Notes

B1: for 2.8 (may be seen in a calculation but must be seen somewhere) oe

M1: for a correct expression for \(s^2\) Can ft their mean or implied by correct answer.

A1: for 1.23 oe

Notes

B1: for mention of “normal” and “equal variance” (or standard deviation)

Notes

1^st B1: for correct hypotheses, both in terms of \(\mu\). Can be \(\mu_1\) and \(\mu_2\) etc

1^st M1: for correct attempt at \(s_p^{\,2}\) ft their value for \(B\) ;

1^st A1: for awrt 1.45 (implied by 2^nd A1)   \(\left(\frac{1599}{1100}\right)\)

2^nd M1: for correct ft expression for test statistic using 3.7 allow (\(\pm\)) standardisation ;

2^nd A1: for awrt (\(\pm\)) 3.93 or awrt (\(\pm\)) 3.94

2^nd B1: for (\(\pm\))1.982 (or better)

3^rd A1: dep on all previous Ms and \(1 \lt |\text{cv}| \lt \) “3.93” for correct conclusion mentioning “mean delivery rate”. Do not allow contradictory statements. Incorrect comparisons A0.

Notes

1^st B1: for both hypotheses in terms of \(\mu\) (or \(\mu_d\)) but not \(\mu_1\) and \(\mu_2\) etc

1^st M1: for attempting differences (at least 6 correct values) implied by mean or \(s_D^{\,2}\) or \(s_D\)

2^nd M: for a correct express’n for \(t\) (ignore df) with their values for \(\bar{x}_D\) and \(s_D\) (correct method)

1^st A1: for awrt \(\pm 2.20\) or \(\pm 2.205\) (accept 2.2 if correct expression or \(\bar{x}_D\) and \(s_D\) seen)

2^nd B1: for correct cv i.e. \(\pm 2.262\) or better

2^nd A1: dep on all previous Ms and “2.20” \(\lt |\text{cv}| \lt 3\) for correct conclusion in context. Do not allow contradictory statements. “mean delivery rate” or incorrect comparison is A0.

SC: If they carry out a \(t\)-test for two independent samples then allow 2nd M1 for standardising with diff of their means and pooled variance and 2nd B1 for cv of \(\pm 2.101\)

Notes

B1: for awrt 0.683 or awrt 68.3% o.e. Do not accept a fraction

Notes

1^st B1: for correct hypotheses in terms of \(\sigma\) or \(\sigma^2\) Do not accept in words.

1^st M1: for a correct expression for \(s\) or \(s^2\)   Implied by \(\dfrac{181}{12}\) or 15.0833…

2^nd M1: for a correct expression for ts (ft their 15.0833…) May be implied by awrt 10.1

1^st A1: for awrt 10.1 or may see \(\dfrac{181}{18}\)

2^nd B1: for the correct cv of 13.848 (accept 13.8 or better or allow 13.85)

2^nd A1: for a correct conclusion in context (independent of hypotheses but dep on M2 and their ts < cv where 12 < cv < 15). Incorrect comparison or contradictory statement is A0

Notes

M1: for an attempt at a correct formula with 130 or their 3.88… and \(\sqrt{25}\) and \(t \gt 2\)

1^st A1ft: for a correct expression using \(t = 2.064\) (or better) can ft their 3.88… = their \(s\)

2^nd A1: for awrt (128, 132) provided M1 clearly scored.

Notes

(i) B1: dep on a CI in (c) which includes 131 and concluding variance lower in (b). For stating treatment was successful (o.e. condone e.g. yes) and mentioning 131 (or referred to as the mean) is inside CI. Do not allow reference to critical region for CI.

(ii) M1: for evidence of a suitable \(\sigma\) used (ft their \(s\)) and a value of \(\mu\) from their CI.
Must see their values for \(\mu\) and \(\sigma\) to score.

A1: dep on \(\sigma\) = awrt 3.9 and \(\mu = [\text{awrt}\,128,\ \text{awrt}\,132]\) for an answer in the range 0.79 to 0.88 inclusive (decimal or %)

NB: \(\sigma = 3.9,\ \mu = 128.4\) gives 79.5%;    \(\sigma = 3.9,\ \mu = 131.6\) gives 87.2%
\(\sigma = 3.9,\ \mu = 131\) gives 87.6%

Notes

B1: for both hypotheses correct in terms of \(\sigma\) or \(\sigma^2\)
May use labelling of e.g. 1 and 2 or e.g. “with” and “without”. Condone \(x\) and \(y\).

M1: for a correct numerical expression for \(F\) (must have squares but does not need to be evaluated) May be implied by awrt 5.05
Could be reciprocal.

B1: for 6.26 (or better)
If reciprocal (0.198…) is used as test statistic then need \(\dfrac{1}{5.19}\) or awrt 0.193

A1: dep on a critical value such that awrt \(5.05 \lt \text{cv} \lt 7\) for a correct test statistic awrt 5.05, and correct conclusion. Condone e.g. variances are the same. Do not accept conclusions which do not refer to the variances. If an incorrect comparison or statement is made A0

Notes

(i) B1: for recognising that the test (for difference of means) requires the populations to have the same variance oe. Must indicate that it is a requirement (not just that you can assume the variances are equal). Must be in words.
Allow explanations suggesting that it enables a pooled estimate for \(\sigma^2\) or a better estimate of \(\sigma^2\) to be found (so that a \(t\)-test can be carried out on the difference between means of two independent distributions with unknown variances)
Do not allow references to the normal distribution being used.

(ii) B1: for a correct conclusion in context mentioning increase in mean yield.
Does not require the comment about variances.

Notes

B1: The idea that samples are not independent. Condone other irrelevant comments provided they do not contradict this.

Notes

B1: For one suitable comment on twins being identical relating to selecting the sample. Condone start weight. Do not accept age / diets of the lambs

B1: For a second suitable comment.

Notes

B1: Need the emboldened words.

Notes

M1: attempting differences (at least 4 correct) implied by awrt 0.304 or 0.551 but not 0.2125

M1: attempt to find \(\bar{w}\) and \(s\) or \(s^2\) for their differences implied by 0.2125 and either awrt 0.304 or 0.551

M1: For using the correct formula their \(\bar{w} \pm t \times \sqrt{\dfrac{\text{their } s^2}{8}}\) where \(|t| \gt 2\) all values need to be substituted in.

A1ft: for their \(\bar{w} \pm\) awrt \(2.998 \times \sqrt{\dfrac{\text{their } s^2}{8}}\) all values need to be substituted in

A1: dependent on all previous method marks (awrt -0.372, awrt 0.797)

SC: If they have carried out a CI for two independent samples allow 3^rd M for using their difference of their means and a pooled variance and A1ft using correct formula with awrt 2.624

Notes

B1: For both hypotheses correct in terms of \(\mu\) or \(\mu_w\)   Condone 200

M1: For changing 200 g to 0.2 kg (ignore units for this mark) and comparing to their CI

A1ft: Independent of hypotheses. Drawing a correct inference following through on their CI provided 0.2 is within their confidence interval, with no contradictory statements. Does not need to be in context. Accept “insufficient evidence to support the (researcher’s) belief”.

Notes

B1: For realising a normal distribution must be used as a model and finding the correct value 1.6449 or better.

M1: For \(30.03 \pm \dfrac{0.868}{\sqrt{15}} \times \text{“}z\ \text{value}\text{”}\). May be implied by a correct CI

A1: awrt 29.7 and 30.4 from correct working (B0M1A1 is possible)

Notes

B1: For realising a chi squared distribution must be used as a model and CV = awrt 21.06

M1: Correct method comparing \(\dfrac{14S^2}{0.868^2}\) to \(19 \lt \chi^2_{14} \lt 24\) (condone equals instead of >)
Ignore any additional calculations outside of this range.

A1: Correct CR allow awrt 1.13 only

Notes

M1: Dependent on an answer to part (a) or part (b) in order to draw an inference. Drawing a correct inference that there is insufficient evidence to suggest that the machine is not working properly following through one of their CR or CI with one correct comparison. Must mention the machine at least once.
If they have either 30.06 outside their CI or 1.02² in their CR then allow an inference that the machine is not working properly for this mark following a correct comparison and no contradictory statements relating to that comparison.
This mark can still be scored if there is a second comparison which is incorrect.

A1ft: Dependent on 30.06 in their confidence interval and 1.02² not in the CR. For drawing a correct inference following through their CR or CI with both correct comparisons and no contradictory statements. Must mention the machine at least once. Do not accept a comparison of 30.06 with just one of the limits for the CI.

Notes

B1: correct explanation

Notes

B1: correct modelling assumption

Notes

M1: for setting up paired \(t\)-test with at least 5 correct differences consistent in direction

M1: complete method for \(\bar{d}\) and \(s_\mathrm{d}\) or \(s_d^{\,2}\)

B1: Use of \(t\)-distribution for differences with both hypotheses correct in terms of \(\mu\) / \(\mu_\mathrm{d}\)
Allow \(\mu_\mathrm{d} \lt 0\) if candidate does test with negative test statistic

M1: method for finding test statistic with their values

A1: awrt \(\pm\)2.30

B1: correct critical \(\pm\)1.895 (or better) with compatible sign

A1ft: drawing a correct inference in context
Must be consistent with their CV and a standardised test statistic from \(t\)-distribution
Must mention “percentage” and “words” or “psychologist’s claim”

SC: If they have carried out a \(t\)-test for two independent samples they can potentially score: M0M0B0M1A0B1A1ft
If they do follow this path, they can score:
3^rd M1 for standardising with difference of their means and a pooled variance
2^nd B1 for a CV of 1.761
A1ft for a correct contextual conclusion for their test statistic and 1.761

Notes

M1: correct expression for \(s_A^2\)

B1: for selecting \(\chi^2\) with \(v = 8\). May indicated in notation or either of 17.535, 2.180

M1: setting up 95% CI with their sample variance, chi-squared values, and 8

A1: correct CI with awrt 9.35 and awrt 75.2

Notes

B1: both hypotheses correct using \(\sigma\) or \(\sigma^2\)

M1: using the \(F\)-distribution as the model eg \(\dfrac{s_A^2}{s_B^2}\)

B1: awrt 3.07

A1: Drawing a correct inference following through their CV and value for
Allow \(\sigma_B^2 = \sigma_A^2\)
Allow standard deviation instead of variance

NB: Allow candidates to use their \(\dfrac{s_B^2}{s_A^2}\) with \(\dfrac{1}{3.35}\) = awrt 0.299 for the final M1B1A1 in (b)

Notes

B1: 2.3263 or better

M1: for \(128 \pm \sqrt{\dfrac{3.5^2}{80}} \times \text{“}z\ \text{value}\text{”}\). \(|z| \gt 1\)   May be implied by a correct CI

A1: awrt 127.09 and awrt 128.91

Notes

B1: Both hypotheses (oe) correct with correct notation (if using \(\mu_x\) and \(\mu_y\) these must be defined)

Notes

M1: setting up normal model with correct mean or variance
May be shown in standardisation with correct mean or variance and \(|z| \gt 1\)

A1: correct variance and mean, may be seen in standardisation

M1: setting up an equation or inequality by standardising with their mean and their standard deviation and \(|z| \gt 1\)

B1: Correct critical value 1.6449 or better, allow 1.64 or better

M1: rearranging to solve for \(n\) (may be implied by 73.9…)

A1: \(n = 74\) cao

Notes

B1: correct explanation mentioning large sample size and the means being distributed normally

Notes

B1ft: drawing a correct inference in context, must be consistent with their value of \(n\)

Notes

B1: Correct modelling assumption. Allow air pollutant samples are independent.
Samples of air pollutant are normally distributed is B0

Notes

B1: Setting up either \(F\) statistic

M1: Either correct inequality (condone equation)

M1: Understanding both correct inequalities are needed (condone equation)

A1: Correct range of values with awrt 1.13 and awrt 28.8
\(s_B^2 \lt 1.12698\ldots\) and \(s_B^2 \gt 28.755\) is A0

Notes

M1: For realising a \(\chi^2\) distribution must be used as a model with at least one value correct. Allow \(\chi^2_{12,\,0.01} = 26.217\) or \(\chi^2_{12,\,0.99} = 3.571\) for this mark.

M1: Setting up 99% CI

A1: Correct CI with awrt 2.71 and awrt 24.9

Notes

M1: For understanding paired \(t\)-test is required and attempting to find differences at least 5 correct, allow \(\pm\)   (may be implied by correct \(\bar{d}\) and \(s_\mathrm{d}\))

M1: Complete method for \(\bar{d}\) and \(s_\mathrm{d}\) or \((s_\mathrm{d})^2\)

B1: Correct model for differences with both hypotheses correct in terms of \(\mu\) / \(\mu_\mathrm{d}\) (sign of \(\mathrm{H}_1\) must be compatible with their \(\bar{d}\))

M1: Method for finding test statistic with their values

A1: awrt \(\pm 2.80\) (allow awrt \(\pm 2.8\) from correct working)

B1: Correct critical \(\pm 1.943\) (or better) with compatible sign

A1: Correct comparison to deduce that the doctor’s belief is supported. Must be consistent with their CV and their test statistic (dependent upon all M marks).

SC: Difference of means test apply scheme but also allow 2^nd B1 for \(t_{12} = 1.782\)

Notes

B1: Correct modelling assumption

Notes

B1: Understanding that sample mean can be modelled using a Normal distribution with \(z = 1.96\)    Allow 1.959… or better from calculator

M1: Setting up confidence interval \(57.2 \pm z \times \dfrac{1.2}{\sqrt{120}}\)

A1: awrt (57.0, 57.4)   Must come from correct \(z\) value = 1.96 (or better).

Notes

M1: Translating context into a Normal distribution model

A1: Correct mean

A1: Correct variance (allow awrt 0.0178 or exact fraction \(\dfrac{249}{14000}\))

B1: Correct explanation about the distributions of \(Y\) and \(W\) or \(\overline{Y}\) and \(\overline{W}\)

Notes

B1: Both hypotheses (oe) correct with correct notation (if using \(\mu_x\) and \(\mu_y\) these must be defined).

M1: Standardising using normal distribution test statistic for difference of two means with known variance

A1: awrt 2.25

B1: Correct critical value 1.6449 or better, [or \(p\) = 0.01 or better from correct working]

A1: Drawing a correct inference in context. Do not allow contradictory statements, e.g. ‘Do not reject \(\mathrm{H}_0\), so Jamie’s claim is supported’

Notes

B1: For both hypotheses in terms of \(\sigma\)

M1: Correct method for \(s_r^2\) or \(s_{mb}^2\)

A1: awrt 2.79 (allow \(\tfrac{39}{14}\))

A1: 4 cao

M1: Using correct model for test statistic with correct ratio

B1: Correct CV

A1ft: Drawing a correct inference in context using their CV and their test statistic (dep on both M marks)

Notes

M1: Either correct attempt at \(\chi^2_{5,\alpha}\) with \(\nu = 5\)

M1: Using tables to find appropriate probability

A1: 99 cao

Notes

M1: Correct expression for \(s_p^{\,2}\)

B1: Correct 95% \(t\)-value

M1: CI in the correct form (may be implied by either A mark)

A1: awrt 0.115

A1: awrt 4.39

Notes

B1: Both hypotheses (oe) correct with correct notation (if using \(\mu_x\) and \(\mu_y\) these must be defined).

M1: Calculation of standard error

M1: Standardising using normal distribution test statistic for difference of two means with known variance

A1: awrt 2.09

B1: Correct critical value 2.3263 or better

A1: Drawing a correct inference in context

Notes

B1: Correct explanation

Notes

B1: Understanding that the assumptions required for the hypothesis test, by CLT sample means follow a normal distribution

dB1: (dep on previous B1) Correct evaluation

Notes

1^st B1 for finding mean from CI or calculation of width of CI

1^st M1 for using the given \(t\) model to form an equation in \(s\). (Allow \(t\) for 2.064 where \(2 \lt t \lt 3\))

1^st A1 for correct use of \(t_{24} = 2.064\)

2^nd A1 for \(s = 0.21\) or better

2^nd B1 for correct hypotheses in terms of \(\sigma\).

2^nd M1 for selecting the appropriate model for this test

3^rd A1 for test statistic awrt 34.8

3^rd B1 for at least one correct critical value

4^th A1 for a correct conclusion confirming that assuming st. dev = 0.175 is OK

Notes

M1 for use of correct formula with \(1.6 \lt z \lt 2\) (ft \(\bar{x}\) if found in (a))

B1 for \(z = 1.96\) or better used

A1 for an interval awrt (1.21, 1.35)

Notes

1^st B1 for both hypotheses in terms of \(\sigma\).

1^st M1 for at least one of the \(s^2\) calculations correctly attempted
NB \(s_a = 155.72\ldots\) accept awrt 156   and \(s_b = 79.137\ldots\) accept awrt 79.1

2^nd M1 for a correct calculation of the test statistic (ft their \(s^2\))

1^st A1 for awrt 3.87

2^nd B1 for correct cv awrt 3.37

2^nd A1 for a correct conclusion mentioning Gina’s belief (o.e.)

Notes

B1: For realising a \(\chi^2\) distribution must be used as a model and finding a correct value

M1: For realising the need to set \(\dfrac{9s^2}{\text{“}\text{smallest } \chi^2\text{”}} = 0.2328\) or \(\dfrac{9s^2}{\text{“}\text{largest } \chi^2\text{”}} = 0.01712\)

dM1: correct method used to solve equation to find \(s^2\)

B1: awrt 4.84

B1: Both hypotheses correct using the notation \(\mu\)

B1: \(\pm\) 1.833

M1: For us of correct formula ie \(\pm\dfrac{\text{“}\text{their } 4.84\text{”} - 5}{\sqrt{\text{“}\text{their } 0.0449\text{”}/10}}\)   If “4.84” not shown it must be correct here

A1: \(-2.39\)

A1ft: Drawing a correct inference following through their CV and test statistic (must have matching signs)

NB if chi squared values not shown
\(s^2 = 0.045\) or 0.0449 award B0 M1M1 for awrt 0.04487 award B1 M1 A1

Use of \(2(2.5758)\dfrac{\sigma}{\sqrt{10}} = 0.21568\) gives \(\sigma = \sqrt{0.0175}\) could get B0M0M0B1B1B1M0A0A0
Unless continue to get \(s^2 = \dfrac{10}{9}0.0175 = 0.0194\ldots\)

Use of \(2(1.833)\dfrac{s}{\sqrt{10}} = 0.21568\) gives \(s = 0.1860\) could get B0M0M0B1B1B1M1A0A1

Notes

M1: for realising that the model to use is the paired \(t\)-test and finding the differences \((\pm)\) At least 3 correct

M1: correct method for finding \(\bar{d}\) and \(s_\mathrm{d}\).

B1: Using a correct model for difference and both hypotheses correct using the notation \(\mu_\mathrm{d}\) or \(\mu\)
Condone \(\mu_I = \mu_{II}\) and \(\mu_I \neq \mu_{II}\)

M1: Using the correct method to find test statistics ie \(t = \pm\dfrac{\text{“}\text{their } 4.5\text{”}\sqrt{8}}{\text{“}\text{their } 7.09\ldots\text{”}}\)

A1: awrt 1.79 or 1.8

B1: for correct critical value \(t = \pm 3.499\) with compatible sign

A1ft: Drawing a correct inference in context using their CV and their value of \(t\)

NB difference of means test gets M0M0B1M0A0B0A0

Notes

B1: both hypotheses correct using \(\sigma\) or \(\sigma^2\)

M1: Using a correct method for either \(s_A^2\) or \(s_B^2\). May be implied by a correct value

A1: awrt 5.04

A1: awrt 1.62

M1: Using the F-distribution as the model eg \(\dfrac{s_A^2}{s_B^2}\)   \(\left(\text{allow } \dfrac{s_B^2}{s_A^2}[= 0.321\ldots]\right)\)

B1: awrt 1.96 or 0.506 must match their method

A1ft: Drawing a correct inference following through their CV and value for \(\dfrac{s_B^2}{s_A^2}\) or \(\dfrac{s_A^2}{s_B^2}\)
Allow \(\sigma_B^2 \neq \sigma_A^2\)   Allow standard deviation instead of Var. Do not allow \(\sigma_B^2 = \sigma_A^2\)

Notes

B1: recalling the fact that the variable yield needs to be normally distributed

Notes

B1 both hypotheses. allow \(\mathrm{H}_0 : \sigma_T = \sigma_G\) against \(\mathrm{H}_1 : \sigma_G \gt \sigma_T\). Must use \(\sigma\) or \(\sigma^2\) and make clear which is \(\mathrm{H}_0\) and which is \(\mathrm{H}_1\). Do not allow in words

M1 allow 0.31 and 0.66 rather than 0.31² and 0.66² if they write the formula down

B1 correct CV for their \(F\) or a correct comparison if use \(p\)

Final A1: – All previous marks must be awarded. Variances are the same or var are not different

Notes

M1 Allow use of 0.31 and 0.66. May be seen in part (a)

B1 both hypotheses using \(\mu\). Do not allow \(\geqslant\) sign instead of >. May use different letters eg \(A\) and \(B\) but they must be defined.

B1 correct CV but must match \(t\)-value or a correct comparison if use \(p\)

M1 use of correct formula with their \(s_p\) – condone missing 4

M1 use of correct formula with their \(s_p\). (which must have been attempted)

A1 A correct statement or longhand of suppliers claim with the word force and mean and is more than 4 greater oe   Do not allow contradicting statements.

Notes

M1 correct LHS with 1.833 ( or their CV used in (b)) NB subst in 4.55 for is M0

Notes

B1 for a comment that mentions “differences” and “normal” distribution

Notes

M1 attempting differences

M1 attempt to find \(\bar{d} = \dfrac{\sum \text{"their } d\text{"}}{8}\)

M1 attempting \(s\) or \(s^2\ \dfrac{1}{7}\left(\sum \text{"their } d^2\text{"} - \dfrac{\left(\sum \text{"their } d\text{"}\right)^2}{8}\right)\)   \(s = 0.8192\ldots\)

M1 for attempting the correct test statistic \(\dfrac{\bar{d} - m}{s/\sqrt{8}}\), allow any letter

B1 Both critical values correct (ignore sign)

M1d dependent on previous M being awarded. Having a pair of equations with the same sign and one of each CV. Ft their test statistic and CV

A1ft ft their CV four equations, may be implied by both ranges correct

A1 awrt \(0.049 \lt m \lt\) awrt 0.377   allow \(\leqslant\) instead of <

A1 awrt \(1.75 \lt m \lt\) awrt 2.08   allow \(\leqslant\) instead of <

NB if test stat the wrong way round remove one of the A marks awarded at the end

(corrected from the printed mark scheme: the scheme prints \(t_7(0.005\%)\); 3.499 is the 0.5% point)

Notes

1st B1 Sampling frame in context. Accept list of all standard and premium ticket holders at the concert.

2nd B1 Use of random selection eg simple random sampling within strata

3rd B1 Accept description of \(n_s, n_p\).

Notes

1st & 2nd B1 for hypotheses. Accept \(\mu_1, \mu_2\) or \(\mu_A, \mu_B\) etc if it is clear which is which.

1st M1 for an attempt at se with 3 out of 4 values correct.

Condone switching 10 and 8: \(\sqrt{\dfrac{10^2 \text{ or } 8^2}{60} + \dfrac{8^2 \text{ or } 10^2}{55}}\)

2nd dM1 dependent on 1^st M1 for a correct numerator (must have - 6) and ft their se.

1st A1 for awrt 1.19

3rd B1 for \(\pm\) 1.6449 seen or probability of awrt 0.117, Sign must match their test statistic.

3rd dM1 dep. on 1^st M1 for a correct statement based on their normal cv and their test statistic. Ignore their hypotheses. Allow accept \(\mathrm{H}_0\) but reject \(\mathrm{H}_1\) is M0. Can be implied by correct conclusion.

2nd A1cso for correct comment in context dependent upon all other marks being awarded.
Must mention merchandise, standard and premium ticket holders and 6 or manager and belief or claim

NB Use of cv for difference in means \(D\) will have \(D = 6 + 1.6449 \times \text{s.e.} =\) awrt 8.33 and requires sight of \(d = 8\) with a comment for the 3^rd M1

Notes

1st B1 for mentioning large samples and CLT

2nd dB1 dependent on 1^st B1 for stating no need to assume normality. Require merchandise sold not mean merchandise sold.

Notes

1^st B1 Both hypotheses with \(\mu\).

1^st M1 Allow \(\pm\dfrac{202 - 200}{\frac{s}{\sqrt{10}}}\)

1^st A1 awrt 3.33

2^nd B1 allow \(p\) value of awrt 0.00438 in place of critical value. CV must follow from \(\mathrm{H}_1\), sign must match \(t\)-value or be \(\pm\)

2^nd A1ft ft \(t\)-value if awarded B marks. Need correct conclusion in context containing the words mean weight, almonds or packets and 200g

Notes

1st B1 Accept equivalent rearranged equation. Definitions of parameters must be clear e.g. use of 1 and 2 without definitions scores B0. Accept ‘\(x\)’ for ‘new’ and ‘\(y\)’ for ‘old’ as defined in the question.

2nd B1 Accept equivalent rearranged strict inequality. Definitions of parameters must be clear e.g. use of 1 and 2 without definitions scores B0. Accept ‘\(x\)’ for ‘new’ and ‘\(y\)’ for ‘old’ as defined in the question.

M1 for attempting standard error. Condone swapping 7 and 6. Accept 6.86 and 5.85 for 7 and 6.

dM1 for 1/ “their standard error”
A1 for awrt 1.86
NB -1.86 is A0.
If 8 missing from \(\mathrm{H}_1\) then accept \(z = \dfrac{9}{0.5385\ldots} =\) awrt 16.7 and must be consistent with their \(\mathrm{H}_1\).

B1 Accept \(\pm\) or probability of 0.9686

(Or 0.95<0.9686)

A1cso Correct comment in context. Must mention “engineers claim” or “battery”, “old”, “new” and “8”.

Notes

2nd B1 Must mention means and normal. No assumptions are being made so B0 if key to answer.

Notes

(i) B1 Allow because the same person has been used. Do not allow 2 data sets.

(ii) B1 for a comment that mentions “differences” and “normal” distribution

Notes

SC for two sample test they may get M0M0 M0B1M0A0B1A0
\(\mathrm{H}_0 : \mu_{\mathit{first}} = \mu_{\mathit{final}} + 1, \quad \mathrm{H}_1 : \mu_{\mathit{first}} \gt \mu_{\mathit{final}} + 1, \quad \pm 1.761\)

M1 for attempting the \(d\)s, at least 2 correct implied by the figures (\(\Sigma d = 17\), \(\Sigma d^2 = 103\), \(\bar{d} = \pm 2.125\), \(s_d = 3.09\ldots\))

M1 for attempting \(\bar{d}\)

M1 for \(s_d\) or \(s_d^{\,2}\)

B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\) (allow a defined symbol) Must match their differences

M1 for attempting the correct test statistic \(\dfrac{\bar{d} - 1}{s_d/\sqrt{8}}\)

A1 awrt 1.03

B1 awrt 1.895 sign must match their \(t\)-value

A1ft ft \(t\)-value if awarded both B marks. A correct comment in context – bold words needed.

Notes

B1 for \(\mathrm{H}_1\) is rejected when it is in fact true

B1 Correct contextual statement. (corrected from the printed mark scheme: the scheme prints “B2” here, but the part is marked B1 B1)

Notes

B1 Both hypotheses

B1 131

B1 10 or awrt 3.16

M1 Allow \(\pm\dfrac{\text{"their 131"} - 135}{\sqrt{\text{"their 10"}/5}}\)

A1 awrt \(-2.83\) or \(-2\sqrt{2}\)

B1 \(\pm 1.533\) sign must match \(t\)-value or be \(\pm\)

A1ft ft \(t\)-value if awarded 1^st and 4^th B marks. The words ‘mean length’ and ‘135’ must be included in the context

Notes

M1 \(\dfrac{4 \times \text{"their 10"}}{\chi^2\,\textit{value}}\)

B1 awrt 9.49

B1 awrt 0.711

A1 awrt 4.22/4.21 and awrt 56.3

Notes

B1 both hypotheses. Must use \(\sigma\) or \(\sigma^2\) and make clear which is \(\mathrm{H}_0\) and which is \(\mathrm{H}_1\). Do not allow in words

M1 correct method for either \(s_B^2\) or \(s_G^2\)

A1 Both \(s_B^2\) and \(s_G^2\) correct to 3sf allow sd’s

M1 allow use of \(s_B\) and \(s_G\) instead of \(s_B^2\) or \(s_G^2\)

A1 awrt 1.56 or 0.643

B1 correct CV for their \(F\) or a correct comparison if use \(p\)

A1 cso – All previous marks must be awarded. Variances are the same or var are not different

Notes

B1 both hypotheses using \(\mu\). Do not allow \(\geqslant\) sign instead of >. May use different letters eg \(A\) and \(B\) but they must be defined.

M1 only allow use of \(s_B\) and \(s_G\) instead of \(s_B^2\) or \(s_G^2\) – May be seen in part (a)

M1 use of correct formula with their \(s_p\) – condone missing 5

M1 use of correct formula with their \(s_p\). (which must have been attempted)

B1 correct CV but must match \(t\)-value or a correct comparison if use \(p\)

A1 A correct statement with either the word Headteacher/Teacher/Head or time and not more than 5 oe do not allow contradicting statements.

Notes

B1 both hypotheses

M1 Allow use of 6.76 and 5.42 instead of 6.76² and 5.42²

A1 awrt 1.56

Notes

B1 both hypotheses

M1 allow use of 6.76 and 5.42 instead of 6.76² and 5.42²

A1 awrt 39.4 or 6.28

B1 allow p value 0.650 instead of critical value

M1 use of correct formula with their \(S_p\) – condone missing 5
M1 use of correct formula with their \(S_p\)

(Corrected from the printed mark scheme: the critical value is printed as \(t_{13}(2.5\%) = 1.771\); for this one-tailed 5% test it is \(t_{13}(5\%) = 1.771\).)

Notes

Both hyps, one tailed only oe.
Accept \(\mu_1, \mu_2\) or \(\mu_A, \mu_B\) etc if there is some indication of which is which.

M1 for correct method for standard error

M1 for whole expression

A1 awrt 1.95

B1 1.6449 or \(p = 0.974\ldots\) (>0.95)

A1 must mention doctor and claim or description of claim that includes ‘mean lung capacity’ and ‘exercise’.

ALT (a)

Notes

M1 Attempt \(\sum x^2 = 34 \times 12.2 + 35 \times 26.3^2\)
or \(\sum (x - \bar{x})^2 = 34 \times 12.2 + 35(26.45 - 26.3)^2\ (= 415.5875)\)

dM1 \(s^2 = \dfrac{\sum x^2 + 31.7^2 - 36 \times 26.45^2}{35}\) or \(s^2 = \dfrac{415.5875 + (31.7 - 26.45)^2}{35}\)

A1 awrt 12.7

Notes

B1 both hypotheses

M1 for attempting the correct statistic

A1ft follow through their s²

A1 awrt 1.51

Notes

B1 both hypotheses, must be two tail

B1 awrt 0.0311

B1 NB allow 2.733 for one tail hypotheses. (no hypotheses gains B0)

M1 for a correct test statistic

NB one tail test can get B0 B1 B1 (2.733)B0 M1 A1 A1

(Corrected from the printed mark scheme: the upper critical value is printed as \(\chi^2_8(0.25) = 17.535\); it is the upper 2.5% point, \(\chi^2_8(0.025)\).)

Notes

M1 for attempting the \(d\)s
M1 for attempting \(\bar{d}\)
M1 for \(s_d\) or \({s_d}^2\)
B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\). (allow a defined symbol)

M1 for attempting the correct test statistic \(\dfrac{\bar{d}}{s_d/\sqrt{10}}\)

A1 awrt 2.68
B1 awrt 1.83
A1ft for a correct comment in context

Notes

B1 for a comment that mentions “differences” and “normal” distribution

Notes

B1 124
B1 43

Notes

M1 \(\dfrac{8 \times \text{their } 43}{25}\)

A1 awrt 13.8

B1 15.507

B1 dep on previous M1 being awarded. Allow the standard deviation of the IQ scores is 5 oe. Must have IQ

Notes

M1 \(\dfrac{11 \times 8.17}{3.816 \text{ or } 21.92}\)

A1 both correct

Notes

B1ft their interval from part(c). Gurdip’s claim may not be true
NB, no interval in (c) then B0

Notes

M1 for \(\dfrac{12 \times 161 + 9 \times 48}{13 + 10 - 2}\)

A1cso need to get awrt112.57 or \(\dfrac{2364}{21}\) then write 112.6

Notes

M1 \(\dfrac{195 - 186}{\sqrt{112.6\left(\frac{1}{10} + \frac{1}{13}\right)}}\)

2nd B1 alternate method, \(p\) value of 0.0566 in place of critical value

Final A1 requires correct conclusion in context

Notes

1st B1 allow \(\mathrm{H}_0 : \sigma_s = \sigma_a\) against \(\mathrm{H}_1 : \sigma_s \ne \sigma_a\)

M1 allow \(\dfrac{161^2}{48^2}\) if they write the formula down

Final A1 requires correct conclusion

Notes

1st B1 equal variances is necessary assumption (may be implied by saying not equal)

2nd B1d but not supported in (c)/(variances not equal) therefore (b) result invalid

Notes

1st M1 attempting \(s\) or \(s^2\)    1st A1 awrt 54.3

2nd M1 for \(\dfrac{7s^2}{\chi^2}\)

B1 14.067 & 2.167

3rd M1d Dept on previous M mark. Rearranging leading to interval for σ- must square root

A1 awrt 5.20 and 13.2 (allow 5.2)

NB a correct interval gains full marks

Notes

B1ft on their CI must have Fred/He/employee (do not allow empoloyees) and training. They must have an interval in part(a)

Notes

1st B1 Both hypotheses with \(\mu\). Allow one-tail

1st M1 \(\dfrac{102.6 - 100}{\frac{s \text{ or } s^2}{\sqrt{20}}}\)    1st A1 awrt 2.64

2nd B1 allow \(p\) value of 0.0161 in place of critical value. CV must follow from H₁

2nd A1ft a correct statement – do not allow contradicting non context statement.

3rd B1cso need correct conclusion in context containing the words in bold from a fully correct solution. For one tail need “more than 100g”

Notes

1st & 2nd B1 for hypotheses. Accept \(\mu_1, \mu_2\) or \(\mu_A, \mu_B\) etc if there is some indication of which is which e.g. \(G \sim \mathrm{N}(\mu_g, 8.7)\)

1st M1 for an attempt at se with 3 out of 4 values correct. Condone switching 2.1 and 1.4 \(\sqrt{\dfrac{2.1^2 \text{ or } 1.4^2}{80} + \dfrac{1.4^2 \text{ or } 2.1^2}{65}}\)

2nd dM1 dependent on 1st M1 for a correct numerator (must have \(-1.5\)) and ft their se.

1st A1 for awrt 2.05

3rd B1 for \(\pm 2.3263\) or better seen or probability of awrt 0.02

3rd dM1 dep. on 1st M1 for a correct statement based on their normal cv and their test statistic

2nd A1ft for correct comment in context. Must mention “plan” and “group or individual” and “1.5” or “researcher” and “belief or claim”

NB Use of cv for difference in means \(D\) will have \(D = 1.5 + 2.3263 \times \text{s.e.}\) = awrt 2.18 and requires sight of \(d = 2.1\) with a comment for the 3rd M1

Notes

1st B1 for mentioning “large samples” and “CLT”

2nd dB1 dependent on 1st B1 for stating no need to assume normality (since CLT assures it)

Notes

1st B1 for differences all correct (o.e.)

1st M1 attempt to find \(\bar{d} = \dfrac{\sum \text{"their } d\text{"}}{8}\)

2nd M1 attempting \(s_d\) or \({s_d}^2\ \dfrac{1}{7}\left(\sum \text{"their } d^2\text{"} - \dfrac{\left(\sum \text{"their } d\text{"}\right)^2}{8}\right)\)

2nd B1 both correct in terms of \(\mu\) or \(\mu_d\) (allow a defined symbol) condone \(\mu_{July-Jan}\)

3rd M1 for attempting the correct test statistic \(\dfrac{\bar{d}}{s_d/\sqrt{8}}\)

1st A1cso awrt 0.822 with no errors.

3rd B1 alternate method, \(p\) value of 0.219. Allow 2.365 for 2-tail test

Final A1 need conclusion in context, need tickets July and January, ft their test stat and critical value

NB difference of 2 means test gains no marks

Notes

B1 need differences to be normally distributed, not just normal distribution

Alternative

Notes

1st B1 needs both \(\mathrm{H_0}\) and \(\mathrm{H_1}\), can be in words

2nd B1ft on their \(\mathrm{H_1}\)

1st M1 for attempt @ both means (\(\bar{x}_E\) may be in (a))

2nd M1 for attempt at correct test statistic, ft their values

3rd M1 for attempt to compare their test stat and critical values

A1 ft on their test and critical values but must include comment in context

Notes

Require mention of mean of \(E\) or \(F\) and normal distribution

Notes

M1 requires use of random numbers and attempt to find correct sample sizes

A1 both 115 and 75 found.

Notes

M1 for use of the correct test statistic

Notes

M1 for use of a correct expression (LHS) only

Notes

M1 for a correct probability expression involving \(S^2\) or \({\chi^2}_{29}\). Ft their CR, may be implied by a correct answer

1st A1ft for a correct probability expression with \({\chi^2}_{29}\) but ft their CR, may be implied by a correct answer

Notes

1st B1 allow \(\sigma\) or \(\sigma^2\)
M1 for use of the correct test statistic

Notes

1st M1 for attempting \(s_p\) or \({s_p}^2\)
1st A1 for awrt 6.90 or 2.63
2nd M1 for use of a correct test statistic
2nd A1 for awrt 0.77 (accept \(\pm\))
2nd B1 for 2.093 (allow \(\pm\) 1.729 for one-tailed \(\mathrm{H}_1\))

Notes

1st M1 for attempting the \(d\)s
2nd M1 for attempting \(\bar{d}\)
3rd M1 for attempting \(s_d\) or \({s_d}^2\)
1st B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\)

4th M1 for attempting the correct test statistic \(\dfrac{6.75}{s_d/\sqrt{8}}\) or \(p = \text{awrt } 0.06\) or \(\dfrac{c}{10.7/\sqrt{8}} = t\) value

1st A1 1.78 or awrt 0.06 or awrt 7.2
2nd B1 1.895 or awrt 0.06
2nd A1ft for a correct comment in context based on their test statistic and their cv.

Notes

B1 for a comment that mentions “differences” and “normal” distribution

Notes

2nd B1 allow \(\sigma\) or \(\sigma^2\)

3rd B1 allow awrt 16.3 or \(s_1 = \text{awrt } 4.03\)

M1 for use of the correct test statistic

5th B1 allow “assumption is correct”

Notes

1st M1 for attempting \(s_p\) or \({s_p}^2\)

1st B1 for 30.33

2nd M1 for use of a correct test statistic

2nd A1 for awrt 0.49 (accept \(\pm\)) or 0.495

2nd B1 for 2.145 (allow \(\pm\) 1.761 for one-tailed \(\mathrm{H}_1\))

Notes

B1 for a comment that mentions “differences” and “normal” distribution

Notes

1st M1 for attempting the \(d\)s

2nd M1 for attempting \(\bar{d}\)

1st M1 for \(s_d\) or \({s_d}^2\)

1st B1 for both hypotheses correct in terms of \(\mu\) or \(\mu_d\). (allow a defined symbol) Do not allow 10 instead of 1/6 (awrt 0.167) unless working in minutes throughout

3rd M1 for attempting the correct test statistic \(\dfrac{\bar{d} - \frac{1}{6}}{s_d/\sqrt{8}}\) or \(p = \text{awrt } 0.016\) or \(\dfrac{c - \frac{1}{6}}{0.901/\sqrt{8}} = t\) value

2nd A1 awrt 2.65 /2.655 or awrt 1.12 or awrt 0.016

2nd B1 2.998 or 0.0164

3rd A1ft for a correct comment in context based on their test statistic and their cv. Do not allow contradictions.

(Corrected from the printed mark scheme: the critical value is printed as 2.988 in the CR working and in the 2nd B1 note; \(t_7(1\%) = 2.998\).)

Notes

B1 both hypotheses

M1 either \(\dfrac{|92.875 - 100|}{8.3055/\sqrt{8}}\) or \(p = 0.0228\) or \(\dfrac{c - 100}{8.3055/\sqrt{8}} = -(\text{a } t \text{ value})\)

A1 awrt 2.43 or awrt 94.4 or awrt 0.0228

B1 \(\pm 1.895\) or \(0.0228 \lt 0.05\) (must have correct comparison for hypotheses)
A1ft Do Not allow contradictions

(Corrected from the printed mark scheme: the conclusion printed “100mm” for 100 ml.)

Notes

1st B1 for both hypotheses in terms of \(\mu\) not words.
Accept \(\mu_1, \mu_2\) etc if there is some indication of which is which e.g \(X \sim \mathrm{N}(67, 25^2)\) implies \(X\) is “before”.

1st M1 for attempt at s.e. - condone one number wrong or mis-matched variances
i.e. \(\sqrt{\frac{p}{q} + \frac{r}{s}}\) (3 of \(p, q, r\) & \(s\) correct) or \(\sqrt{\frac{10^2}{100} + \frac{25^2}{150}}\)

2nd dM1 Dep on 1st M1 for using their s.e. in correct formula for test statistic. Num of \(\pm(67 - 60)\)
or for correct expression for CR

3rd dM1 dep. on 2nd M1 for a correct statement based on their normal cv (\(|\text{cv}| \gt 1.5\)) and their test statistic

2nd A1ft for correct comment in context. Must mention “lead” or “soil” and “factory”. Allow ft
If hypotheses are the wrong way round score A0
If hypotheses are not for a difference between 2 means award A0

Notes

B1 must mention mean and normal. In words or symbols e.g. \(\bar{X} \sim \mathrm{N}(\ldots\)

Notes

M1 attempt at correct statistic

1st A1 awrt 2.93

2nd A1ft correct contextual comment including amount, water and 500

Notes

1st B1 Both hypotheses, must use \(\sigma\)

2nd B1 for critical value, this should be compatible with their alternative hypothesis

3rd A1cso cso. contextual comment, include standard deviation/ variance and water

Notes

1st M1 for attempting differences

2nd M1 for attempting \(\bar{d}\)

3rd M1 for attempting \(s_d\) or \({s_d}^2\), correct expression with their \(\sum d^2\) and \(\bar{d}\) or correct calculation (to 2 sf or better)

4th M1 for use of \(\dfrac{\bar{d} - 1}{s/\sqrt{8}}\), ft their values.

1st A1 awrt 1.91

2nd B1 for 1.895

2nd A1 contextual conclusion ft their values.

SC if they use a 2 sample test they may get the first B1 for \(\mathrm{H}_0 : \mu_\mathrm{I} - \mu_\mathrm{II} = 1\) and \(\mathrm{H}_1 : \mu_\mathrm{I} - \mu_\mathrm{II} \gt 1\)

Notes

(i) M1 for attempt to calculate \(s^2\)

(ii) M1 use of correct formula for \({s_p}^2\)
A1 awrt 0.349 / 0.3495

Notes

1st B1 allow \(\mathrm{H}_0 : \sigma_1 = \sigma_2,\ \mathrm{H}_1 : \sigma_1 \ne \sigma_2\)

M1 For use of a correct formula

Notes

B1 must use \(\mu\). If not use \(A\) and \(B\) it must be clear which is which

M1 for attempt at correct test statistic – matching their hypotheses

1st A1 correct test statistic for their hypotheses

Notes

M1 for using 0.025

Notes

2nd B1 either \(\dfrac{1}{9.15}\)   or   awrt 0.109

Notes

(corrected from the printed mark scheme: printed as “= (0.0284…)²”; \(\sqrt{0.04343\ldots} = 0.2084\ldots\)) CHECK

Notes

1st & 2nd B1 for hypotheses. Accept \(\mu_1, \mu_2\) or \(\mu_A, \mu_B\) etc if there is some indication of which is which e.g. \(A \sim \mathrm{N}(\mu_A, 0.5)\)

1st M1 for an attempt at se. Condone switching 0.5 and 0.75 \(\sqrt{\dfrac{0.5 \text{ or } 0.75}{60} + \dfrac{0.75 \text{ or } 0.5}{70}}\)

1st A1 for a correct expression for denominator of test statistic or 0.138… or \(\sqrt{0.0190\ldots}\)

2nd A1 for a correct numerator of test statistic (must have the \(-1\))

3rd A1 for awrt 3.62
[Allow – 3.62 from numerator of 5.5 – 7 − − 1 and compatible \(\mathrm{H}_1\)]

3rd B1 for \(\pm\) 1.6449 seen or probability of 0.0002 (tables) or 0.000145…(calc) [allow 0.0001]

2nd dM1 dep. on 1^st M1 for a correct statement based on their normal cv and their test statistic

2nd A1 for correct comment in context. Must mention “yield” and “varieties” or “old” and “new” and “1”
If second B mark is B0 award A0 here

ALT Pooled estimate: If they calculate \(s_p = \sqrt{0.41845\ldots} = 0.64688\ldots\) allow 1^st M1, 1^st A1 for expression (or awrt 0.114) and 2^nd A1 if numerator correct but A0 for test statistic (4.39)

Notes

1st B1 for mention of mean (yield) and normal (distribution)

2nd B1 for mention of sample (size) being large in this case

Notes

M1 for use of a correct formula Allow \(F = \dfrac{s_A^{\,2}}{s_B^{\,2}} = \dfrac{4.24^2}{4.37^2} = 0.941\ldots\) with 0.1295..

Notes

B1 if \(A\) and \(B\) not used it must be clear which is \(A\) and which is \(B\)
1^st M1 for attempt to calculate \(s_p\) or \(s_p^{\,2}\)
2^nd M1 for attempt correct test statistic
2^nd A1 ft need archaeologist’s or compression

Notes

Need to refer to ‘allows us to assume variances the same’ and this is needed in for test. oe

Notes

1^st M1 for attempting differences
2^nd M1 for attempting \(\bar{d}\)
3^rd M1 for attempting \(s_d^{\,2}\), correct expression with their \(\sum d^2\) and \(\bar{d}\) or correct calculation (to 2 sf or better)
4^th M1 for use of a correct CI formula, using a value for \(t\) and ft their values.
1^st A1 for lower limit of -1.57 or -2.32
2^nd A1 for corresponding upper limit

S.C. Allow A1A1 for (0, 2.32)

(corrected from the printed mark scheme: the first line is printed as “d = Jan - June”; the second set of times is July) CHECK

Notes

B1 for both hypotheses using \(\mu_D\)
M1 for a comment about 0 being in (or out) of their interval
A1 contextual conclusion – must include assemble components

S.C. If they have used difference in means test in part (a) to get the confidence interval then award the B1 for \(\mathrm{H}_0 : \mu_x - \mu_y = 0 \quad \mathrm{H}_1 : \mu_x - \mu_y \neq 0\) or the correct hypotheses.

Notes

1^st B1 Both hypotheses, must use \(\sigma\). Allow \(\mathrm{H}_0 : \sigma = 2.4 \quad \mathrm{H}_1 : \sigma \neq 2.4\)
1^st M1 correct method used
1^st A1 awrt 14.2
2^nd M1 \(\chi^2 = \dfrac{9 \times \text{"their }s^2\text{"}}{2.4^2}\)
2^nd A1 awrt 22.2
2^nd B1 for critical value, this should be compatible with their alternative hypothesis (16.919 for one tail test)
3^rd A1ft fully correct solution only

Notes

B1 both correct. Also allow \(\mathrm{H}_0 : \sigma = 6;\ \mathrm{H}_1 : \sigma \gt 6\)
B1 36.415
M1 use of \(\dfrac{(n-1)S^2}{\sigma^2}\)
A1 awrt 36.7

Notes

M1 \(\pm\dfrac{455 - 450}{\sqrt{\dfrac{55}{25}}}\)
A1 awrt 3.4

A1ft any statement – no conflicting
A1ft contextual statement must include “weight of chocolate” and is “greater than 450” (corrected from the printed mark scheme: printed as “greater than 50”) CHECK

Notes

1st M1 for attempt at s.e. (condone one number wrong) and for using their s.e. in correct formula for test statistic.

1st A1 for correct expression for se

2nd dM1 dep. on 1st M1 for a correct statement based on their normal cv and their test statistic

3rd A1 for correct comment in context. Must mention “scores” and “students / groups/classes” Award A0 for a one-tailed comment.

Notes

1st B1 Alternative hyp should be clearly defined

Notes

B1 both correct. Must use \(\sigma\). May use different notation to \(A\) and \(B\)
M1 \(\dfrac{225}{36}\) or \(\dfrac{36}{225}\) allow \(\dfrac{15}{6}\) or \(\dfrac{6}{15}\)
A1 either 6.25 or 0.16
B1 CR must match their method
A1 context must include “lengths of slabs”

Notes

B1 – awrt 0.61
B1 – awrt 0.935
Both may be implied by correct \(t\) value or \(S_p\)
B1 allow rearrangements eg \(\mu_F - \mu_M = 5\). If \(M\) and \(F\) not used then they must make clear what each letter is.
B1 CV (if using one tail test allow 1.746)
M1 \(\dfrac{5 \times \text{their }0.61 + 11 \times \text{their }0.93545\ldots}{16}\)
A1 awrt 0.834
M1 \(\pm\left(\dfrac{19.6 - 13.7 - 5}{\sqrt{p\left(\frac{1}{6} + \frac{1}{12}\right)}}\right)\) where \(p\) is either their 0.61 or 0.94 or their \(S_p^2\) (awrt 0.834) (Allow 13.7 - 19.6 - 5)
A1 ft their \(S_p^2\)
A1 awrt 1.97

Notes

(i) B1 1.96
M1 must use z value

(ii) M1 writing or using N(6, 0.225)
M1 finding correct area and standardising (must use 6 but allow use of 0.9 and (0.9/18) for var)

(In the printed notes these two sets are labelled (b) and (c).)