Histograms and Sampling

1. 1 Calculate [3 marks]

   The table shows the ages of some people. For the class \(0 < a \leq 10\) the frequency is 5, for \(10 < a \leq 30\) it is 30, and for \(30 < a \leq 50\) it is 60. Calculate the frequency density for each class. [3 marks]

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   Model answer

   The widths are 10, 20 and 20. The frequency densities are \(\dfrac{5}{10} = 0.5\), \(\dfrac{30}{20} = 1.5\) and \(\dfrac{60}{20} = 3\).

   Mark scheme

   • Class widths 10, 20 and 20 used — M1
   • At least two frequency densities correct — A1
   • 0.5, 1.5 and 3 — A1
2. 2 Calculate [4 marks]

   The histogram shows the ages of some people at a village event. (a) Calculate the number of people aged more than 40 and up to 60. [2 marks] (b) Calculate the number of people aged over 60. [2 marks]

   

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   Model answer

   (a) The frequency is the area of the bar: \(2 \times 20 = 40\). (b) The bar from 60 to 100 has a frequency density of 0.5 and a width of 40, so \(0.5 \times 40 = 20\).

   Mark scheme

   • (a) \(2 \times 20\) — M1
   • (a) 40 — A1
   • (b) \(0.5 \times 40\) — M1
   • (b) 20 — A1
3. 3 Calculate [3 marks]

   On a histogram the bar for the class \(20 < x \leq 40\) has a frequency density of 2.5. (a) Calculate the frequency of the class. [2 marks] The class \(40 < x \leq 50\) has a frequency of 30. (b) Calculate the height of its bar. [1 mark]

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   Model answer

   (a) The width is 20, so the frequency is \(2.5 \times 20 = 50\). (b) The width is 10, so the frequency density is \(\dfrac{30}{10} = 3\).

   Mark scheme

   • (a) \(2.5 \times 20\) — M1
   • (a) 50 — A1
   • (b) 3 — B1
4. 4 Calculate [3 marks]

   A club has 90 juniors, 60 seniors and 30 veterans. A stratified sample of 36 members is taken. Calculate the number of members from each group in the sample. [3 marks]

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   Model answer

   There are 180 members. Juniors: \(\dfrac{90}{180} \times 36 = 18\). Seniors: \(\dfrac{60}{180} \times 36 = 12\). Veterans: \(\dfrac{30}{180} \times 36 = 6\).

   Mark scheme

   • \(\dfrac{90}{180} \times 36\) or the total 180 seen — M1
   • At least two numbers correct — A1
   • 18, 12 and 6 — A1
5. 5 Calculate [3 marks]

   A farmer marks 25 sheep in a flock. Later he selects 50 sheep, and 5 of them are marked. (a) Calculate an estimate for the number of sheep in the flock. [2 marks] (b) State one assumption you have made. [1 mark]

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   Model answer

   (a) \(\dfrac{5}{50} = \dfrac{25}{N}\), so \(N = \dfrac{25 \times 50}{5} = 250\). (b) The marked sheep have mixed evenly with the others, and the size of the flock has not changed.

   Mark scheme

   • (a) \(\dfrac{25 \times 50}{5}\) — M1
   • (a) 250 — A1
   • (b) The marked sheep have mixed evenly, or the population has not changed — B1
6. 6 Describe [2 marks]

   There are 800 students in a school. Describe how to take a random sample of 20 of them. [2 marks]

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   Model answer

   Give each student a different number from 1 to 800. Use a random number generator to pick 20 different numbers, and choose the students with those numbers.

   Mark scheme

   • Numbers each of the students — B1
   • Uses random numbers to choose 20 students — 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.