Estimating from Samples (Foundation) - Worksheets, Questions and Revision

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F09 Estimating from Samples (Foundation)

EDEXCEL 1ST0 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A gym manager surveyed a sample of 40 gym members and asked which activity they use most. The results are shown in the table.

Activity	Cardio	Weights	Swimming	Classes
Frequency	14	10	6	10

The gym has 600 members in total.

(a)Write down the fraction of the sample who said Swimming was the activity they use most.(1)

(b)Estimate the number of the gym's 600 members who use Swimming as their most-used activity.(2)

(Total for Question 1 is 3 marks)

A factory takes a sample of 8 boxes of matches from a delivery and counts the number of matches in each box.

48, 50, 47, 49, 50, 49, 49, 50

The full delivery contains 5000 boxes of matches.

(a)Work out the mean number of matches per box in the sample.(2)

(b)Estimate the total number of matches in a delivery of 5000 boxes like this.(2)

(Total for Question 2 is 4 marks)

For each statement about estimating from samples, write down whether it is True or False.

(i)To estimate a characteristic of a population, a sample must include every member of the population.(1)

(ii)Multiplying a sample mean by the total number of items in the population gives an estimate of the population total.(1)

(iii)A sample taken from only one part of a population, for example only one street in a town, is likely to give a reliable estimate for the whole population.(1)

(Total for Question 3 is 3 marks)

A film critic recorded the running time, in minutes, of a sample of 25 films. The median running time in the sample was 108 minutes.

A streaming library holds 4000 films in total.

(a)Write down what a median running time of 108 minutes tells you about the sample of 25 films.(1)

(b)Use the median to estimate how many of the 4000 films in the library have a running time of more than 108 minutes.(2)

(Total for Question 4 is 3 marks)

In a sample of 60 customers, 35% said they had used a shop's app to place an order. The shop has 2450 customers in total.

(a)Work out 35% of 60 to find the number of customers in the sample who used the app.(2)

(b)Use the sample to estimate the number of the shop's 2450 customers who have used the app. Give your answer to the nearest whole number of customers.(2)

(Total for Question 5 is 4 marks)

A biologist catches 30 fish from a lake, tags each one, and releases them back into the lake. Later, she catches a second sample of 50 fish, of which 6 are already tagged.

Capture-recapture formula:
estimated population = (number tagged in 1st catch x total in 2nd catch) / (number tagged in 2nd catch)

(Total for Question 6 is 3 marks)

Two students each estimate the mean height of trees in the same forest.

Student A measures a sample of 5 trees. Student B measures a sample of 50 trees.

(a)Whose estimate of the mean height is more likely to be reliable?(1)

(b)Give a reason for your answer.(1)

(Total for Question 7 is 2 marks)

The bar chart shows the payment method used by a sample of 50 shoppers at a shopping centre.

The shopping centre has 12000 shoppers each week.

(a)Write down the number of shoppers in the sample who paid by Contactless.(1)

(b)Work out the proportion of the sample who paid using a Mobile app.(1)

(c)Estimate the number of the shopping centre's 12000 weekly shoppers who pay by Card.(2)

(Total for Question 8 is 4 marks)

A factory makes a batch of 5000 light bulbs. Two inspectors each test a different sample of 40 bulbs from the batch.

Inspector A finds 3 defective bulbs in her sample. Inspector B finds 5 defective bulbs in his sample.

(a)Use Inspector A's sample to estimate the number of defective bulbs in the batch of 5000.(2)

(b)Use Inspector B's sample to estimate the number of defective bulbs in the batch of 5000.(2)

(c)The two estimates are different, even though both samples came from the same batch of bulbs. State the name given to this kind of difference between sample estimates, and explain why it happens.(2)

(Total for Question 9 is 6 marks)

A local council wants to estimate what proportion of all residents in a town support building a new skate park. A researcher stands outside the existing skate park on a Saturday afternoon and surveys 60 people leaving it. 54 of them say they support the new skate park.

(a)Work out the percentage of the sample who support the new skate park.(2)

(b)Explain why this sample is unlikely to give a reliable estimate of the proportion of all town residents who support the new skate park.(2)

(Total for Question 10 is 4 marks)

A sample of 15 households recorded the weight of recycling, in kg, they put out in one week. The results, in order, were:

3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 14

There are 3000 households in the area.

(a)Find the median weekly recycling weight, in kg, for this sample.(1)

(b)Use the median to estimate how many of the 3000 households in the area recycle less than 7 kg of waste per week.(2)

(Total for Question 11 is 3 marks)

A conservation officer wants to estimate the number of red squirrels living in a woodland, using the capture-recapture method.

(a)Plan: Give one reason why it would not be practical for the officer to count every red squirrel living in the woodland.(1)

(b)Collect: The officer traps 24 squirrels, tags each one, then releases them back into the woodland. Give two things she should do to make sure the second catch gives an accurate population estimate.(2)

(c)Process: Two weeks later, the officer catches a second sample of 30 squirrels. Of these, 4 are already tagged. Use the capture-recapture method to estimate the number of red squirrels living in the woodland.(3)

(d)Interpret: Give one assumption that must be true for this estimate of 180 to be reliable.(2)

(Total for Question 12 is 8 marks)

A researcher sampled pupils in Year 10 and Year 11 and asked whether they owned a smartphone.

Group	Year 10	Year 11
Owns a smartphone	28	34
Does not own a smartphone	12	6

The school has 500 pupils in Year 10 and 480 pupils in Year 11.

(a)Work out the proportion of the Year 10 sample who own a smartphone.(1)

(b)Use the sample to estimate the number of the school's 500 Year 10 pupils who own a smartphone.(2)

(c)Use a similar calculation to estimate the number of the school's 480 Year 11 pupils who own a smartphone, and hence estimate the total number of smartphone owners in Year 10 and Year 11 combined.(3)

(Total for Question 13 is 6 marks)

A dairy farmer records the milk yield, in litres, of a sample of 12 cows on one day:

22.5, 24.0, 23.5, 21.0, 25.5, 22.0, 23.0, 24.5, 22.5, 23.5, 24.0, 20.0

The farmer's whole herd has 250 cows.

(a)Work out the mean daily milk yield, in litres, for the sample of 12 cows.(2)

(b)Estimate the total daily milk yield, in litres, for the whole herd of 250 cows.(2)

(Total for Question 14 is 4 marks)

Three ecology students each estimate the total number of daisies growing on a school field by counting daisies in a number of 1 square metre quadrats, then scaling up.

Sample size (number of quadrats)	2	10	40
Estimate of total daisies on the field	3400	2650	2500

(a)State which of the three estimates is likely to be the most reliable.(1)

(b)Explain why using a larger number of quadrats makes the estimate of the total number of daisies more reliable.(2)

(Total for Question 15 is 3 marks)

A ranger uses the capture-recapture method and estimates there are 250 deer living in a national park. In the first catch, the ranger tagged 20 deer before releasing them. In the second catch, 50 deer were caught in total.

Capture-recapture formula:
estimated population = (number tagged in 1st catch x total in 2nd catch) / (number tagged in 2nd catch)

(Total for Question 16 is 3 marks)

A student is estimating the number of oak trees in a woodland of 8000 trees, using a sample of 40 trees in which 6 are oak. Here is the student's working:

Proportion of oaks in the sample = 6/40 = 0.15
Estimated number of oaks in the woodland = 0.15 + 8000 = 8000.15

(a)Identify the error in the student's working.(1)

(b)Work out the correct estimate for the number of oak trees in the woodland.(2)

(Total for Question 17 is 3 marks)

A gym has 900 members in total. The manager wants to estimate what proportion of all 900 members would use a new sauna if one were built. She emails a survey link to the 150 members who attended a 6am spin class in the last month. 80 of these members reply to the survey, and 72 of the 80 say they would use the sauna.

(a)Work out the percentage of the 80 replies who said they would use the sauna.(2)

(b)Use the sample to estimate the number of the gym's 900 members who would use the sauna.(2)

(c)Give two reasons why this sample may not give a reliable estimate for all 900 gym members.(2)

(Total for Question 18 is 6 marks)

Mark scheme · F09 Estimating from Samples

Question 1

(a) B1 6/40 oe (e.g. 3/20, 0.15) cao
(a) Answer: 6/40 (= 3/20)
(b) M1 6/40 x 600, oe
(b) A1 90 cao
(b) Answer: 90

Question 2

(a) M1 (48+50+47+49+50+49+49+50) / 8, oe
(a) A1 49 cao
(a) Answer: 49
(b) M1 49 x 5000, oe (ft their mean from (a))
(b) A1 245000 cao
(b) Answer: 245 000

Question 3

(i) B1 False cao
(i) Answer: False
(ii) B1 True cao
(ii) Answer: True
(iii) B1 False cao
(iii) Answer: False

Question 4

(a) B1 half of the sample of films have a running time of 108 minutes or less, and half have a running time of 108 minutes or more, oe
(a) Answer: Half the sample runs 108 minutes or less; half runs 108 minutes or more.
(b) M1 0.5 x 4000, oe
(b) A1 2000 cao
(b) Answer: 2000

Question 5

(a) M1 0.35 x 60, oe
(a) A1 21 cao
(a) Answer: 21
(b) M1 0.35 x 2450 (=857.5), oe
(b) A1 858 (awrt), rounded to the nearest whole customer
(b) Answer: 858

Question 6

M1 30 x 50 (=1500), oe
M1 1500 / 6
A1 250 cao
Answer: 250

Question 7

(a) B1 Student B cao
(a) Answer: Student B
(b) B1 a larger sample is more likely to represent the whole forest / is less affected by one or two unusually tall or short trees, oe
(b) Answer: A larger sample is more representative and less affected by unusual values.

Question 8

(a) B1 17 cao
(a) Answer: 17
(b) B1 5/50 oe (1/10, 0.1) cao
(b) Answer: 5/50 (= 1/10)
(c) M1 20/50 x 12000, oe
(c) A1 4800 cao
(c) Answer: 4800

Question 9

(a) M1 3/40 x 5000, oe
(a) A1 375 cao
(a) Answer: 375
(b) M1 5/40 x 5000, oe
(b) A1 625 cao
(b) Answer: 625
(c) B1 sampling error (or sampling variation) cao
(c) B1 different samples happen to contain different bulbs by chance, so the proportion defective found can vary from sample to sample even though the true proportion in the batch is fixed, oe
(c) Answer: Sampling error: different random samples contain different bulbs, so estimates vary by chance.

Question 10

(a) M1 54/60, oe
(a) A1 90% cao
(a) Answer: 90%
(b) B1 the sample only includes people who already use the skate park, who are much more likely to support it than a resident chosen at random from the whole town (the sample is biased), oe
(b) B1 so the estimate of 90% is likely to overestimate the true level of support among all town residents, oe
(b) Answer: The sample is biased towards skate-park users, so it is likely to overestimate town-wide support.

Question 11

(a) B1 7 cao
(a) Answer: 7 kg
(b) M1 0.5 x 3000, oe
(b) A1 1500 cao
(b) Answer: 1500

Question 12

(a) B1 squirrels move around and are difficult to see/find, so counting every one would be very difficult, take a very long time, or be unreliable, oe
(a) Answer: Squirrels are mobile and hard to find, so a full count would be impractical.
(b) B1 wait long enough for the tagged squirrels to mix back thoroughly (and randomly) with the rest of the population, oe
(b) B1 catch the second sample from the same woodland/area, using a similar method to the first catch, oe (accept: make sure no squirrels have been born, died, or moved in/out between catches)
(b) Answer: Allow time for tagged squirrels to mix back in; catch the second sample the same way, from the same area.
(c) M1 24 x 30 (=720), oe
(c) M1 720 / 4
(c) A1 180 cao
(c) Answer: 180
(d) B1 the population is closed: no squirrels were born, died, or moved into/out of the woodland between the two catches, oe
(d) B1 the tagged squirrels mixed back randomly/thoroughly with the rest of the population, so the second catch was a fair, representative sample, oe
(d) Answer: The squirrel population must stay unchanged between catches, and tagged squirrels must mix back in randomly.

Question 13

(a) B1 28/40 oe (7/10, 0.7) cao
(a) Answer: 28/40 (= 7/10)
(b) M1 28/40 x 500, oe
(b) A1 350 cao
(b) Answer: 350
(c) M1 34/40 x 480 (=408), oe
(c) M1 350 + 408 (ft their answer to (b))
(c) A1 758 cao
(c) Answer: 758

Question 14

(a) M1 276 / 12, oe (sum of the 12 values found first)
(a) A1 23 (litres) cao
(a) Answer: 23 litres
(b) M1 23 x 250, oe (ft their mean from (a))
(b) A1 5750 (litres) cao
(b) Answer: 5750 litres

Question 15

(a) B1 2500 (from the sample of 40 quadrats) cao
(a) Answer: 2500 (the 40-quadrat estimate)
(b) B1 a larger sample is less affected by chance, e.g. by one or two unusually crowded or unusually empty quadrats, oe
(b) B1 a larger sample gives a more representative picture of how the daisies are spread across the whole field, reducing sampling error, oe
(b) Answer: A larger sample is less affected by chance and better represents how daisies are spread across the field.

Question 16

M1 250 = (20 x 50) / m, oe (correct equation set up for the unknown number tagged, m)
M1 m = (20 x 50) / 250 (=1000/250)
A1 4 cao
Answer: 4

Question 17

(a) B1 the student added the proportion and 8000 instead of multiplying them (should be 0.15 x 8000, not 0.15 + 8000), oe
(a) Answer: The student added instead of multiplied.
(b) M1 0.15 x 8000, oe
(b) A1 1200 cao
(b) Answer: 1200

Question 18

(a) M1 72/80, oe
(a) A1 90% cao
(a) Answer: 90%
(b) M1 0.9 x 900, oe (ft their percentage from (a))
(b) A1 810 cao
(b) Answer: 810
(c) B1 the sample only includes members who attend an early 6am spin class, who may be more likely than typical members to want extra facilities such as a sauna (the sample is biased towards a particular type of member), oe
(c) B1 only the members who chose to reply are counted, not all 150 who were emailed, so members who feel less strongly either way may be under-represented (non-response), oe
(c) Answer: The sample is biased towards keen early-class attendees, and only responders (not all 150) are counted.

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Estimating from Samples (Foundation) - Worksheets, Questions and Revision

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