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

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H19 Estimating from Samples (Higher)

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A mobile phone network takes a random sample of 80 customers in a city, from its 45000 customers in that city, and asks each one whether they experienced a dropped call last week. 18 of the sample said yes.

(a)Write down the proportion of the sample who experienced a dropped call last week.(1)

(b)Use the sample to estimate the number of the network's 45000 customers in the city who experienced a dropped call last week.(2)

(Total for Question 1 is 3 marks)

A bakery takes a random sample of 10 loaves from a batch of 4500 loaves baked in one day. The mass, in grams, of each loaf in the sample is:

812, 805, 798, 810, 807, 795, 802, 809, 800, 806

(a)Work out the mean mass of the loaves in the sample.(2)

(b)Estimate the total mass, in kg, of all 4500 loaves in the batch.(2)

(Total for Question 2 is 4 marks)

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

(i)Scaling up a sample statistic to estimate a population value relies on the assumption that the sample is representative of the population.(1)

(ii)If a sample is representative, increasing its size can never improve the accuracy of an estimate.(1)

(iii)A sample chosen so that every member of the population has an equal chance of being included is more likely to be representative than one chosen by convenience (e.g. asking whoever is nearby).(1)

(Total for Question 3 is 3 marks)

A PE teacher records the time, in seconds, taken by a sample of 15 pupils to run 100 m. The times, in order, are:

13.1, 13.4, 13.6, 13.9, 14.0, 14.2, 14.3, 14.5, 14.7, 14.9, 15.0, 15.3, 15.6, 15.9, 16.4

The school has 480 pupils in the year group.

(a)Find the median running time for this sample.(1)

(b)Use the median to estimate how many of the 480 pupils in the year group ran 100 m in under 14.5 seconds.(3)

(Total for Question 4 is 4 marks)

An e-scooter hire company takes a random sample of 65 of the 5200 rides taken in a city last month. 24 of the sampled rides ended with the scooter parked outside a designated bay.

(a)Work out the percentage of the sample of rides that ended outside a designated bay. Give your answer to 1 decimal place.(2)

(b)Use the sample to estimate the number of the 5200 rides last month that ended outside a designated bay.(2)

(Total for Question 5 is 4 marks)

A student wants to estimate the average number of hours per week that adults in her town spend exercising. She plans to stand outside a gym on a weekday morning and ask people leaving the gym how many hours they exercise each week.

(a)Give one reason why this sample of gym-goers is unlikely to be representative of all adults in the town.(1)

(b)Give two ways the student could improve her sampling method so that it better represents all adults in the town.(2)

(Total for Question 6 is 3 marks)

A wildlife ranger uses the capture-recapture method to estimate the number of otters living along a stretch of river. She catches 18 otters, tags each one, and releases them. Two weeks later she catches a second sample of 24 otters, of which 4 are already tagged.

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

Estimate the number of otters living along this stretch of river.

(Total for Question 7 is 3 marks)

A rail company surveys a random sample of 60 passengers at a station about which ticket type they bought. The results are shown in the bar chart.

(a)Write down the number of passengers in the sample who bought a Peak single ticket.(1)

(b)Work out the proportion of the sample who bought an Off-peak return ticket.(1)

(c)The station has 9020 passengers on a typical weekday. Use the sample to estimate how many of these 9020 passengers buy an Off-peak return ticket. Give your answer to the nearest 10 passengers.(3)

(Total for Question 8 is 5 marks)

A city council takes stratified samples of residents from two districts to estimate how many residents support a new recycling scheme.

District	Number of residents	Sample size	Number in favour
District A	3000	50	32
District B	5000	80	68

(a)Use the District A sample to estimate the number of District A's 3000 residents in favour of the scheme.(2)

(b)Use the District B sample to estimate the number of District B's 5000 residents in favour of the scheme.(2)

(c)Combine your estimates from (a) and (b) to estimate the total number of residents, across both districts, in favour of the scheme.(1)

(Total for Question 9 is 5 marks)

Two market researchers each estimate the mean daily spend of customers at a shopping centre. Researcher A samples 10 customers and gets a mean spend of £42.50. Researcher B samples 200 customers and gets a mean spend of £38.20.

(a)Write down which researcher's estimate of the mean daily spend is more likely to be reliable.(1)

(b)Explain why estimates based on small samples, like Researcher A's, tend to vary more from the true population mean than estimates based on large samples.(2)

(c)Give the statistical name for the difference between a sample estimate and the true population value that occurs simply because a sample, rather than a census, was used.(1)

(Total for Question 10 is 4 marks)

A magazine wants to estimate what percentage of adults in the UK read printed books regularly. It prints a survey in one issue of the magazine and asks readers to fill it in and post it back. 340 people reply, and 250 of them say they read printed books regularly.

(a)Work out the percentage of the people who replied who read printed books regularly. Give your answer to 1 decimal place.(2)

(b)Identify the sampling method used by the magazine.(1)

(c)Explain why this method is likely to overestimate the percentage of all UK adults who read printed books regularly.(2)

(Total for Question 11 is 5 marks)

A marine biologist estimates there are 640 turtles nesting on a beach, using the capture-recapture method. In the first catch, she tagged 40 turtles. In the second catch, 32 turtles were caught in total, of which t were already tagged.

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

Work out the value of t.

(Total for Question 12 is 3 marks)

A courier company takes a random sample of 60 delivery times, in minutes, from one depot's daily deliveries. The times are grouped in the table below.

Time (t minutes)	Frequency
0 < t ≤ 10	8
10 < t ≤ 20	22
20 < t ≤ 30	18
30 < t ≤ 40	9
40 < t ≤ 50	3

The depot makes 3600 deliveries in a typical week.

(a)Work out an estimate for the mean delivery time for this sample.(3)

(b)Use your answer to part (a) to estimate the total delivery time, in minutes, for all deliveries in a typical week.(2)

(Total for Question 13 is 5 marks)

A gym recorded the time, in minutes, that a random sample of 100 members spent on cardio machines during one visit. The grouped results are shown in the table.

Time (m minutes)	Frequency
0 < m ≤ 15	12
15 < m ≤ 30	28
30 < m ≤ 45	34
45 < m ≤ 60	20
60 < m ≤ 75	6

The gym has 2400 members in total.

(a)Use linear interpolation to estimate the median cardio time for this sample.(3)

(b)Use your median from part (a) to estimate how many of the gym's 2400 members spend longer than this median time on cardio machines during a visit.(3)

(Total for Question 14 is 6 marks)

A national coffee chain has three types of store: 40 City centre stores, 25 Retail park stores and 60 Local high street stores (125 stores in total). The company wants to estimate the total number of loyalty-app sign-ups across all its stores last month.

They decide to take a stratified sample of 25 stores, proportional to store type.

(a)Plan: Show that a stratified sample of 25 stores should include 8 City centre stores, 5 Retail park stores and 12 Local high street stores.(3)

(b)Collect: Give one reason why the loyalty-app sign-up data should be collected from every sampled store during the same week, rather than from different stores in different weeks.(1)

(c)Process: The mean number of loyalty-app sign-ups per store last month, from each part of the sample, was:

Store type	Sample size	Mean sign-ups per store
City centre	8	46
Retail park	5	30
Local high street	12	18

Use this data to estimate the total number of loyalty-app sign-ups, across all 125 stores, last month.(4)

(d)Interpret: Explain why it would not be appropriate simply to find the mean of the three sample means (46, 30 and 18) and multiply by 125 to estimate the total number of sign-ups.(2)

(Total for Question 15 is 10 marks)

A student is asked to estimate the number of a school's 900 pupils who travel to school by bicycle. A sample of 40 pupils found that 6 travel by bicycle. Here is the student's working:

Proportion who cycle = 6/40 = 0.15
Estimated number who cycle = 900 / 0.15 = 6000

(a)Identify the error in the student's working, and explain what should have been done instead.(2)

(b)Work out the correct estimate for the number of pupils who travel to school by bicycle.(2)

(Total for Question 16 is 4 marks)

A charity wants to estimate the proportion of a city's 60000 households that recycle food waste. Over one week, three different volunteer teams each survey a different sample of households.

Team	Sample size	Number recycling food waste
Team 1	40	26
Team 2	60	33
Team 3	100	61

(a)Work out the percentage of each team's sample that recycles food waste, giving each answer to 1 decimal place.(3)

(b)Using the sample with the largest number of households sampled, estimate the number of the city's 60000 households that recycle food waste.(2)

(c)The three teams' percentages range from 55.0% to 65.0%, even though every sample came from the same city. Comment on what this tells you about using a single sample to estimate a population percentage, and suggest what the charity could do to get a more precise estimate.(2)

(Total for Question 17 is 7 marks)

Mark scheme · H19 Estimating from Samples

Question 1

(a) B1 18/80 oe (9/40, 0.225) cao
(a) Answer: 18/80 (= 9/40)
(b) M1 18/80 x 45000, oe
(b) A1 10125 cao
(b) Answer: 10125

Question 2

(a) M1 (812+805+798+810+807+795+802+809+800+806) / 10, oe
(a) A1 804.4 (g) cao
(a) Answer: 804.4 g
(b) M1 804.4 x 4500, oe (ft their mean from (a))
(b) A1 3619.8 (kg) cao
(b) Answer: 3619.8 kg

Question 3

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

Question 4

(a) B1 14.5 (seconds) cao
(a) Answer: 14.5 seconds
(b) M1 identifies that 7 of the 15 sample times are less than 14.5 (the median itself is the 8th value, not below it)
(b) M1 7/15 x 480, oe
(b) A1 224 cao
(b) Answer: 224

Question 5

(a) M1 24/65 (x 100), oe
(a) A1 36.9% awrt
(a) Answer: 36.9% (awrt)
(b) M1 24/65 x 5200, oe (using the unrounded proportion)
(b) A1 1920 cao
(b) Answer: 1920

Question 6

(a) B1 people leaving a gym are likely to exercise more than a typical adult in the town, so the sample is biased towards people who already exercise regularly, oe
(a) Answer: The sample only includes gym-goers, who probably exercise more than typical adults in the town.
(b) B1 select a random sample of adults from across the whole town (e.g. from the electoral register), not just gym users, oe
(b) B1 survey people at a range of different times and locations across the town, not only one place at one time, oe
(b) Answer: Take a random sample of adults from across the whole town, and sample at a range of times and locations.

Question 7

M1 18 x 24 (=432), oe
M1 432 / 4
A1 108 cao
Answer: 108

Question 8

(a) B1 9 cao
(a) Answer: 9
(b) B1 21/60 oe (7/20, 0.35) cao
(b) Answer: 21/60 (= 7/20)
(c) M1 21/60 x 9020, oe (ft their proportion from (b))
(c) A1 3157 cao
(c) B1 3160 (correctly rounded to the nearest 10 passengers)
(c) Answer: 3160 (nearest 10 passengers)

Question 9

(a) M1 32/50 x 3000, oe
(a) A1 1920 cao
(a) Answer: 1920
(b) M1 68/80 x 5000, oe
(b) A1 4250 cao
(b) Answer: 4250
(c) B1 6170 (ft the sum of their (a) and (b)) cao
(c) Answer: 6170

Question 10

(a) B1 Researcher B cao
(a) Answer: Researcher B
(b) B1 a small sample is more strongly affected by a few unusually high or low spenders, so chance variation has a bigger effect on the sample mean, oe
(b) B1 as the sample size increases, unusually high and low values are more likely to balance out, so the sample mean settles closer to the true population mean, oe
(b) Answer: Small samples are affected more by a few unusual values; larger samples average out this variation, giving estimates closer to the true population mean.
(c) B1 sampling error (or sampling variation) cao
(c) Answer: Sampling error

Question 11

(a) M1 250/340 (x 100), oe
(a) A1 73.5% awrt
(a) Answer: 73.5% (awrt)
(b) B1 self-selecting (voluntary response) sample cao
(b) Answer: Self-selecting (voluntary response) sample
(c) B1 only people who read the magazine, and so may already have an interest in reading, were able to take part, oe
(c) B1 people who feel strongly about reading books are more likely to volunteer to reply than people who rarely read, so keen readers are over-represented among the 340 replies, oe
(c) Answer: Only magazine readers could take part, and keen readers were more likely to bother replying, so book-readers are over-represented.

Question 12

M1 640 = (40 x 32) / t, oe (correct equation set up for the unknown t)
M1 t = (40 x 32) / 640 (=1280/640)
A1 2 cao
Answer: t = 2

Question 13

(a) M1 uses midpoints 5, 15, 25, 35, 45
(a) M1 (8x5)+(22x15)+(18x25)+(9x35)+(3x45), all over 60, oe (=1270/60)
(a) A1 21.2 (minutes) awrt (or exact 127/6)
(a) Answer: 21.2 minutes (awrt)
(b) M1 (1270/60) x 3600, oe (ft their mean from (a))
(b) A1 76200 cao
(b) Answer: 76200 minutes

Question 14

(a) M1 identifies the median class as 30 < m ≤ 45 (cumulative frequency reaches 50 within this class: cf before = 40, cf after = 74)
(a) M1 30 + (50-40)/34 x 15, oe (correct interpolation set up)
(a) A1 awrt 34.4 (minutes)
(a) Answer: 34.4 minutes (awrt)
(b) B1 recognises that, by definition, half of the (continuous) population lies above the median and half lies below, oe
(b) M1 0.5 x 2400, oe
(b) A1 1200 cao
(b) Answer: 1200

Question 15

(a) M1 40/125 x 25 (=8), oe (City centre)
(a) M1 25/125 x 25 (=5) and 60/125 x 25 (=12), oe (Retail park and Local high street)
(a) A1 8, 5 and 12 stated, with 8+5+12=25 checked
(a) Answer: 8 City centre, 5 Retail park, 12 Local high street stores
(b) B1 collecting all sampled stores' data in the same week means the comparison is not distorted by sign-ups changing over time (e.g. a new promotion, a seasonal trend or a public holiday), oe
(b) Answer: Collecting all sampled stores' data in the same week avoids the estimate being distorted by sign-ups changing over time.
(c) M1 46 x 40 (=1840), oe (City centre total)
(c) M1 30 x 25 (=750) and 18 x 60 (=1080), oe (Retail park and Local high street totals)
(c) M1 1840 + 750 + 1080 (adds the three stratum totals)
(c) A1 3670 cao
(c) Answer: 3670 sign-ups
(d) B1 the three store types are not equal in number (40, 25 and 60 stores), so an unweighted mean of the three sample means treats each type as if it were equally common, oe
(d) B1 this would over-represent the smaller store types (e.g. Retail park) and under-represent the larger ones (e.g. Local high street), giving a biased estimate of the total, oe
(d) Answer: The store types have different numbers of stores, so an unweighted mean of means would misrepresent smaller and larger store types and bias the total.

Question 16

(a) B1 the student divided 900 by 0.15 instead of multiplying 900 by 0.15, oe
(a) B1 to scale a sample proportion up to a population, the proportion must be multiplied by the population size (0.15 x 900), not divided into it, oe
(a) Answer: The student divided instead of multiplying; the correct method is 0.15 x 900.
(b) M1 0.15 x 900, oe
(b) A1 135 cao
(b) Answer: 135

Question 17

(a) B1 Team 1: 65.0% cao
(a) B1 Team 2: 55.0% cao
(a) B1 Team 3: 61.0% cao
(a) Answer: Team 1: 65.0%, Team 2: 55.0%, Team 3: 61.0%
(b) M1 0.61 x 60000, oe (using Team 3, the largest sample)
(b) A1 36600 cao
(b) Answer: 36600
(c) B1 different samples give different estimates purely by chance (sampling error), even though the true percentage for the city's households is fixed, so a single sample estimate should be treated as approximate, not exact, oe
(c) B1 the charity could take a much larger sample (or pool/combine all three teams' data into one larger sample) to reduce sampling error and get a more precise estimate, oe
(c) Answer: The spread shows sampling error - a single sample estimate is only approximate. Taking a larger or combined sample would reduce this error and improve precision.

Question 16

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Question 17

7 marks

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