$ frequency-density units. Compute the frequency density and the actual bar height (in grid units) for each class.

Data (reaction time in s):

Class interval	Frequency
$0.0\text{--}0.2$	$5$
$0.2\text{--}0.5$	$18$
$0.5\text{--}0.8$	$12$
$0.8\text{--}1.4$	$9$

Report both the frequency density and the bar height for each class.

[6]

Question 4

The following table shows the data for time in minutes:

Class interval	Frequency
0–5	4
5–8	7
8–12	12
12–20	9

Compute the frequency density for each class and hence specify the bar heights for the histogram.

[4]

Question 5

The following table shows the mass ( $m$ kg) of a collection of objects.

Mass ( $m$ kg)	Frequency
$50 \le m < 55$	6
$55 \le m < 60$	9
$60 \le m < 70$	20
$70 \le m < 90$	15

Calculate the frequency density for each class and construct a histogram to represent the data.

[4]

Question 6

A histogram uses a vertical axis labeled $k \times \text{frequency density}$ (i.e., heights are scaled by a constant $k$ ). The classes and plotted heights are as follows. If the actual frequency in $20\text{--}30$ is $24$ , determine $k$ and reconstruct all class frequencies.

Class interval	Width	Plotted height ( $k \times$ density)
$10\text{--}20$	$10$	$1.5$
$20\text{--}30$	$10$	$2.4$
$30\text{--}50$	$20$	$0.9$
$50\text{--}70$	$20$	$0.6$

Find $k$ and the frequency in each class.

[5]

Question 7

A data set of ages (years) is grouped as follows.

Class interval	Frequency
$0\text{–}5$	$5$
$5\text{–}10$	$9$
$10\text{–}15$	$15$
$15\text{–}25$	$12$

Calculate the frequency density for each class.

[3]

Question 8

For the grouped data below, calculate the frequency density for each class to construct the histogram.

Data (speed in km/h):

Class interval	Frequency
40–50	16
50–52	8
52–60	24
60–80	12

State the frequency density for each class.

[4]

Question 9

One class width in a histogram is unknown. The table below shows class intervals (one with an unknown upper bound), bar heights (frequency densities), and one known frequency.

Class interval	Frequency density	Frequency
0–5	1.4	?
5–?	2.0	18
?–20	1.0	?

Assume the three classes are contiguous with no gaps.

Determine the missing upper bound and the two missing class frequencies.

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Question 10

A histogram is summarized by class intervals and frequency densities. Also given is the total frequency $N=20$ . Complete the frequency table and verify the total.

Class interval	Frequency density
$0\text{--}4$	$1.0$
$4\text{--}6$	$3.0$
$6\text{--}10$	$1.25$
$10\text{--}12$	$2.5$

Find the frequency in each class and check that they sum to $N$ .

[4]

Question 11

A histogram shows the following classes and bar heights (frequency densities). The total frequency is $N=120$ . Find the missing bar height and complete the frequencies.

Class interval	Frequency density
0–10	1.0
10–20	2.5
20–40	0.75
40–50	?

Compute the missing density and all class frequencies.

[6]

Question 12

A histogram has bars with the following class intervals and frequency density heights. Construct the frequency distribution table by computing the frequency in each class.

Class interval	Frequency density
0–5	1.2
5–10	2.0
10–20	0.8

Fill in the frequency for each class.

[2]

Question 13

Construct a histogram for the data by computing the frequency density for each class.

Data (length $l$ in cm):

Class interval	Frequency
$0 \le l < 10$	8
$10 \le l < 20$	12
$20 \le l < 30$	20
$30 \le l < 50$	10

State the frequency density (bar height) for each class.

[3]

Question Type 1: Constructing histograms given the frequency distribution table Exercises