Browse Statistical sampling

$A$ is the number of the $120$ inspected street lights that are not working correctly
$B$ is the mean age of all $1850$ street lights
$C$ is the cost of replacing one street light

(d) State which one of

A

B

and

C

is a statistic. Give a reason for your answer.

(1 mark)

●●●●●Level 35 marksStart

A factory makes a very large number of floor tiles, and each tile is stamped with a serial number when it is made. Each tile is graded

A

B

C

and sold for

£3

£2

£1

respectively. Tiles of grades

A

B

and

C

are made in the ratio

1 : 6 : 3

respectively.

A quality inspector checks a random sample of the tiles made each day, rather than inspecting every tile.
(a) (i) Suggest a suitable sampling frame for the inspector to use.

(ii) Identify the sampling units.

(2 marks)

A random sample of

3

tiles is taken. The random variable

M

represents the median price, in pounds, of the

3

tiles.
(b) Show that

P(M = 1) = 0.216

(2 marks)
(c) Find the sampling distribution of

M

(4 marks)

On each of

150

days, a random sample of

3

tiles is taken. The random variable

Y

represents the number of these

150

samples that have a median price of

£1

(d) Using a suitable approximation, find the probability that fewer than

25

of these samples have a median price of

£1

(5 marks)

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Random sampling — simple, systematic and stratified

12 questions

LessonNot started

Maya has to choose one person at random from the 9 members of a quiz team. Maya makes the choice using a single throw of a fair six-sided dice and a fair four-sided dice, whose faces are numbered 1, 2, 3 and 4, together with the following table.

Total score on the two dice	2	3	4	5	6	7	8	9	10
Member chosen	A	B	C	D	E	F	G	H	I

(a) Show that this sampling method is not random.

(2 marks)

Leo suggests making the choice using a single throw of the same two dice, together with the following table. In each pair of scores, the first number is the score on the six-sided dice and the second number is the score on the four-sided dice.

Scores on the two dice	1, 1	1, 2	1, 3	1, 4	2, 1	2, 2	2, 3	2, 4	3, 1
Member chosen	A	B	C	D	E	F	G	H	I

Maya says that a further instruction is needed to complete the method.
(b) (i) Write a suitable further instruction.

(ii) Using Leo's method, state the probability of choosing member G.

(2 marks)

●●●●●Level 24 marksStart

SECTION B: STATISTICS

A college has

1200

students:

500

in Year

12

and

700

in Year

13

. A researcher wants to select a sample of

60

students, stratified by year group.
(a) Explain briefly how the researcher should select the sample. (2 marks)
(b) Find the number of Year

12

students that should be in the sample. (2 marks)
(c) Give one advantage of using a stratified sample rather than a simple random sample in this situation. (2 marks)

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●●●●●Level 23 marksStart

Describe how to take a simple random sample of

5

students from a class of

30

●●●●●Level 22 marksStart

A company employs

600

staff. A researcher wants a sample of

40

staff to complete a wellbeing survey.
(a) Describe how the researcher could select the sample using systematic sampling. (2 marks)
(b) A colleague suggests instead surveying the first

40

staff to arrive at the office one morning. Explain why this method is likely to produce a biased sample. (2 marks)

The annual salaries, £

x

, of the

40

sampled staff are coded using

y = \dfrac{x - 20000}{1000}

.

The coded values have mean

\bar{y} = 8.4

and standard deviation

s_y = 2.5

.
(c) Find the mean and the standard deviation of the annual salaries. (4 marks)

●●●●●Level 38 marksStart

A meteorologist is studying daily maximum temperatures, in °C, recorded at two coastal weather stations,

A

and

B

, on each of the

184

days from 1 May to 31 October in a particular year.
(a) The meteorologist first wants a systematic sample of

23

of the

184

days at station

A

. Describe how such a sample could be taken. (2 marks)

The box plot below summarises the full 184 days at station

A

. The highest temperature recorded was

31.3

°C and the lowest was

11.2

°C.(b) An observation is classed as an outlier if it lies more than

1.5

times the interquartile range above the upper quartile or below the lower quartile. Show that the highest temperature is an outlier but the lowest is not. (3 marks)
(c) For station

B

, the corresponding summary is: lower quartile

15.8

°C, median

17.6

°C, upper quartile

19.7

°C, with no outliers. Compare the distributions of daily maximum temperature at the two stations, making two comparisons in context. (3 marks)

●●●●●Level 38 marksStart

A systematic sample of

50

is to be taken from a population of

800

people, listed in order.
(a) Find the sampling interval.
(b) Describe how the first member is chosen.

●●●●●Level 32 marksStart

A school has

1200

students, of whom

480

are in Year

12

. A stratified sample of

60

students is taken. Find how many should be from Year

12

●●●●●Level 32 marksStart

A company has

300

employees:

150

men,

90

women in the office and

60

in the warehouse. A stratified sample of

40

is required. Find how many should be taken from each group.

●●●●●Level 32 marksStart

A systematic sample of

80

is taken from

2000

items on a production line. Find the sampling interval.

●●●●●Level 31 markStart

Give one advantage of stratified sampling over simple random sampling.

●●●●●Level 31 markStart

A factory makes candles of a particular type. Each day the candles are made in 11 batches, numbered 2 to 12. Ravi, the quality manager, chooses one batch each day for checking. He throws two fair six-sided dice and chooses the batch whose number is the total of the two scores.
(a) Show that this method of choosing a batch is not random.

(2 marks)

To check the burning time of a candle, the candle is lit and left to burn until it goes out.
(b) State one reason why Ravi tests a sample of candles rather than every candle that is made.

(1 mark)

In the past the burning times of candles of this type had mean 32.5 hours. The factory has recently changed to a cheaper wax, and Ravi wishes to test whether the mean burning time has decreased. He now uses a correct method to choose a random sample of 45 candles made with the cheaper wax, and finds that their mean burning time is 31.9 hours. It is given that the standard deviation of the burning times is 2.2 hours.
(c) Carry out the test at the 1% significance level.

(7 marks)

●●●●●Level 510 marksStart

Non-random sampling — quota and opportunity

10 questions

LessonNot started

Total score on the two dice	2	3	4	5	6	7	8	9	10
Member chosen	A	B	C	D	E	F	G	H	I

Scores on the two dice	1, 1	1, 2	1, 3	1, 4	2, 1	2, 2	2, 3	2, 4	3, 1
Member chosen	A	B	C	D	E	F	G	H	I

●●●●●Level 24 marksStart

Describe how a quota sample might be taken when surveying

100

shoppers.

●●●●●Level 22 marksStart

Explain what is meant by opportunity (convenience) sampling.

●●●●●Level 21 markStart

Mean	Standard deviation	Median	Lower quartile	Upper quartile
28.4	9.6	26.0	21.5	33.0

(c) Kofi, a member of the club who was not in the sample, tells Ffion that he ran 50 km in the previous week.

Discuss briefly whether this value should be considered an outlier.

(3 marks)

●●●●●Level 35 marksStart

SECTION B: STATISTICS

A researcher records the time,

x

seconds, taken by each of

40

students at a school to complete a puzzle. The times are coded using

y = \dfrac{x - 60}{5}

, and the coded values are summarised by

\sum y = 72 \qquad\qquad \sum y^2 = 394

(a) Find the mean and the standard deviation of the

40

times. (3 marks)
(b) The slowest student took

118

seconds. An outlier is defined as a value more than

2

standard deviations from the mean. Determine whether this time is an outlier. (1 mark)
(c) The

40

students were the first

40

to arrive at the school library one morning. Name this method of sampling, and explain why the sample may not be representative of the students at the school. (2 marks)

●●●●●Level 36 marksStart

State one advantage and one disadvantage of quota sampling.

●●●●●Level 32 marksStart

A student surveys her friends about a new app and reports the result as representative of all students. Identify the sampling method and explain why the conclusion may be unreliable.

●●●●●Level 32 marksStart

A researcher interviews the first

30

people leaving a supermarket. State the sampling method used and one disadvantage.

●●●●●Level 32 marksStart

Give one reason a market researcher might prefer quota sampling to a stratified random sample.

●●●●●Level 31 markStart

A survey carried out in a town in 2020 found that

30\%

of adults regularly cycled to work. Since then a network of cycle lanes has opened in the town. A council officer believes that the proportion of adults in the town who regularly cycle to work has changed. To investigate, she asks

25

adults leaving the town's leisure centre one Saturday morning whether they regularly cycle to work, and uses

X

, the number who say yes, to carry out a hypothesis test.
(a) Explain why the officer should use a two-tailed test. (1 mark)
(b) Stating your hypotheses clearly, determine the critical region for the test at the

5\%

significance level, showing the probabilities you use. Find the actual significance level of the test. (5 marks)
(c) In fact

12

of the

25

adults say that they regularly cycle to work. State, with a reason, the conclusion of the test in context. (2 marks)
(d) Explain why the way the officer chose her sample makes the conclusion of the test unreliable. (1 mark)

●●●●●Level 49 marksStart