Teach Me Maths

Types of Data, Sampling and Bias

Question 11 mark

A survey records each person’s favourite fruit. What type of data is this?

Hint

Decide whether the responses are categories, counts or measurements.

Worked solution

A survey records each person’s favourite fruit. What type of data is this?

  1. Categorical
  2. Discrete numerical
  3. Continuous numerical
  1. Fruit names are categories rather than numerical measurements. These data are categorical.

Question 21 mark

A survey records the number of bicycles owned by each household. What type of data is this?

Hint

Can the count take every value between two whole numbers?

Worked solution

A survey records the number of bicycles owned by each household. What type of data is this?

  1. Categorical
  2. Discrete numerical
  3. Continuous numerical
  1. The number of bicycles is a count: 0, 1, 2 and so on. It is discrete numerical data.

Question 31 mark

The time taken to complete a swim is measured. What type of data is time?

Hint

Consider the underlying measurement, even if it is rounded when recorded.

Worked solution

The time taken to complete a swim is measured. What type of data is time?

  1. Categorical
  2. Discrete numerical
  3. Continuous numerical
  1. Time can take values between whole seconds, such as 42.6 seconds. It is continuous numerical data. Rounding the recorded times does not change the underlying type.

Question 41 mark

A council wants to find out how all residents of a town usually travel around the town. It asks 120 residents. What is the population for this investigation?

Hint

The population is the whole group the investigation is about.

Worked solution

A council wants to find out how all residents of a town usually travel around the town. It asks 120 residents. What is the population for this investigation?

  1. All residents of the town
  2. Only the 120 residents asked
  3. Only residents who use buses
  1. The investigation concerns all residents of the town. The 120 people asked form a sample from that population.

Question 51 mark

A school has 840 pupils. A researcher asks 35 pupils from each of four year groups. How many pupils are in the sample?

Hint

Count the people actually asked.

Worked solution

A school has 840 pupils. A researcher asks 35 pupils from each of four year groups. How many pupils are in the sample?

  1. Sample size =35×4=140 pupils. The 840 pupils form the school population.

Question 61 mark

Mina uses rainfall figures already collected and published by a weather station. For Mina, these figures are:

Hint

Who collected the figures for this investigation?

Worked solution

Mina uses rainfall figures already collected and published by a weather station. For Mina, these figures are:

  1. Primary data
  2. Secondary data
  3. Not data because they are measurements
  1. The weather station collected the figures. Mina is reusing existing data, so for her they are secondary data.

Question 71 mark

Which method gives every pupil on a complete school register an equal chance of being selected?

Hint

Think about who could be missed by each method.

Worked solution

Which method gives every pupil on a complete school register an equal chance of being selected?

  1. Ask the first 30 pupils arriving at school
  2. Number every pupil and use random numbers to select 30 different pupils
  3. Ask the 30 pupils nearest the office
  1. Using the complete register and selecting different numbers at random gives every pupil an equal chance. Arrival order and location can favour particular pupils.

Question 81 mark

In a random sample of 40 households, 14 have a dog. What percentage of the households in the sample have a dog?

Hint

Write the number with a dog as a fraction of the number of households in the sample.

Worked solution

In a random sample of 40 households, 14 have a dog. What percentage of the households in the sample have a dog?

  1. Fraction of the sample with a dog =14/40.
  2. Percentage =14÷40×100=35\%.

Question 91 mark

A random sample is selected correctly. Which statement must still be remembered?

Hint

Random selection reduces selection bias but does not remove chance variation.

Worked solution

A random sample is selected correctly. Which statement must still be remembered?

  1. The sample must match the population exactly
  2. The sample can differ from the population by chance
  3. Every answer in the sample will be the same
  1. A random sample can differ from the population by chance. It does not guarantee exactly the same proportions.

Question 101 mark

To estimate how often all town residents cycle, Jo asks people outside a cycle shop. What is the main problem?

Hint

Consider which residents are especially likely to visit that location.

Worked solution

To estimate how often all town residents cycle, Jo asks people outside a cycle shop. What is the main problem?

  1. The sample may include unusually many people interested in cycling
  2. Cycling frequency cannot be recorded as data
  3. A survey must ask every resident
  1. People outside a cycle shop may cycle more often than other residents. This selection could make the estimate too high.

Question 111 mark

A survey asks, “Don’t you agree that our excellent library should open longer?” Which replacement is least leading?

Hint

A neutral question should not suggest a preferred response.

Worked solution

A survey asks, “Don’t you agree that our excellent library should open longer?” Which replacement is least leading?

  1. Why should the excellent library open longer?
  2. Should the library open for more hours, fewer hours, or the same hours?
  3. Do you want the library to stop being useful?
  1. The second question gives balanced options and does not praise one answer. The other questions steer people towards longer opening hours.

Question 122 marks

A random sample of 60 members chooses an activity. The club has 450 members. Use the sample to estimate how many members would choose walking.

Hint

Use the sample proportion for walking.

Worked solution

A random sample of 60 members chooses an activity. The club has 450 members. Use the sample to estimate how many members would choose walking.

  1. Walking proportion =24/60=0.4.
  2. Estimated number =0.4×450=180 members. This assumes the sample represents the club.

Question 131 mark

A form asks for age in completed whole years using the groups “under 12”, “12 to 16”, and “16 or over”. What should be corrected?

Hint

Check whether every age has exactly one place.

Worked solution

A form asks for age in completed whole years using the groups “under 12”, “12 to 16”, and “16 or over”. What should be corrected?

  1. Age 16 fits two groups
  2. The groups must all contain equal numbers of people
  3. Ages cannot be grouped
  1. Age 16 is included in both the second and third groups. For example, changing the second group to “12 to 15” removes this overlap.

Question 143 marks

Two random samples are used to estimate support for a new sports centre. Sample A has 24 supporters out of 40 people. Sample B has 33 supporters out of 60 people.

(a)

What percentage of sample A supports the centre?

1 mark

(b)

What percentage of sample B supports the centre?

1 mark

(c)

Which sample has the greater proportion of supporters?

1 mark

Hint

Convert both results to the same kind of measure.

Worked solution

Two random samples are used to estimate support for a new sports centre. Sample A has 24 supporters out of 40 people. Sample B has 33 supporters out of 60 people.

Part (a)

What percentage of sample A supports the centre?

  1. 24÷40×100=60\%.

Part (b)

What percentage of sample B supports the centre?

  1. 33÷60×100=55\%.

Part (c)

Which sample has the greater proportion of supporters?

  1. A
  2. B
  3. They are equal
  1. A has 60% support compared with B’s 55%. B has more supporters counted, but a lower proportion.

Question 153 marks

In a sample of 50 customers, 18 prefer a morning delivery. The business has 600 customers.

(a)

Estimate how many customers prefer a morning delivery.

2 marks

(b)

Which assumption supports this estimate?

1 mark

Hint

Link the calculation to its sampling assumption.

Worked solution

In a sample of 50 customers, 18 prefer a morning delivery. The business has 600 customers.

Part (a)

Estimate how many customers prefer a morning delivery.

  1. 18/50×600=216 customers.

Part (b)

Which assumption supports this estimate?

  1. The sample’s preferences are representative of all customers
  2. Every customer must prefer mornings
  3. The sample contains 600 customers
  1. The estimate assumes the sample has similar preferences to the full customer population. If it does not, the estimate may be too high or too low.

Question 16Challenge2 marks

A council emails a survey to 200 randomly selected residents. Only 80 reply, and 60 of those support a proposal.

(a)

What percentage of the people who reply support the proposal?

1 mark

(b)

Why might this percentage be a poor estimate of support among all residents?

1 mark

Hint

Distinguish the invited sample from the people who responded.

Worked solution

A council emails a survey to 200 randomly selected residents. Only 80 reply, and 60 of those support a proposal.

Part (a)

What percentage of the people who reply support the proposal?

  1. 60÷80×100=75\%.

Part (b)

Why might this percentage be a poor estimate of support among all residents?

  1. People with strong opinions may be more likely to reply
  2. A random invitation guarantees that all replies are representative
  3. The denominator should be 60
  1. Even with random invitations, non-response can introduce bias: people who reply may differ from those who do not. The 75% correctly describes the replies, but may not represent all residents.

Question 17Challenge3 marks

A college has 320 first-year students and 480 second-year students. A sample of 50 is to contain the same proportion from each year as the college.

(a)

How many first-year students should be selected?

2 marks

(b)

How should the required students be selected within each year?

1 mark

Hint

Account for both group sizes and selection within groups.

Worked solution

A college has 320 first-year students and 480 second-year students. A sample of 50 is to contain the same proportion from each year as the college.

Part (a)

How many first-year students should be selected?

  1. 320/(320+480)×50=20 first-year students.

Part (b)

How should the required students be selected within each year?

  1. Ask only students attending one optional club
  2. Choose randomly from the complete list for that year
  3. Choose only the students with the highest attendance
  1. Random selection from each complete year list avoids deliberately favouring one club or attendance group. It still allows chance variation.

Question 18Challenge3 marks

A school surveys random samples from two year groups. Use the results to estimate the total number of pupils in these two years who travel by bus.

Hint

Make one estimate for each year, then combine them.

Worked solution

A school surveys random samples from two year groups. Use the results to estimate the total number of pupils in these two years who travel by bus.

  1. Year 8 estimate =12/40×360=108.
  2. Year 9 estimate =16/50×300=96.
  3. Combined estimate =108+96=204 pupils.

Question 19Challenge3 marks

A shop asks 80 randomly selected customers about a new product. Of these, 28 say they would buy it. The shop expects 1200 customers next week.

(a)

Use the survey to estimate the number who would buy the product.

2 marks

(b)

Sam works out 1200÷28×80 for part (a). Why is Sam’s method wrong?

1 mark

Hint

Check the direction of the proportion and whether the result is sensible.

Worked solution

A shop asks 80 randomly selected customers about a new product. Of these, 28 say they would buy it. The shop expects 1200 customers next week.

Part (a)

Use the survey to estimate the number who would buy the product.

  1. 28/80×1200=420 customers.

Part (b)

Sam works out 1200÷28×80 for part (a). Why is Sam’s method wrong?

  1. It uses the sample fraction the wrong way round and predicts more buyers than customers
  2. It gives a fraction, so it must be wrong
  3. A prediction always equals the sample size
  1. The fraction needed is 28/80, not 80/28. Sam’s calculation gives more than 1200 buyers, which is more than the number of customers expected.

Question 20Challenge4 marks

A town has 900 adults and 300 children. A random sample within each group gives the results shown. The council wants to estimate how many residents want a new playground.

(a)

Estimate the total number of residents who want a playground.

3 marks

(b)

Why would using 44/100×1200 give a different estimate?

1 mark

Hint

Account for different population group sizes before combining estimates.

Worked solution

A town has 900 adults and 300 children. A random sample within each group gives the results shown. The council wants to estimate how many residents want a new playground.

Part (a)

Estimate the total number of residents who want a playground.

  1. Adult estimate =18/60×900=270.
  2. Child estimate =26/40×300=195.
  3. Total estimate =270+195=465 residents.

Part (b)

Why would using 44/100×1200 give a different estimate?

  1. The two samples do not contain adults and children in the same proportions as the town
  2. The two sample counts must never be added
  3. Every random sample has identical preferences
  1. Adults make up 60% of the combined sample but 75% of the town. Using the overall sample fraction gives too much weight to the children’s preferences here.

More on this topic: Types of Data, Sampling and Bias worksheet with full solutions

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