5 A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination.
The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726 .
Test at the \(5 \%\) significance level whether there is a difference in the ranks of the students taught by the two professors.
[0pt]
[10]
Total Marks for Question Set 3: 38
\section*{Mark scheme}
\section*{Marking Instructions}
a An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed.
b The following types of marks are available.
\section*{M}
A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words "Determine" or "Show that", or some other indication that the method must be given explicitly.
\section*{A}
Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded.
\section*{B}
Mark for a correct result or statement independent of Method marks.
\section*{E}
A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
c When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
d The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'.
e We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.
- When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
- When a value is not given in the paper accept any answer that agrees with the correct value to \(\mathbf { 3 ~ s } . \mathbf { f }\). unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range.
Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
Candidates using a value of \(9.80,9.81\) or 10 for \(g\) should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric.
f Rules for replaced work and multiple attempts:
- If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
- If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
- if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.
For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors.
If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
h If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold "In this question you must show detailed reasoning", or the command words "Show" or "Determine". Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Abbreviations}
| Abbreviations used in the mark scheme | Meaning |
| dep* | Mark dependent on a previous mark, indicated by *. The * may be omitted if only one previous M mark |
| cao | Correct answer only |
| ое | Or equivalent |
| rot | Rounded or truncated |
| soi | Seen or implied |
| www | Without wrong working |
| AG | Answer given |
| awrt | Anything which rounds to |
| BC | By Calculator |
| DR | This question included the instruction: In this question you must show detailed reasoning. |
\end{table}
| Question | Answer | Mark | AO | Guidance |
| 1 | (a) | 0.8392... | B1 [1] | 1.1 | Awrt 0.839 | \(\begin{aligned} S _ { x x } | = 1.7449 \ldots , S _ { y y } = 41.2 \ldots , |
| S _ { x y } | = 7.116 \ldots \end{aligned}\) |
| 1 | (b) | \(y = - 1.180 + 4.0781 x\) | | | | Both coeffs, awrt -1.18 and 4.08 | | Letters correct, needs 1 correct coefficient |
| |
| 1 | (c) | Value of PMCC suggests that there is strong correlation, or 0.75 shown close to mean 0.399 | | 3.5a | | E.g. " \(r\) high so points lie close to line". " \(r\) is high" alone is enough. | | No wrong extras |
| Not "0.75 is close to mean", unless properly justified, e.g. SD (= 0.264) calculated |
| 1 | (d) | Whether \(x = 0.75\) is within the data range | | 3.5b | | E.g. "maximum and minimum values of \(x\) "; not "all data points". | | No wrong extras |
| Or clear reference to interpolation. NB: 95\% CI for \(x\) is ( \(- 0.156,0.954\) ) |
| 2 | (a) | | Po(497) | | \(\mathrm { P } ( \geq 520 ) = 1 - \mathrm { P } ( \leq 519 )\) used correctly | | \(= 0.1564 \ldots\) |
| | | | Stated or implied | | Allow 0.146(08) from 1 \(\mathrm { P } ( \leq 520 )\) | | In range [0.156,0.157] |
| | SC: Normal approx.: | | N(497, 497) B1 | | In range [0.156, 0.157]: B2 |
|
| | | | | | |
| 2 | (b) | Occurrence of a bus is not a random event if it runs on or close to a schedule. | | 2.4 | | Needs context (not just "events"). | | Allow just "buses not random", or "buses not independent because time between buses is regulated" |
| Not "not independent" without such justification. Not "not constant rate". No extras. |
| Question | Answer | Mark | AO | Guidance |
| \multirow[t]{4}{*}{3} | (a) | | \(\mathrm { H } _ { 0 } : \mu = 500 , \mathrm { H } _ { 1 } : \mu < 500\) | B1 | 1.1 | One error, e.g. \(\mathrm { H } _ { 1 } : \mu \neq 500\), or \(\mu\) not defined, or all in words: B1 | \(x\) or \(\bar { x } : 0\) unless defined as population mean (then B1) |
| | | \(\begin{aligned} | \bar { X } \sim \mathrm {~N} \left( 500 , \frac { 80 ^ { 2 } } { 40 } \right) = \mathrm { N } ( 500,160 ) \text { and } \bar { X } |
| \mathrm { P } ( \bar { X } < 473 ) = 0.01640 \text { or } z = - 2.13 ( 45 ) |
| \text { or } \mathrm { CV } = 470.6 \end{aligned}\) | М1 | 3.3 | \(p\) or \(z\) correct to 3 sf . | Can be implied by 0.0164, 0.9836, 0.433, 0.198, 0.000 but not 0.3679 or 0.00127 |
| | | \(p > 0.01\) or \(z > - 2.326 \quad\) or \(473 > 470.6\) | A1 | 1.1 | Compare \(p\) with 0.01 or \(z\) with -2.326, or 2.326 used in CV | Must be like-with-like, Not e.g. 0.9836 > 0.01 or \(p < 2.326\) |
| | | Do not reject \(\mathrm { H } _ { 0 }\). Insufficient evidence that greatest weight that new design can support is less than the greatest weight that the traditional design can support. | | 1.1 | Correct first conclusion, needs correct method and like-with-like, ft on test statistic if method correct Contextualised, not too definite | But BOD if no explicit comparison of \(p\) with 0.01 Not "the new design does not have a smaller greatest weight . . ." |
| 3 | (b) | | Standard deviation/variance remains unchanged, or sample must be random | B1 [1] | 1.2 | No extras. Not "same distribution". | Not "assume normal"; this is not needed |
| 3 | (c) | | | Either: Yes as we do not know that the distribution of weights for the new design is normal | | Or: \(\quad\) No as the population distribution known to be normal |
| B1 [1] | 2.1 | Allow "population distribution assumed to be normal". No extras, e.g. "and sample size is large". | Allow "yes as we do not know that the distribution for the new design is normal" only if clearly refers to the new design only |
| Question | Answer | Mark | AO | Guidance |
| 4 | (a) | \(9 \times \frac { 40 } { 9 } = 40\) | B1 [1] | 1.1 | 40 or awrt 40.0 only | |
| 4 | (b) | \(\frac { 1 - p } { p ^ { 2 } } = \frac { 40 } { 9 }\) | М1 | 3.1b | Use correct formula for variance | SC: insufficient working, \(\frac { 3 } { 8 }\) only: M0B1 for \(\frac { 3 } { 8 }\), then B0 |
| | \(\begin{aligned} | \mathrm { E } ( D ) = 1 / p \quad \left[ = \frac { 8 } { 3 } \right] |
| \mathrm { E } ( 3 D + 5 ) = 3 \times \frac { 8 } { 3 } + 5 \quad [ = 13 ] \end{aligned}\) | B1ft | 2.3 | - formula for \(\mathrm { E } ( D )\) | | Allow for explicit rejection of a solution even if both are wrong | | \(p\) doesn't need to be between 0 and 1 for either of these marks |
|
| | | A1ft [6] | 1.1 | \(3 \times (\) their \(\mathrm { E } ( D ) ) + 5\) | |
| SC: \(\frac { 1 - p } { p ^ { 2 } } = 40\) (their 40), \(p = \frac { - 1 \pm \sqrt { 161 } } { 80 }\), reject negative solution, \(\mathrm { E } ( D ) = \frac { 1 + \sqrt { 161 } } { 2 } = 6.844 , \mathrm { E } ( 3 D + 5 ) = 25.53 : \quad \mathrm { M } 1 , \mathrm { M } 1 \mathrm {~A} 0 , \mathrm {~B} 1 , \mathrm {~B} 2\) total \(5 / 6\) |
| 4 | (c) | | | | | |
| | \(\begin{aligned} | \mathrm { P } ( D > \mathrm { E } ( D ) ) = \mathrm { P } ( D \geq 3 ) |
| = ( 1 - p ) ^ { 2 } |
| = \frac { 25 } { 64 } \text { or } 0.390625 \end{aligned}\) | | | | Convert inequality to integer, their \([ 1 / p ] + 1\), allow > | | \(( 1 - p ) ^ { r } , \mathrm { ft }\) on their \(p , r\), e.g. 8/3 or 13 | | Allow \(( 1 - p ) ^ { 3 } = 125 / 512\) or 0.244 | | Answer, exact or art 0.391, www |
| | Not their 13 | | \(( 1 - p ) ^ { 8 / 3 } [ 0.286 ]\) : M0M1A0 | | Need \(0 < p < 1\) here | | Allow \(( 1 - p ) ^ { 6 } = 0.3876\) from SC above |
|
| Question | Answer | Mark | AO | Guidance |
| \multirow[t]{10}{*}{5} | \multirow{10}{*}{} | \(\mathrm { H } _ { 0 } : m _ { Q } = m _ { R } , \mathrm { H } _ { 1 } : m _ { Q } \neq m _ { R }\), where \(m _ { Q }\) and \(m _ { R }\) are the medians of the rankings given to \(Q\) and | B1 | 1.1 | Allow \(m\) undefined. If verbal, must mention medians, \(m\) or distribution. Allow \(m _ { d } = 0\) as opposed to \(m Q = m _ { R }\) | Not anything that might be \(\mu\) unless symbol clearly defined as median. Not "there is no difference in the ranks ..." |
| | Sum of ranks \(= 1 / 2 \times 54 \times 55 = 1485\) | М1 | 1.1 | Find sum of ranks | |
| | \(R _ { m } = 1485 - 726 = 759 \quad\) [or 561] | A1 | 1.1 | Correct value of \(R _ { m }\) seen | Allow even if 726 used later |
| | \(\begin{array} { r } R _ { m } \sim \mathrm {~N} ( 660 , |
| \quad \ldots 3300 ) \end{array}\) | | | normal, mean their \(\frac { 1 } { 2 } \times 24 \times\) 55 | |
| | | | | | Allow SD/Var muddle |
| | \(\begin{aligned} \mathrm { P } \left( R _ { m } \geq 759 \right) | = 0.0432 \text { (3 s.f.) } |
| { [ \text { or } z } | = 1.715 ] \end{aligned}\) | | | | |
| | | | | | Both parameters correct Standardise, their \(R _ { m }\) | | Correct test statistic (0.0432) 0.0424 or 0.0416 (no/wrong cc): M1A0 |
| (Same for \(\mathrm { P } \left( R _ { m } \leq 561 \right)\) Allow \(z \square \in [ 1.71,1.715 ]\), allow \(z = 1.72\) only if cc demonstrated correct |
| | | Alternatively: \(\operatorname { CV } 660 + 1.96 \sqrt { } 3300 [ = 772.6 ]\) | | 758.5 < 772.6 |
| M1 A1 | | Not 759 - or 726 - ...; not wrong tail for comparison, but allow ± Needs correct cc | Or 561.5 > 547.4 Wrong \(z\)-value: M1A1ft B0 |
| | \(p > 0.025,2 p > 0.05 , z < 1.96\), or 1.96 used in CV | B1 | 1.1 | Explicit correct comparison | Needs like-with-like (e.g. \(p\) must be < 0.5) |
| | Do not reject \(\mathrm { H } _ { 0 }\). Insufficient evidence of a difference between the ranks. | | | Correct first conclusion, needs correct method and like-with-like Contextualised, not too definite | ft on wrong ts, or 1-tail/2-tail confusions, e.g. \(p\) compared with 0.05 or not explicit, or \(z \geq 1.645\) |
| \includegraphics[max width=\textwidth, alt={}]{6cdb3135-90ca-42f1-bab1-a4b35451cea2-10_54_1750_1703_611} |