Notes

B1: for systematic (ignore other non-contradictory descriptions). Condone misspelling.
If a clear choice is given e.g. stratified or systematic we take the final answer.

Notes

B1: for mention of “tr” or “trace” entries in rain(fall) [ or \(y\)] data. Ignore “n/a”.
Ignore any comment about what to do with trace.

Notes

(i) B1: for mm/h or equivalent in words.

(ii) B1: for the idea of amount of rain when no sun. “Minimum rain” is B0
Don’t need value or units but if given must be correct or consistent with (i)

Notes

B1: for a suitable reason and saying not consistent (o.e.) e.g. “false” or “untrue” or “no”
Reason only needs to be about the line and may be a description e.g. as \(x\) increases \(y\) increases or at least 2 values of \(x\) substituted and \(y\) values evaluated

Notes

B1: for both hypotheses correct in terms of \(\rho\) (condone attempt at \(\rho\) that looks like \(p\))

B1: for a correct conclusion in context with no contradiction.
Using words “support” and “teacher’s claim”
or “negative correlation” and “sun or \(x\)” and “rain or \(y\)”
[If a comparison is given it must be \(0.015 < 0.05\)]

Notes

(i) B1: for recognising that all the data/days for these variables in LDS for 2015 is the sample.
Condone omitting “Leeming”
B0 for “sunshine and rainfall for Leeming in 2015” It could be the population.

(ii) B1: for a suitable attempt to describe a population. The teacher’s sample must be a subset.

Notes

Ignore units in this question

B1: for 55.1 or \(55\frac{1}{10}\) only (may be seen in the question) ISW e.g. 55.1 followed by 55 etc
Do not accept \(\frac{551}{10}\) that scores B0

Notes

Ignore units in this question

M1: for a correct expression (values seen not just a formula)(including \(\sqrt{\phantom{x}}\))

A1: for 2.2 or \(\frac{11}{5}\) o.e. (allow \(s = 2.209\ldots\) so awrt 2.21)
Correct answer is 2 marks

Notes

Ignore units in this question

Ignore comments about different number of runners. Don’t need to state means are the same.
If there is no answer given for (b) then no marks can be scored in (c)

“2.2” < 3.6

B1: for coach \(B\) and states s.d. of \(B\) is greater o.e. Must compare. Allow implied choose \(B\)
e.g. \(B\) has a greater range or \(B\)s values are more spread out or \(3.6 > \text{``}2.2\text{''}\) or \(\sigma_B > \text{``}2.2\text{''}\)
May say s.d. of \(A\) is smaller which is equivalent to saying s.d. of \(B\) is larger

dB1: (dep on 1^st B1 scored) for clear reason why greater s.d. means choose coach \(B\)
Idea of greater range and more extreme values (faster times) in \(B\)
e.g. \(B\) has more (or greater chance of or greater % of) runners at lower times
(may compare mean – 1 or 2 or 3 standard deviations to show that \(B\) has the quicker time)
e.g. coach \(A\) (52.9) > (51.5) coach \(B\) so coach \(B\) likely to be quickest

NB: “coach \(B\) since \(B\) has greater range /s.d. and so more extreme values” scores B1B1

SC: “2.2” > 3.6 (A0 in (b) and answer > 3.6)

B1: equivalent reasons as above but choose \(A\)

dB1: same rules as above but applied to \(A\)

Notes

Answers may appear next to the question.

B1 for Rain[fall] or precipitation (e.g. allow Daily Total [or Mean or max] Rainfall etc)
(or allow Mean Windspeed or just “windspeed” BUT not max windspeed or “gust”)
If they give more than one answer we take the last one.
(NB Actual windspeed mean is 8.2, sd 2.38. No other quantitative variables available)

Notes

B1 for awrt 15.2 (Do not accept fractions or mixed numbers)

M1 for a correct expression including square root (ft their mean)
May be implied by an answer of 3.3 or better.

A1 for awrt 3.34
[Allow \(s = 3.3471\ldots\) i.e. awrt 3.35 if correct formula/expression is seen]

Notes

If answer in (b)(i) \(\gt 19.4\) and an attempt is made in (c) please send to review.

M1 for a reason mentioning that mean or temperature is higher (o.e.)
e.g. it is a warmer/hotter month is OK
or sight of \(19.4 \gt\) (their) 15.2
Only ft their 15.2 if it is less than 19.4
OR suggesting a summer/spring month.
Ignore incorrect statements that are irrelevant or don’t contradict
For incorrect statements that contradict score M0

A1cso dep on M1 scored for inferring October
Must choose just October not a range like August~October
(NB actual mean for Sep is 15.6 and sd 3.19 and this scores A0)
Can accept for example “high mean so December” for M1A0

SC M1A1 for “October since Perth is in the southern hemisphere/Australia”
M1A0 for “Sep or Nov or Dec or Jan or Feb and “Perth is in the southern hemisphere/Australia”
M0A0 just “Perth is in the southern hemisphere/Australia” without a month M0A0

Typical solution

Mean = 3

Typical solution

Variance = 1.5

Typical solution

The mean is the same but the variance is different, so this is not a suitable model.

Typical solution

(d)(i)

Opportunity sampling.

(d)(ii)

Easy to do.

Typical solution

(a)(i)

Total area of rectangles

Total number of individuals who travel between 3 km and 5 km

(a)(ii)

The actual distance for each person is unknown

Typical solution

Negative

Typical solution

Class widths are different.

Typical solution

2700

Typical solution

Typical solution

The cost of energy for households in Village A is less on average

The variation in cost of energy is higher in Village A.

Typical solution

3

Typical solution

(i)

Hence 2040 should be removed

(ii)

Outlier

Typical solution

Typical solution

(i)

Select 20 of each of the five makes of car in each of the two years.

(ii)

Could produce a biased sample

Typical solution

Height is continuous data and 95% of heights lies within two standard deviations of the mean so normal may be a suitable model.

Typical solution

(i)

0

(ii)

0.335

(iii)

Typical solution

Mean = 1.73

Standard deviation \(= \sqrt{\dfrac{2.81}{40}}\)

\(=\) 0.265

Typical solution

Summer athletes are taller on average than Winter athletes.

Summer athletes’ heights are less varied than the heights of Winter athletes.

Typical solution

Adults on the electoral register

Typical solution

(i)

Systematic sampling

(ii)

Cheap to collect

Typical solution

The Large Data Set is only a sample of vehicles in the UK.

The Large Data Set only has data for 2002 and 2016

Typical solution

Positively skewed

Notes

B1: Accept e.g.:

• “these ranges are the same width and 10 years apart”
• “will include the same cohort for comparison”

Notes

B1: Accept e.g.:

• “too old”
• “not a stable group for comparison”
• “there are not two [appropriate] ranges that are 10 years apart [and non-overlapping etc.]”
• “ageing population means that the number of residents will be decreasing”
• “ranges of 14[or 15] years are not useful for comparison” [as not everyone will have moved to the next class after 10]

But not “age gaps too large” or “classes too wide” B0

Notes

B1: Bristol stated and explanation in words or values given (must refer to ‘decrease’ or ‘less’ or provide an inequality oe).
If values are quoted then they may be rounded (to 2sf or better) but must be correct.
Accept e.g.

• Decrease in population of 1750

Notes

(d)(i)

B1: For a correct statement in words or symbols. Accept e.g.

• “the population is approximately the same”
• “the population has only gone up slightly”

(NB candidates who obtain incorrect values could score B1B0)

B1: For correctly using correct values from the table e.g.

• Scale factor 1.03.
• Accept 5103 and 5249 seen anywhere
• Accept \(1732 + 3517\) for 5249 (ISW incorrect addition)

The statement \(5103 \approx 5249\) scores B1B1.

(d)(ii)

B1: Or equivalent, e.g. may not be the same people
Accept e.g.

• “could have been large movement in and [equally] large movement out”

Notes

M1: Two appropriate values in the following ranges (selected and used):
Value in range [26,32]
Value in range [128,134]

A1: Value in range [96,108]

If no working seen, award 2/2 marks for a correct answer in [96,108]

Notes

B1: Must include a value in range [76,80]
Condone ‘the maximum mark [scored by a student] was 78’ but not ‘the maximum possible mark was…’

Notes

B1: Value in range [68,70]

Notes

B1: Must be referring to the easier questions but could refer to e.g. ‘minimum mark’ or ‘starting score’ or ‘bottom of graph’
Accept e.g.:

• The easier questions on Test 2 are harder [than the easier questions on Test 1]
• Lower-attaining (weaker) students scored better on Test 1
• On Test 2 some people have marks lower than… whereas on Test 1 the minimum mark was…
• Test 1 had more easier questions (or Test 2 had fewer easier questions)

But not e.g.:

• Test 1 had easier questions.
• References to A* boundary (i.e. harder questions)

Notes

(f)(i)

M1: Correct method for mean soi, using attempted midpts soi (allow incorrect midpoints but not use of boundary values)
Should see midpoints multiplied by frequencies and summation implied. NB \(\Sigma x = 8045\), \(\Sigma x^2 = 445467.5\)
May be implied by correct answer (A1)

A1: awrt 53.6

B1: awrt 9.66

(f)(ii)

M1: FT their (f)(i) i.e. \(\bar{x} \pm 2 \times s\). Allow this mark for correct values seen.

A1: Dependent on correct values to 2sf (awrt 34 and 73)
Accept “there are outliers at both ends”
Condone ‘there are outliers’

SCB1 for \(53.6 \pm 3 \times 9.66 = 24.6\) and \(82.6\) (3sf) so no outliers (1/2)

Notes

M1: Must be seen. Condone a correct statement e.g. “The probability of obtaining 2 is not the same as the probability of getting 3” – must refer to specific scores, need not compute probabilities (but if given these must be correct).

A1: A generalised conclusion must be present:

• Allow “probabilities are not the same”
• May come at the beginning e.g. “the chance of each student being chosen is not equal because…”

Alternative using combinations

M1: Must be seen. Must refer to specific scores but need not compute the number of combinations/ways (but if given these must be correct).

A1: A generalised conclusion must be present:

• Accept “student __ is more likely to be chosen than student __, therefore it is not random”
• Accept “some scores are more likely than others”

Notes

(b)(i) B1: Any valid correction to the method, e.g.:

• “If obtain a pair not included, throw again.”
• “make the two dice distinguishable” (because 1,3 and 3,1 have different students) or “throw in order”

Condone extending to the entire sample space (e.g. “assign a further 2 pairs to each student and the remaining 3 to ‘throw again’”) but the method must remain random.

(b)(ii) B1: Accept \(\frac{1}{36}\)

Notes

(a)(i) B1: Must specify Train and give a reason. Any sensible reason related to chart, e.g.:

• “Train has two of highest category and three of the 2^nd highest”
• May refer to colours (e.g. dark, darker, darkest)

(a)(ii) B1: Any sensible reason relating to either ‘exact values’ or ‘actual/relative sizes of each LA’. Examples:

• “we do not know exact values because the data is grouped” B1
• “grouped” (without reference to exact values) B0
• “increases could be much higher than 90%” B1
• “low % of high number may be greater than high % of low number” B1
• “not all workers accounted for” B0
• “may be errors in data” B0

NB throughout this question, do not credit reasons that don’t relate to the question or that attempt to criticise the data set (rather than the chart/presentation and reasoning from it).

Notes

B1 B1: Any 2 distinct correct reasons. Examples:

• “% ranges are wide” B1
• “Do not know the upper limit for +90% PI” B1
• “Do not know values within -10 to +10%” B1
• “Do not know baseline values for each mode” B1
• “size of the LAs” (not relevant here) B0

Do not credit criticism of underlying data or grouping of modes of transport (e.g. walking/working from home) B0

Notes

B1: Must refer to numbers or absolute values, not just %, e.g.:

• “actual increases small (so hard to see trend)” B1
• Do not credit references to grouped data / exact values here (not relevant) B0

Notes

M1: M1 for sight of midpoints: 20% and 40%

• Allow 0.2 and 0.4
• Condone 1.2 and 1.4

Alternatively may see comparison of corresponding lower and upper bounds e.g. 10% -> 30% and 30% -> 50% (this leads to an estimate of 8.15) – M1

M1: For setting up two expressions and correctly using 3.5 (must be on the correct side). Do not accept 1.2, 1.4 here (NB \(3.5 \times 0.4 = 1.4\))
Allow = instead of \(\approx\)
Must be using two estimates for midpoints or considering both sets of lower/upper bounds

A1: Accept 3.5:0.5. If lower and upper bounds used, accept 8(.15):1
Need not be given as a ratio: accept \(\frac{D_{01}}{H_{01}} \approx 7\) oe but not \(D = 7H\)
Note that 7:2 often comes from not using any information from the chart (M0M0A0)

Notes

(a)(ii)

B1FT: Strictly FT their \(k\) (i.e. this must be \(100 -\) their \(k\) only)

Notes

(b)(i)

B1: For a clear explanation that must both identify the group unambiguously (e.g. ‘top left’, ‘least public transport use’) and give a reason.
Accept equivalent justifications e.g. ‘each individual uses more fuel’

(b)(ii)

B1 B1: For two sensible distinct suggestions. Acceptable answers include:

• The type or amount of fuel used (by different modes of transport)
• Proportion or usage of Electric Vehicles
• Types of vehicle used (or available)
• Proportion of the different modes of transport within each category
• The occupancy of each mode (or e.g. car sharing)
• Proportion of full-time vs part-time working patterns

Do not accept:

• Population size (because \(F\) is per person)
• Number of those not in work (because \(F\) is for employees)
• References to emissions e.g. ‘given off’ (because \(F\) is the amount of fuel used)

Notes

(c)(i)

B1: For the shared characteristic (any of walk/cycle/work from home). Condone ‘these have the lowest total proportion using public or motorised private transport combined’ but do not accept only ‘lower percentage using motorised private transport’ or ‘lower percentage using public transport’ – need both

B1: For a justification (must be related to the LA). Acceptable answers include:

• ‘less need to travel to work’
• ‘shorter journeys’
• ‘better walking/cycling infrastructure’

But not just ‘more work from home’ (this is the characteristic)

(\(A\) is Scilly Isles, \(B\) is City of London)

(c)(ii)

B1: B1 for identifying the difference in environment (must make a comparison e.g. ‘\(A\) is more rural than \(B\)’). Accept clearly equivalent statements e.g. ‘city’ or ‘countryside’

B1: B1 for justification.
Cannot just restate the data so do not accept e.g. ‘\(A\) has low public transport use’ – must give a justification as to why this might be.
Accept e.g. ‘less availability of public transport’
Other sensible answers may be seen e.g. ‘motorised private transport is almost absent in \(B\) but more widely used in \(A\)’

Notes

B1: Must make a specific criticism of the graph related to reading values. Acceptable answers include:

• ‘the scale is not precise enough to read detailed values’
• ‘closely clustered points mean it is hard to read’
• ‘there are no gridlines so cannot read exact values’

Do not accept generic statements such as ‘the data may not be accurate’ or references to information not included on the graph.

Notes

B1: Allow ‘\(n = 20, p = \frac{1}{3}\)’ or just \(\left(20, \frac{1}{3}\right)\). Accept \(p = 0.\dot{3}\) but not 0.333

Notes

(b)(i)

B1FT: FT their values of \(n\) and \(p\) and allow \(\dbinom{20}{x}\) as alternative to \({}^{20}\mathrm{C}_x\)
(If decimals used both must be correct to 3sf)

(b)(ii)

B1FT: FT their ‘20’ only
Any acceptable alternative in set notation
Indication that \(x\) is an integer must be present (but condone ‘integer’ or ‘whole number’ or \(x \in \mathbb{N}\))

Notes

M1: Condone \(\mathrm{P}(X \leqslant 9) - \mathrm{P}(X \leqslant 5)\), or \(0.90810 - 0.29721\) or 0.611
May be implied by correct working or answer.
Allow sum of \(\mathrm{P}(X = x)\) for \(x\) from 5 to 9.

A1: BC cao

Notes

B1: For an explanation (which must be in context) showing clear recognition that a random sample (of this size) would not be representative across the year groups.
Acceptable similar answers in context include:

• ‘should include roughly equal numbers from each year’
• ‘not representative of each year group’
• ‘may not include students from every year group’

Do not accept ‘may end up with more students from Y7-11 than from Y12-13’ (because a representative sample would have this)
Must be in context – so do not accept generic answers without context such as just: ‘It might be biased’ or ‘It’s not representative’ or ‘it should be a stratified sample’

Do not accept answers solely referring to the suitability of the individual students within the sample to represent the school

Notes

B1: (Median) Stated or shown on diagram (need not be labelled)

B1: (Quartiles) Stated or shown on diagram (need not be labelled)

B1FT: Correct diagram, FT their median & quartiles & correct end points shown (2, 62) – all correct to +/−0.5
(Must see a box with a vertical line and two whiskers)

B1: An appropriate linear scale, consistently labelled. (This mark can only be awarded if a box-and-whisker plot is drawn)

Notes

M1: Attempting to calculate either with their quartiles (may be implied by 1.5 or 77.5)

A1: Must see both correct calculations or both 1.5 and 77.5

A1: www (so must be from correct calculations)
Accept a single statement of ‘no outliers’ following correct calculations oe e.g. ‘no heights should be ignored’

Alternative using mean and sd

M1: Calculating either \(\mu \pm 2\sigma\) with sensible values \(\mu \in [35, 40]\) \(\sigma \in [14, 19]\)

A1: Must see correct calculation or value and statement

A1: Must see correct calculation or value and statement

Notes

All parts of Q10: Allow “percentage” or “value” or “number” or “rate” etc for proportion in all parts of qu 10

(a)(i) B1: eg “many 18-24”   Ignore any LA mentioned
Ignore extras only if they don’t contradict High 18-24 only

(a)(ii) B1: or high proportion of younger. Ignore any LA mentioned
Ignore extras

(a)(iii) B1: or low proportion of younger. Ignore any LA mentioned
eg “LA F because low % in younger ages” B1
Ignore extras

Notes

All parts of Q10: Allow “percentage” or “value” or “number” or “rate” etc for proportion in all parts of qu 10

(b)(i) B1: No extras or omissions

(b)(ii) B1: No extras or omissions

Notes

All parts of Q10: Allow “percentage” or “value” or “number” or “rate” etc for proportion in all parts of qu 10

B1: Low 0-17 & 18-24 not \(\Rightarrow\) attractive to older
High % of young people does not necessarily imply low % of older people
Older people may want live near young relatives

Eg May be reasons for low % younger people eg no schools

Notes

All parts of Q10: Allow “percentage” or “value” or “number” or “rate” etc for proportion in all parts of qu 10

NB. No ft for either mark

B1: Or \(16.76 + 1.5 \times (16.76 - 14.56)\)   \((= 20.06)\)
Ignore attempt at lower limit

B1: Independent mark. But must mention (c)

Notes

All parts of Q10: Allow “percentage” or “value” or “number” or “rate” etc for proportion in all parts of qu 10

or mean is in 4^th quartile
Ignore all else
Not Mean skewed by F, N, R so median better
Not Median not skewed by F, N, R so better
Not Mean because need take account of outliers (or F,N,R)

Notes

M1: attempted, using any units, eg small squares or cm²

A1: Correct calculation seen and answer 0.147 seen
Not any method starting with 0.147, eg \(0.147 \times 150 = 22.05\)

Notes

(b)(i)

B1: or Roughly symmetrical and peaks in middle or has one peak and tails off at each end, or drops off either side
All 3 of these must be seen (except “Bell-shaped” scores B1)
Not “Shape is like normal curve”   Ignore all else

(b)(ii)

B1: or calculate mean and obtain \(\frac{5915}{150}\) or 39.4
Allow 40 has the highest frequency or frequency density
Ignore all else

B1: or calculate sd and obtain 10.3
Most data is within \(6\sigma\)
Must see correct fraction and \(\approx \frac{2}{3}\), or 68% or 0.68

Notes

B1: BC

Notes

B1: Allow \(39.1 \leqslant m \leqslant 39.7\) Ignore method   BC

B1: Allow \(105.5 \leqslant s^2 \leqslant 108.5\) or \(10.27 \leqslant s \leqslant 10.42\) Ignore method
(Use of denominator \(n\) or \((n - 1)\) is OK for full marks)
OR if neither mark scored,
M1 for attempting find frequencies or areas (NOT heights) or at least five of these seen: 4, 22, 23, 28, 29, 22, 20, 2
or 20, 110, 115, 140, 145, 110, 100, 10

B2: cao Correct with unclear or no working; B1B1B1B1
or B1 for 0.145 to 0.158

NB No retrospective marks if 0.151 seen in table for (e)(i)

Notes

(e)(i)

No FT

B1 for middle row correct \(\pm 0.001\)   NB

B1 for bottom row correct \(\pm 0.003\)   NB

(e)(ii)

Allow “more accurate” or “less accurate” or similar

BUT SC: “Both less good fit for >60” alone: B1 only
NOT “Both are fairly good fit” B0B0
Ignore all else   NB No ft

Notes

Allow B1 for A or B or C with one correct factor only. Ignore else

All answers can be implied
Ignore all else

Notes

NOT: Bar charts don’t show proportions unless also state pies do.
Assume “They” means pie charts.
NOT: It’s easier to compare data
Pie charts don’t easily show which is greatest
Allow percentages instead of proportions

Notes

(c)(i)

B1: Both, no others

B1: Dep applied to at least one of C and D, even if other LA mentioned

B-marks are independent

(c)(ii)

B1: "Disagree" may be implied

B1: independent    Ignore all else

Alternatives, assuming (incorrectly) users of P&R will report both methods or just public:

Other sensible answers may be seen not covered in this MS

Notes

or, eg Would take too long to contact all students.
NOT “Easier”

Notes

or: Different years may have different numbers of students

NOT “It’s more representative” or “Takes all students into account”
“You get a range of people” “It avoids bias”

Notes

B1: Allow “30 is more than 2 sds away from the mean”

B1: Allow “30 is less than 1.5×IQR from UQ”

B1: or eg "It depends which definition you use."
Any comment implying uncertainty
Ignore comments about mean ± 3 sds or mean ± 1 sd

Notes

NB Allow 2 sf throughout

B1 B1: Allow other letters, not \(X\) unless defined. Not \(\overline{X}\)
B1B0 for 1 error eg 2-tail or:

• 2-tail: B1B0
• undefined \(\mu\): B1B0
• not in terms of parameter: B1B0
• \(\mu\) = sample mean implied: B1B0
• Not include value 20: B0B0
• eg \(\mathrm{H}_0 = 20\) etc: B0B0

M1: Correct distribution and value of \(\overline{X}\).
stated or implied eg by 0.0326 or 0.967 or 20.9 or 1.84 or 0.000335 even if within incorrect statement eg \(\mathrm{P}(X = 21) = 0.0326\)
Condone \(\dfrac{4.2^2}{\sqrt{60}}\) or \(\dfrac{4.2^2}{60^2}\) or \(\dfrac{4.2}{60}\)

A1: BC Allow 2 sf, ie 0.033

A1: Dep 0.0326 or 1.84 or 0.9674 or \(\mathrm{P}(X \gt 21)\) or \(\mathrm{P}(X \geqslant 21)\) soi
Must compare like with like,
eg NOT prob cf \(z\)-value or large prob cf small prob or CV cf wrong end of acceptance region

M1: Dependent on clearly valid comparison of like with like.
Dep 0.0326 or 0.9674 or 20.9 or 1.84 or \(\mathrm{P}(X \gt 21)\) or \(\mathrm{P}(X \geqslant 21)\) soi (corrected from the printed mark scheme: it printed “0.0485 or 0.951”, values that do not arise in this question)
May be implied by conclusion,
eg “There is evidence that mean time is \(\gt 20\) hours” M1A1

A1f: In context, not definite;
eg “Mean time is \(\gt 20\) hours": A0
But “There is evidence to reject \(\mathrm{H}_0\) and that mean time is \(\gt 20\) h” M1A1
Allow opposite conclusion, ft their values, if above conditions met and loses 2^nd A1. But potentially can score all the other 6 marks

Alternative methods for M1A1A1

For each M1: Condone \(\dfrac{4.2^2}{\sqrt{60}}\) or \(\dfrac{4.2^2}{60^2}\) or \(\dfrac{4.2}{60}\)
\(\mathrm{P}(\overline{X} \lt 21) = 0.9674\): BC

Notes

(i) B1: allow convenience (sampling)
do not allow eg convenient sampling; eg opportunistic sampling

(ii) B1: allow so
any eg which alludes to sample possibly giving unrepresentative values

Notes

B1: ignore adjectives such as weak, strong etc
allow
eg as mean hours spent running increases, resting pulse rate decreases;
eg negative relationship;
do not allow
eg (negative) correlation;
eg as pulse rate increases, mean hours per week spent running decreases

Notes

B1: approximately vertical line at about 2.5; B0 if more than one line

Notes

B1: allow eg \(r_s\) is closer to 0 than to 1;
eg low value of \(r_s\); eg allow \(r_s\) close to 0

M1: allow eg \(p\)-value \(\gt\) 0.05 (or 0.01 or 0.1)
eg \(p\)-value not small
do not allow
eg \(p\)-value (relatively) large
eg high \(p\)-value

A1: may infer no association

B1: allow
eg \(r_s\) close to \(-1\)
eg \(r_s\) is closer to –1 than 0;
eg \(r_s\) is not small
B0 for eg \(r_s\) is large oe

M1: allow eg \(p\)-value \(\lt\) 0.05 (or 0.01 or 0.10)
allow eg \(p\)-value is (very) small/low

A1: may infer negative association

Notes

B1: allow
eg because he/she/they used more data
eg the sample is bigger

Notes

B1: mark the final answer

Notes

B1: FT \(\dfrac{\text{their } 0.76}{4}\)

Notes

B1: for 1 correct frequency to 2 d.p.

B1: for 3 correct frequencies to 2 d.p.

B1: for all 5 correct to 2 d.p.

FT use of B(4, their 0.19);
allow SC2 for all 5 frequencies awrt 22, 20, 7, 1, 0 or better
allow SC1 for 3 of the above frequencies awrt nearest whole number or better
if B0B0B0 allow SC1 for probabilities: 0.430467…, 0.403895…, 0.142111…, 0.022223…, 0.001303 given to 3 dp or more; FT use of B(4, their 0.19) with \(0 \lt p \lt 1\)

Notes

B1: allow comparison of one pair of frequencies or one pair of probabilities;

Notes

B1: allow eg yes
do not allow
eg 53.125 is within range
eg positive correlation
eg follows the same trend
eg the value for 1975 is between the values for 1970 and 1980

Notes

B1: must refer to life expectancy;
must refer to UK or Europe or Sudan or Africa;
ignore superfluous speculation
do not allow
eg not likely to be close because data is different
eg life expectancy varies between countries

Notes

B1: do not allow “cell” or “entry”
do not allow ignore instead of remove
ignore superfluous comments

Notes

B1: do not allow
eg because it’s not linear

Notes

M1: allow if seen on diagram for part (b)

A1: if M0 allow B1 for 62, 63 or 64

Notes

M1: allow if seen on diagram for part (a)

A1: if M0 allow B1 for 14, 15 or 16

Notes

(ii) B1: NB \(\sqrt{14.289} = 3.7800\ldots\)
may see \(\sqrt{\frac{6247066.6}{55} - 337^2} \times \sqrt{\frac{55}{54}}\) or \(\sqrt{\frac{6247066.6}{54} - \frac{55 \times 337^2}{54}}\) oe;
AG
must see substitution of at least three of 6247066.6, 337, 55 and 54

Notes

M1: may be implied by 0.032; allow a more precise value for 3.78 if found in part (a)(ii)
NB eg N(−9E999,330,337,3.78);
NB may see \(\sigma^2 = \frac{643}{45}\) or \(z = \frac{330-337}{3.78}\ (= -1.85\ldots)\)

A1: mark the final answer

Notes

B1: to 2 or more dp

A1: must be correct to 3 sf
A0 for \(\mu \geqslant 339\) or \(\mu \gt 339\)

Alternatively using calculator

M1: allow slip in calculation if intent is clear; allow for any value between 338 and 338.78 inclusive
must see correct distributions if probabilities are wrong

M1: allow for any value between 338.8 and 339.5
NB critical value is 338.79

A1: must be correct to 3 sf; A0 for \(\mu \geqslant 339\) or \(\mu \gt 339\)

Notes

B1: do not allow \(\bar{X}\) or \(X\), but allow other symbol if defined as [population] mean volume;
allow equivalent in words

M1*: may be implied by \(0.0477 - 0.048\)
may see N\((340, 0.378^2)\)
may see \(\sigma^2 = \frac{643}{4500}\)

A1: allow slip such as \(X\) for \(\bar{X}\) but do not allow \(\mu\)

A1: dependent on award of all other marks apart from second B1
do not allow eg conclude / prove / indicate or other assertive statement instead of suggest

Alternatively

B1: do not allow \(\bar{X}\) or \(X\), but allow other symbol if defined as [population] mean volume;
allow equivalent in words

M1*: may be implied by \(339.378 - 339.38\)
may see N\((340, 0.378^2)\)
may see \(\sigma^2 = \frac{643}{4500}\)

A1: or [critical value is \(\bar{X} =]\ 339.378 - 339.38\);
allow slip such as \(X\) for \(\bar{X}\) but do not allow \(\mu\)

M1dep*: allow eg so 339.37 is in the critical region if critical region explicitly identified

A1FT: A0 if \(339.37 \gt\) their 339.378

A1: dependent on award of all other marks apart from second B1
do not allow eg conclude / prove / indicate or other assertive statement instead of suggest

Alternatively, using standard Normal distribution

B1: do not allow \(\bar{X}\) or \(X\), but allow other symbol if defined as [population] mean volume;
allow equivalent in words

M1*: may be implied by \(z = -1.667\)
may see N\((340, 0.378^2)\)
may see \(\sigma^2 = \frac{643}{4500}\)

A1FT: A0 if their \(z \gt -1.64485\)

A1: dependent on award of all other marks apart from second B1
do not allow eg conclude / prove / indicate or other assertive statement instead of suggest

Notes

B1: LDS advantage
must refer to data not being available
or reference to #N/A

Notes

A1: given correct to 1 dp or better; both regions needed; allow non-strict inequalities

A1: allow eg between 118.6 and 132.2
allow strict or non-strict inequalities

if M0 allow SCB1 for both regions outliers \(\lt 21.175\) or outliers \(\gt 118.575\) unsupported;
allow SCB2 for all outliers in (118.575,132.2] unsupported

Notes

B1: LDS advantage

Notes

B1: allow
eg an average man is heavier than an average woman

do not allow
eg men are heavier than women on average

Notes

B1: allow mean weight for men greater than mean weight for women, so men are heavier than women (by about 10 kg) oe
must refer to mean or average

B1: must refer to standard deviation or variance

Notes

B1: must refer to at least one of the given modes of transport

Notes

M1: allow slip in calculation of 500 if clearly their sum of all pupils; may be implied by 12.84 or 12.8

A1: B2 for 13 unsupported

Notes

B1: allow
eg no because not possible to select every possible sample
eg not simple random sampling because every pupil not equally likely to be selected

do not allow
eg not simple random sampling because it’s systematic sampling

Notes

M1: allow 1 incorrect class width or one incorrect frequency density
may be implied by 5 + 16 + 9 + 14 + 4;

A1: allow SCB1 for correct answer unsupported

Notes

B1: mark the final answer

Notes

B1: ignore extra comments unless they contradict an otherwise correct answer

Notes

M1: NB 62.15567…may be implied by sd = 7.88 or 7.89
NB 0.263047…from \(\sqrt{62.2}\) or 0.263133… from 7.89
NB 0.262871… from 7.88, 0.262973…from unrounded sd

A1: allow B2 for correct answer unsupported

Notes

A1: may be given to 2 sf; allow B2 for correct answer unsupported

Notes

M1: or \(0.18 \times \sqrt{62.2}\)

A1: allow eg \(1.42^2\) for variance

Notes

B1: sample, since not all data used; ignore further reasoning unless contradictory

Notes

B1: do not allow
eg 27085 lies between 2008 and 2010
eg 27085 lies between the values for 2008 and 2010

B1: do not allow
eg positive association
eg positive relationship

Notes

B1: LDS advantage

Notes

B1: allow “not a simple random sample since each plant does not have an equal probability of being selected”
ignore further comments unless contradictory
the essential elements of the comment are in bold

Notes

B1: may be embedded in calculation

B1: may be embedded in calculation

M1: FT their 18 and their 110; allow M1 for \(\frac{91}{120}\) or \(\frac{29}{120}\) oe
or \(\frac{75}{100} \times 120 = 90\) oe

A1: or 92 > 90 so this supports the (owner’s) statement; need comparison and comment

Notes

B1: ignore superfluous comments referring to eg growing conditions, soil type etc
do not allow
eg no, conditions different for other plants
eg no, different plants on different rows
eg no, might be biggest and smallest plants on other rows
eg the first sample is not representative

Notes

B1: B0 for 8.757 explicitly rounded to 8.8

Notes

M1: allow M1 for \(8.8^2\) or \(\sqrt{8.8}\)

A1: \(a = 112.4\) or 112 and \(b = 8.8\) or \(2.97^2\)

Notes

M1: may see \(\mathrm{N}(-\infty, 104.5, 112.4, \sqrt{8.8})\)
NB 0.00387 or 0.0063(06) implies M1
NB 0.00573 or 0.00914 implies M1
NB 0.00379(69..) or 0.00619(81…) may imply M1 FT use of variance = 8.757
NB 0.200(199…) and 0.184(665…) may imply M1 FT use of sd = 8.8
if probability is correctly found to be 0 eg from use of \(\mathrm{N}\left(112.4, \frac{8.8}{205}\right)\) allow M1 only – no further marks available

M1: or compare \(\frac{1}{205}\ (\approx 0.00488)\) with their non-zero 0.00387

A1: or probabilities similar so consistent oe

Alternatively

M1: FT their distribution

Notes

M1: NB awrt 0.0915 or awrt 0.0807 implies M1

A1: unsupported answers score M0

Notes

B1: eg 10 riders rode more than 160 km
or 14 rode 156 km or more
or the reserve should have ridden (approximately) 161 km so (Ali’s complaint is) justified oe

Notes

M1: allow one incorrect frequency density and/or one incorrect class width
NB 8 + 13 + 18 + 20 + 12 soi with four of five correct implies M1
may be implied by 71

A1: if M0 allow SC1 for 8, 13, 18, 20, 12 and no others seen

Notes

M1: use of \(\mathrm{B}(10, p)\) FT their \(p\) to find at least 2 probabilities; may be unsimplified
must see at least 2 probabilities for M1

M1: multiplying their binomial probabilities by 100
must see at least 2 expected values calculated for M1

A1: all correct to 1 dp or better; must add up to 100

Notes

B1: FT dependent on award of M1M1 in (c) and all frequencies calculated

Notes

B1: allow so that the cell is ignored oe
or the software interprets #N/A as no data oe
advantage

Notes

B1: need to refer to BMI being implausible
advantage

B1: advantage

Notes

B1: ignore other comments unless contradictory

Notes

B1: isw

Notes

B1: NB mode is 9

Notes

M1: allow for sight of 11 or 30

Notes

if B0B0 allow SC1 for
true – median smaller in 2019 oe and
false – IQR smaller in 2019 oe

Notes

B1: \(N \times P = 120\) where \(N\) and \(P\) are integers greater than 1.
eg 20 and 6 or 15 and 8

Notes

B1: allow any parameter apart from \(\bar{x}\) for population mean as long as clearly defined as (population) mean

after B marks M1A1M1A1A1 may be earned if working with 2.7054 rounded to 2 or more sf

M1: condone use of 2.6941 (may be rounded) instead of 2.7054 for M1, may be implied by 0.32076 or CR is \(\bar{X} < 4.51797\ldots\), but A1 not available
or \(z = \dfrac{4.8855 - 5}{\frac{2.7054}{\sqrt{120}}}\) for M1

A1: or CR is \((\bar{X}) < 4.5 - 4.6\)
awrt \(-0.46\) A1 (may be implied by \(-0.466\) or \(-0.465567\ldots\) if 2.6941 used)

M1: comparison of their probability with 0.025 or comparison of 4.8855 with their critical value from use of 0.025, as long as previous M1 awarded
their \(-0.4636 > -1.96\) oe (corrected from the printed mark scheme: printed as \(-0.4436\))
M1 dep on award of previous M1

A1FT: may be embedded in conclusion in context

A1FT: do not allow eg conclude / prove / indicate or other assertive statement instead of suggest; A0 if answer spoiled

Notes

B1: from \(5 \times 1.8\)