Notes

1^st M1: for an attempt to rank marks (at least 5 correct in one row) Allow reverse ranking. May be implied by 24

2^nd dM1: dep on 1^st M1 for attempt at calculating \(d^2\) (does not need to be summed). (Allow 1 error squaring their differences)
May be implied by 24

3^rd M1: indep of first two Ms for using correct formula with their \(12 \lt \sum d^2 \lt 240\)
May be implied by 0.8 o.e.

A1: for 0.8 o.e.

Notes

B1: for both hypotheses correct in terms of \(\rho\) or \(\rho_s\) (condone if it appears as \(p\)). Do not allow hypotheses in words on their own.

M1: for selecting the correct critical value. Allow \(\pm\)
If they scored B0 for 2-tail test allow 0.7

A1: dep on \(0.7 \lt \text{“}0.8\text{”} \lt 1\) for a correct conclusion in context stating agreement (or agree) between lecturers. Do not allow correlation for agreement (not contextual)
Can award this mark independently of hypotheses.
Do NOT award this mark if contradictory statements are seen e.g. not significant so lecturers agree.
If a comparison is made then it must be correct.
Note that M0A1 is not possible

Notes

M1: For pairing up the ranks (at least 4 pairs correct) may be implied by a later calculation or awrt 0.571

M1: For an attempt at \(\sum d^2\) condoning slips. Implied by 24 or awrt 0.571 (If \(\sum d^2\) is not correct then look for attempts at the differences proceeding to a value for \(\sum d^2\))

dM1: Dependent on the previous method mark for use of their \(\sum d^2\) in a correct formula (which must be less than 112) Does not need to be evaluated. Implied by awrt 0.571

A1: awrt 0.571 correct answer with no incorrect working seen scores full marks Accept \(\dfrac{4}{7}\)

Notes

B1: For both hypotheses in terms of \(\rho\) or \(\rho_s\) (do not allow \(r\) or \(r_s\))

B1: Correct CV of (\(\pm\)) 0.7143 or better

B1: Correct conclusion in context based on their cv and their \(r_s\) such that \(|r_s| \lt |\text{CV}|\) suggesting there is insufficient evidence to suggest the customers are (generally) in agreement oe / no evidence to suggest a positive correlation between the rankings (of features) by the customers oe. Condone conclusions such as e.g. sufficient evidence to suggest the customers are not in agreement
A comparison statement such as “not sig” is not needed but if seen must be correct. Do NOT award this mark if contradictory comments or working is seen e.g. “reject H₀” or comparison of their 0.571 with 0.05 or e.g. \(0.571 \gt 0.7143\)   (corrected from the printed mark scheme: “their 0.517” should read “their 0.571”)

SC: B0(for 2-tail) B1(for CV \(\pm 0.7587\)) and B1 correct conclusion in context based on their cv and their \(r_s\) such that \(|r_s| \lt |\text{CV}|\) (as in main scheme)

Notes

M1: correct expression for \(r\)

A1: awrt 0.956

Notes

M1: correct expression for gradient, or correct formula stated with \(b\) to more than 3.s.f

M1: correct expression, or correct formula stated with \(a\) to more than 3.s.f
follow through their \(b\)

*A1: correct linear model with \(\boldsymbol{a}\) = awrt 7.42 and \(\boldsymbol{b}\) = awrt 5.12

Notes

B1: correct use of model awrt –5.8

Notes

B1: correct contextual explanation mentioning positive residuals and underestimate o.e.

Notes

B1: cao

Notes

B1: Correct definition. Allow equivalent wording.
Distance from regression line on its own is B0, but allow if vertical distance or \(y\) is referenced

Notes

M1: Use of correct expression for \(r\) or \(r^2\)
Allow use of \(\mathrm{S}_{ty} = 45.5 \times 52.0\ [=2366]\)
or RSS: \(1774155 - \dfrac{(\mathrm{S}_{ty})^2}{52} = 1666567 \rightarrow [\mathrm{S}_{ty} = 2365.2\ldots]\)
and then \(r = \dfrac{\text{awrt 2365 or awrt 2366}}{\sqrt{52.0 \times 1774155}}\)

A1: awrt 0.246   (\(-0.246\) or \(\pm 0.246\) scores M1A0)

Notes

(i) B1: Correct explanation

(ii) B1: Correct evaluation of the fit of the model’s residuals (e.g. variance either side of \(t = 20\) does not appear to be the same)
‘residuals not randomly scattered’ on its own is B0.

Notes

B1: Correct assessment of the conclusion involving the units/size of the variables used to calculate the RSS

Notes

B1: Correct contextual interpretation

Notes

M1: Using pmcc to find \(\mathrm{S}_{xt}\)

M1: Setting up linear model by attempting to find \(b\)
Note: Allow M2 for \(b = r\sqrt{\dfrac{\mathrm{S}_{tt}}{\mathrm{S}_{xx}}}\)

M1: Setting up linear model by attempting to find \(a\)

A1cso*: Correct model \(t = 14.3 - 0.00681x\) with \(a\) = awrt 14.3 and \(b\) = awrt \(-0.00681\) dependent upon all previous M marks.

Notes

B1: Correct model

Notes

B1cso*: Either correct expression

Notes

M1: Using the model to evaluate the squared residual

A1: awrt 82%

B1: Evaluating the result obtained from the model to suggest that this point may be an outlier

Notes

B1: cao

B1: cao

Notes

B1: Correct explanation

Notes

B1: Both hypotheses correct with correct notation (must use \(\rho_s\) or \(\rho\))

B1: Correct critical value 0.7143 or better

M1: Drawing a correct inference using their CV and 0.7106

A1: Drawing a correct inference (condone “marks” instead of ranks) in context using their CV and 0.7106

Notes

B1: Both hypotheses correct written using the notation \(\rho\)

B1: awrt 0.6

M1: Drawing a correct inference using their CV and the value of \(r_s\)

A1: Drawing a correct inference in context using their CV and the value of \(r_s\)

Notes

M1: For realising they need to equate \(1 - \dfrac{6\sum d^2}{9(80)}\) to 0.85 to enable them to find the \(\sum d^2\)

A1: 18

M1: for \(\sum d^2 = 3\)

A1: For using the information in the question with the value for \(\sum d^2\) to deduce that each must contribute 1 to the \(\sum d^2\) and explain why \(B\) must be in position 7

A1: \(C\) 6^th   \(D\) 8^th

Notes

M1: Complete explanation why it decreases

A1: using the information given to deduce that it decreases

Notes

M1: Complete correct method for finding \(r\)

A1: for awrt 0.937

Notes

B1: Correct contextual statement

Notes

M1: For use of a correct model ie a correct expression for \(b\)

M1: For use of a correct model ie a correct expression (ft) for \(a\)

A1: For a correct model \(w = 15.4 + 6.42f\) with awrt 15.4 and awrt 6.42

Notes

B1ft: awrt 3.21 condone \(-3.21\)

Notes

M1: Explaining a reason for their conclusion eg there is a pattern/trend in the residuals
Do not accept residuals not close to zero

A1: concluding it is not valid oe

Notes

B1: A comment about not using a linear line eg use a quadratic model, logarithmic graph exponential

Notes

1st B1 awrt 0.549

Notes

1st B1 Both correct. Require population parameter \(\rho\) and one tailed test.

2nd B1 cv 0.6215

3rd B1 Context required. Must mention concentration of a radioactive element and amount of dissolved solids

Notes

1st M1 for an attempt to rank the concentration of a radioactive element and the amount of dissolved solids with at least 4 correct for each variable. Allow reverse ranks.

2nd M1 for attempt at \(d^2\) row

1st A1 all correct

3rd M1 for use of the correct formula and an attempt to rank, follow through their \(\sum d^2\) if clearly stated
If answer is not correct, a correct expression is required.

A1 awrt 0.714

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H}_1\). Allow use of \(\rho_s\).
Alternative hypothesis compatible with their ranking.

2nd B1 for cv of 0.6429

3rd B1ft for a correct contextualised comment. Must mention concentration of a radioactive element and the amount of dissolved solids. Follow through their \(r_s\) and their cv (provided it is |cv| <1)
Don’t insist on the word “positive” for a one-tailed test.

Notes

B1 for ‘non-linear’ oe, or ‘not normal’

Notes

M1 for an attempt to rank judges lists ( at least 4 correct for each judge)

A1 for correct rankings for both (may be reversed). Can be implied by correct \(d^2\) or \(r_s\)

N.B. Table could be ordered in terms of Senior Judge.

A1 8 or 62 or correct \(d^2\) row

M1 for use of the correct formula, follow through their \(\sum d^2\) (Dependent on 1^st M1)
If answer is not correct, a correct expression is required.

A1 exact fraction or awrt (\(\pm\))0.771

Notes

B1 Both hypotheses in terms of \(\rho\) or \(\rho_s\). Hypotheses just in words e.g. “no correlation” score B0

B1 Accept \(\pm 0.8857\) if 2-tailed \(\mathrm{H}_1\).

M1 Follow through their \(r_s\) and their c.v. if |cv| <1 and \(|r_s| \lt 1\)

A1 ft A correct contextualised comment that includes “judges”.

Notes

B1 ft o.e. Follow through from their cv and \(r_s\)

Notes

Accept ‘already ranked’ oe for ordinal
Accept one variable is not normally distributed

Notes

M1 attempt to find \(d\), \(d^2\) and sum. may be implied by sight of \(\sum d^2 = 14\)

M1 for use of the correct formula, follow through their \(\sum d^2\) if clearly stated. If answer is not correct, a correct expression is required.

A1 0.75 cao

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail. Allow use of \(\rho_s\).
Only award if no errors seen in hypotheses in part(c) and part(d)
Hypotheses just in words e.g. “no correlation” score B0.

B1 0.7143 cao

M1 must correctly reject / not reject the null hypothesis based on their test stat and cv oe

A1 Conclusion must mention ice cream and sunglasses

Notes

M1 for not rejecting / accepting null hyp

A1 must mention ice cream and sunglasses

Notes

1st B1 for labelling/numbering/listing/using sampling frame of all 160 books

2nd B1 for use of random numbers/selection and mentioning the number 10

Notes

1st M1 for an attempt to rank the number of pages (at least 4 correct) Allow reverse ranks

2nd M1 for attempt at \(d^2\) row (may be implied by sight of \(\sum d^2 = 66\) or 264 for reverse ranks)

3rd M1 for use of the correct formula, follow through their \(\sum d^2\) if clearly stated
If answer is not correct, a correct expression is required.

A1 for 0.6 (or \(-0.6\) for reverse ranks)

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H_1}\) (compatible with ranks) Allow use of \(\rho_s\)
Hypotheses just in words e.g. “no correlation” score B0.

2nd B1 for cv of 0.5636   [If they have a two tail \(\mathrm{H_1}\) then allow 0.6485]
Allow \(\pm\) for reverse ranking but must be same sign as \(r_s\)
If hypotheses are the wrong way around this must be B0 but 3rd B1 is possible.

3rd B1ft for a correct contextualised comment. Must mention “librarian” (or he) or “number of pages” and “borrowing”
Follow through their \(r_s\) and their cv (provided it is \(|\text{cv}| \lt 1\))
Don’t insist on the word “positive” or “negative” for a one-tailed test
Use of “association” is B0

Independent of 1st B1 so if \(|r_s| \gt |\text{cv}|\) must say there is sufficient evidence of …….(o.e.) and if \(|r_s| \lt |\text{cv}|\) must say insufficient evidence of … (o.e.) regardless of their hypotheses

Notes

M1 for attempting to rank at least one set of data

A1 for at least one set of data ranked correctly (NB this mark comes after 2nd M1)

M1 for attempting \(\Sigma d^2\)

M1 for correct use of formula for \(r_s\)

Notes

B1 for \(\mathrm{H_0}\) and \(\mathrm{H_1}\) correct (condone \(\leqslant\) for \(\mathrm{H_0}\))

2nd B1 allow 0.7381 if their \(\mathrm{H_1}: \rho_s \neq 0\)

M1 for correct statement relating their test statistic and critical value

A1ft their test statistic, \(\mathrm{H_1}\) and critical value but must be in context.

Notes

B1 require Normal distribution, ignore additional assumptions

Notes

B1 require not Normal and valid reason

Notes

M1 Clear use of \(r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\)

A1 0.840 cao

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H_1}\) must be compatible with their \(r\)
Hypotheses just in words e.g. “no correlation” score B0

2nd B1 for 0.5822 cao

M1 for a statement comparing ‘their r’ with ‘their cv’

A1 for a correct contextualised comment. Must mention positive correlation, be carrying out a 1-tailed test and mention height and weight.
Follow through their \(r\) and their cv (provided their \(|\text{cv}| \lt 1\) and their \(|r| \lt 1\))

Notes

1st B1 for attempt to rank actual weight / Peter’s order with at least 4 correct

2nd B1 for correct rankings for both (one or both may be reversed)

1st M1 for use of \(\sum d^2\) with at least 4 values correct and attempt to add

1st A1 for 70 or 170 with reversed rankings

2nd dM1 for use of the correct formula, follow through their \(\sum d^2\). Dependent on 1st M1
If answer is not correct, a correct expression is required.

2nd A1 for awrt 0.417 or \(\dfrac{5}{12}\)

Notes

1st B1 for both hypotheses in terms of \(\rho\) or \(\rho_s\) One tail \(\mathrm{H_1}\) must be compatible with their ranking
Hypotheses just in words e.g. “no correlation” score B0

2nd B1 for cv of 0.6(00) cao
Their cv must be compatible with their \(\mathrm{H_1}\) which may be in words

M1 for statement comparing ‘their r’ with ‘their cv’

A1 for a correct contextualised comment. Must mention Peter and Men.
Follow through their \(r\) and their cv (provided their \(|\text{cv}| \lt 1\) and their \(|r_s| \lt 1\))

Notes

1st M1 for an attempt to rank no of employees against the populations

2nd M1 for attempting \(\sum d^2\) (must be using ranks) ft their ranks

1st A1 for 22

3rd dM1 dep on 1st M1 for use of the correct formula with their \(\sum d^2\).
If ans. is not correct an expr’ is required.

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H_1}\) must be one tail and compatible with their ranking

M1 for a correct statement relating their \(r_s\) (\(|r_s| \lt 1\)) with their cv but cv must be such that \(|\text{cv}| \lt 1\)

A1 for a correct contextualised comment that is rejecting \(\mathrm{H_0}\)
Must mention “population” and “no. of employees” and “positive correlation”.
Follow through their \(r_s\) and their cv (provided it is \(|\text{cv}| \lt 1\))    Use of “association” is A0

Notes

1st B1 for 0.6319

2nd B1 does not require context just no positive correlation mentioned

Notes

1st B1 for a comment relating to pmcc
(i) no linear relationship   or   (ii) pmcc requires (joint) normal distribution

2nd B1 for a second comment relating to Spearman’s
(i) there is a (non-linear) relationship between ranks.. or (ii) data not (joint) normal so Spearman’s is better

ALT

Notes

1st M1 for an attempt to rank the staff-students ratio or satisfaction ( at least 4 correct)

1st A1 for correct rankings for both (one or both may be reversed)

2nd A1 for \(\sum d^2 = 20\) or correct \(d^2\) row (NB \(\sum d^2 = 92\) for one set of reversed ranks)

2nd dM1 for use of the correct formula, follow through their \(\sum d^2\) (Dependent on 1^st M1)
If answer is not correct, a correct expression is required.

3rd A1 If \(\sum d^2 = 20\) for awrt 0.643 or if \(\sum d^2 = 92\) for awrt – 0.643 (accept \(\pm\tfrac{9}{14}\))

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H}_1\) must be compatible with their ranking
Hypotheses just in words e.g. “no correlation” score B0

2nd B1 for cv of 0.7857 or 0.7143 for one-tailed test (accept \(\pm\))
Their cv must be compatible with their \(\mathrm{H}_1\) which may be in words
If hypotheses are the wrong way around this must be B0 but 3^rd B1 is possible.

3rd B1ft for a correct contextualised comment. Must mention “ratio” or “no. of students per member of staff” and “satisfaction”
Follow through their \(r_s\) and their cv (provided it is |cv| <1)
Don’t insist on the word “positive” for a one-tailed test
Use of “association” is B0
Independent of 1^st B1 so if \(|r_s| \gt |\text{cv}|\) must say there is sufficient evidence of …….(o.e.)
and if \(|r_s| \lt |\text{cv}|\) must say insufficient evidence of … (o.e.) regardless of their hypotheses
Contradictory statements score B0
(This mark is just testing interpretation of comparison of their \(r_s\) and their cv)

Notes

1st M1 for an attempt to rank score \(X\) and score \(Y\)

2nd M1 for attempting d² for their ranks. Must be using ranks.

1st A1 for sum of 18

3rd M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

2nd A1 for awrt 0.786

Notes

1st B1 for null hypotheses in terms of \(\rho\) or \(\rho_s\)

2nd B1 for alt hyp as given

3rd B1 for cv of +0.6429 (or 0.7381 if two tailed from hyp)

M1 for a correct statement relating their \(r_s\) with their cv but cv must be such that |cv|<1

A1ft for a correct contextualised comment. Must mention “scores / rankings” and “manager”
Follow through their \(r_s\) and their cv (provided it is |cv| <1
Use of “association” is A0

(corrected from the printed mark scheme: the second hypothesis is printed as \(\mathrm{H}_0 : \rho \gt 0\); it is the alternative hypothesis \(\mathrm{H}_1 : \rho \gt 0\))

Notes

1st B1 Tied ranks can be implied by 2.5, 6.5 or both 2 or 6 or description.

2nd B1 Average rank implied by 2.5 or 6.5 or description and ‘use of pmcc’.

Notes

1st M1 for an attempt to rank the hardship against calls

2nd M1 for attempting \(d\) for their ranks. Must be using ranks.

3rd M1 for attempting \(\sum d^2\) (must be using ranks)

1st A1 for sum of 28 (or 84)

4th M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

2nd A1 for awrt 0.5 (or -0.5)

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H}_1\) must be two tail.

2nd B1 for cv of \(\pm\)0.7857 (or 0.7143 to ft from 1-tailed \(\mathrm{H}_1\))

M1 for a correct statement relating their \(r_s\) with their cv but cv must be such that |cv|<1

A1ft for a correct contextualised comment. Must mention “Councillor” and “claim” or “hardship” and “number of calls (to the emergency services)”
Follow through their \(r_s\) and their cv (provided it is |cv| <1
Condone use of “association” in conclusion for A1
Condone ‘positive’ in conclusion.

Notes

1st M1 for an attempt to rank the depths against the distances

2nd M1 for attempting \(d\) for their ranks. Must be using ranks.

3rd M1 for attempting \(\sum d^2\) (must be using ranks)

1st A1 for sum of 8 (or 104 for reverse ranking)

4th M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

2nd A1 for awrt (\(\pm\)) 0.857. Sign should correspond to ranking (so use of 104 should get -0.857)

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H}_1\) must be one tail and compatible with their ranking

2nd B1 for cv of 0.8929 (accept \(\pm\))

M1 for a correct statement relating their \(r_s\) with their cv but cv must be such that |cv|<1

A1ft for a correct contextualised comment. Must mention “researcher” and “claim” or “distance (from bank)” and “depth (of water)”
Follow through their \(r_s\) and their cv (provided it is |cv| <1)
Use of “association” is A0

Notes

1st M1 for attempt to rank BMI scores

2nd M1 for attempt at \(\sum d^2\) (must be using ranks)

3rd M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

1st A1ft for a correct expression. ft their \(\sum d^2\) but only if all 3 Ms are scored

2nd A1 awrt \(\pm\) 0.806 (but sign must be compatible with their \(\sum d^2\))

No ranking can score 3^rd M1 only

Notes

2nd B1 for \(\rho \gt 0\) (or <0 but must be one tail and consistent with their ranking)

3rd B1 for critical value that is compatible with their \(\mathrm{H}_1\). If one-tail must be \(\pm\) 0.5636 if two-tail must be \(\pm\) 0.6485 [Condone wrong sign]

M1 for a correct statement relating their \(r_s\) with their cv. e.g. “reject \(\mathrm{H}_0\)”, “in critical region”, “significant result”
May be implied by a correct comment

A1ft for correct comment in context. Must mention low/high BMI and race/fitness or doctor’s belief. Comment should be one-tailed. Allow positive correlation between… but NOT …positive relationship…

No \(\mathrm{H}_1\) assume one-tail for 3^rd B1

Notes

B1 for a correct and relevant comment either based on the fact that the data was originally partially ordered or on the underlying normal assumption
“Quicker” or “easier” score B0

Notes

(i) 1^st B1 for 5 or more points on a straight line of positive gradient

(ii) 2^nd B1 for 4 or more points satisfying \(-1 \lt r \lt 0\)
3^rd B1 for 5 or more points of decreasing ranks not on a straight line

Notes

(i) 1^st M1 for attempting to rank one of the judges (at least 2 correct rankings)
2^nd M1 for ranking both (may be reversed) (at least 2 correct rankings)
3^rd M1 for attempting \(d^2\).
1^st A1 for \(\sum d^2 = 10\)
4^th M1 for correct use of the \(r_s\) formula

(ii) 3^rd B1 for the correct critical value – depends upon their \(\mathrm{H}_1\): \(\rho \gt 0\) needs 0.7143, \(\rho \neq 0\), 0.7857
The \(\mathrm{H}_1\) may be in words so B0B1 is possible. If no \(\mathrm{H}_1\) award for 0.7143 only.

5^th M1 for a correct statement relating their \(r_s\) and their cv (may be implied by correct comment)
3^rd A1ft follow through their \(r_s\) and their cv. Comment in context. Must mention judges.
Don’t insist on “positive” and condone it if they are using \(\rho \neq 0\).

Notes

1^st M1 for attempting to rank both \(P\) and \(Q\).
1^st A1 for both correct (could be reversed)
2^nd M1 for attempting \(d^2\)
2^nd A1 for \(\sum d^2 = 32\).
3^rd M1 for correct use of formula for \(r_S\)

S.C. No ranking Typical answer (−3.82) can get mark for use of \(r_s\) formula and hypotheses in (b) only
(a) M0A0M0A0M1A0     (b) B1B1M0A0

Notes

M1 for a correct comparison or statement about significance (o.e.)
Follow through their \(r_s\) provided \(0 \lt r_s \lt 1\)

A1f.t. for a conclusion in context. Must mention judges or marks or competitor.
If they use correlation they must say it is positive.
Follow through their positive \(r_s\) with their positive c.v. and ignore hypotheses.
So \(r_s = 0.667\) they could say competitor’s claim is not justified etc.

Notes

(−cv from correct table required)   (positive in context)

Notes

Scales & Labels B1; Points B2 (8, 9 points B1)