Histograms and Sampling

1. 1 Work out [3 marks]

   The table shows the lengths of some worms. For the class \(0 < l \leq 2\) the frequency is 6, for \(2 < l \leq 8\) it is 18, and for \(8 < l \leq 20\) it is 24. Work out the frequency density for each class. [3 marks]

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   Model answer

   The widths are 2, 6 and 12. The frequency densities are \(\dfrac{6}{2} = 3\), \(\dfrac{18}{6} = 3\) and \(\dfrac{24}{12} = 2\).

   Mark scheme

   • Class widths 2, 6 and 12 used — M1
   • At least two frequency densities correct — A1
   • 3, 3 and 2 — A1
2. 2 Work out [4 marks]

   The histogram shows the lengths, in centimetres, of some worms. (a) Work out the number of worms with a length more than 15 cm and up to 20 cm. [2 marks] (b) Work out the number of worms with a length of more than 15 cm. [2 marks]

   

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   Model answer

   (a) The frequency is the area of the bar: \(6 \times 5 = 30\). (b) The bar from 20 to 40 has a frequency of \(1.5 \times 20 = 30\), so the total is \(30 + 30 = 60\).

   Mark scheme

   • (a) \(6 \times 5\) — M1
   • (a) 30 — A1
   • (b) \(1.5 \times 20 = 30\) — M1
   • (b) 60 — A1
3. 3 Work out [2 marks]

   On a histogram the bar for the class \(5 < l \leq 15\) has a frequency density of 3. Work out an estimate for the number of values in the class \(5 < l \leq 8\). [2 marks]

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   Model answer

   The width of \(5 < l \leq 8\) is 3, so the estimate is \(3 \times 3 = 9\).

   Mark scheme

   • \(3 \times 3\) — M1
   • 9 — A1
4. 4 Work out [3 marks]

   A company has 120 staff in Sales, 80 staff in Admin and 40 staff in IT. A stratified sample of 30 staff is taken. Work out the number of staff from each department in the sample. [3 marks]

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   Model answer

   There are 240 staff. Sales: \(\dfrac{120}{240} \times 30 = 15\). Admin: \(\dfrac{80}{240} \times 30 = 10\). IT: \(\dfrac{40}{240} \times 30 = 5\).

   Mark scheme

   • \(\dfrac{120}{240} \times 30\) or the total 240 seen — M1
   • At least two numbers correct — A1
   • 15, 10 and 5 — A1
5. 5 Work out [3 marks]

   In a lake, 40 fish are caught, marked and returned. Later, a sample of 90 fish is caught, and 12 of them are marked. (a) Work out an estimate for the number of fish in the lake. [2 marks] (b) State one assumption you have made. [1 mark]

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   Model answer

   (a) \(\dfrac{12}{90} = \dfrac{40}{N}\), so \(N = \dfrac{40 \times 90}{12} = 300\). (b) The marked fish have mixed evenly with the others, and none have joined or left the lake.

   Mark scheme

   • (a) \(\dfrac{40 \times 90}{12}\) — M1
   • (a) 300 — A1
   • (b) The marked fish have mixed evenly, or the population has not changed — B1
6. 6 Explain [2 marks]

   Beth wants the views of residents about a new park. She asks 30 people as they leave a sports centre. (a) Give one reason why her sample may be biased. [1 mark] (b) Describe how she could improve her sampling. [1 mark]

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   Model answer

   (a) People at a sports centre are more likely to want a park, so they do not represent all the residents. (b) Choose people from all the residents at random, for example from the electoral roll using random numbers.

   Mark scheme

   • (a) The sample is not representative of all the residents — B1
   • (b) A random sample from all the residents — B1

Quick check

1. 1

   What is the formula for frequency density?

   1. AFrequency times class width
   2. BFrequency divided by class width
   3. CClass width divided by frequency
   4. DTotal frequency divided by the number of classes
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   B: Frequency divided by class width

   Frequency density \(= \dfrac{\text{frequency}}{\text{class width}}\).

2. 2

   What does the area of a bar in a histogram show?

   1. AThe frequency
   2. BThe frequency density
   3. CThe class width
   4. DThe mean
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   A: The frequency

   Area \(=\) frequency density \(\times\) class width \(=\) frequency.

3. 3

   A histogram bar has class width 20 and frequency density 3. What is the frequency?

   1. A\(23\)
   2. B\(6.7\)
   3. C\(17\)
   4. D\(60\)
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   D: \(60\)

   \(3 \times 20 = 60\).

4. 4

   A class of width 10 has frequency 30. What is its frequency density?

   1. A\(300\)
   2. B\(0.3\)
   3. C\(3\)
   4. D\(40\)
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   C: \(3\)

   \(\dfrac{30}{10} = 3\).

5. 5

   A survey about exercise only asks people leaving a gym. What is wrong with the sample?

   1. AIt is too random
   2. BIt is biased towards people who exercise
   3. CIt is too small to be a sample
   4. DIt is a stratified sample
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   B: It is biased towards people who exercise

   People at a gym are not typical of the whole population.

6. 6

   A school has 600 pupils, 240 in Year 10. A stratified sample of 50 is taken. How many Year 10 pupils are in the sample?

   1. A\(20\)
   2. B\(12\)
   3. C\(24\)
   4. D\(240\)
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   A: \(20\)

   \(\dfrac{240}{600} \times 50 = 20\).

7. 7

   50 fish are marked and released. A second sample of 40 fish contains 8 marked fish. What is the estimate of the population?

   1. A\(10\)
   2. B\(400\)
   3. C\(62.5\)
   4. D\(250\)
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   D: \(250\)

   \(\dfrac{50 \times 40}{8} = 250\).

8. 8

   A histogram bar from 30 to 40 has frequency 40. Estimate the number of values from 30 to 35.

   1. A\(40\)
   2. B\(5\)
   3. C\(20\)
   4. D\(35\)
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   C: \(20\)

   35 is halfway through the class, so about half the frequency: 20.

9. 9

   Why does a larger sample usually give a better estimate?

   1. AIt removes all bias
   2. BIt is more likely to represent the whole population
   3. CIt always gives exact answers
   4. DIt makes the population smaller
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   B: It is more likely to represent the whole population

   Bigger samples are less affected by chance.