June 2022 Paper 2 Q11

Notes

NB. Use of a “continuity correction” loses 1st A1 only

B1 B1: Allow other letter (including \(X\)) only if clearly defined
Subtract B1 for each error eg:

2-tail	B1B0	Undefined \(\mu\)	B1B0
not in terms of parameter	B1B0	Not include 3300	B0B0
\(\overline{X}\) stated or implied	B0B0	\(\mathrm{H}_0 = 3300\) etc:	B0B0

\(\mu =\) sample mean implied B0 & (B1 or B0)

M1*: Correct distribution and \(\overline{X}\) (allow 3359.5, 3360.5, 3659) stated or implied eg by 0.0297 or 0.0307 or 0.0286
even if within incorrect statement eg \(\mathrm{P}(X = 3360) = 0.0297\)
Allow \(450^2 \div \sqrt{200}\) or \(450^2 \div 200^2\)
Not 0.0297 from \(\mu = 3360\), \(\mathrm{P}(\overline{X} \lt 3300)\)

A1: BC cao

A1: Explicit comparison
Allow compare (any value \(\leqslant\) 0.35) with 0.025

M1dep: Allow Reject \(\mathrm{H}_1\) or Accept \(\mathrm{H}_0\)
Dep M1A1A1 or M1A0A1 or M1A1A0
Dep also on comparing like with like, eg not \(0.970 \gt 0.025\)
May be implied by their conclusion, if M1 criterion is met
ft with opposite conclusion if, eg \(\mathrm{P}(\overline{X} \gt 3360) = 0.024\)

A1f: Dep M1A1A1 or M1A0A1 or M1A1A0
In context, not definite; eg not “Mean mass not > 3300 g”
Not “There is evidence that mean mass hasn’t increased”
Not “This suggests that mean mass hasn’t increased”
Not “Unlikely (or unsure) that mean has increased”
ft with opposite conclusion if, eg \(\mathrm{P}(\overline{X} \gt 3360) = 0.024\)

Alternative method 1 for M1A1A1

Scheme	Marks
\(\overline{X} \sim \mathrm{N}\left(3300, \dfrac{450^2}{200}\right)\) and \(\overline{X} = 3360\)	M1*
\(\mathrm{P}(\overline{X} \lt 3360) = 0.970\) (NB 3 sf)	A1
\(0.970 \lt 0.975\)	A1

M1*: Correct distribution and \(\overline{X}\) (allow \(\overline{X} = 3359.5\) or 3360.5) stated or implied eg by 0.970 or 0.969 or 0.971
even if within incorrect statement eg \(\mathrm{P}(X = 3360) = 0.970\)
Allow \(450^2 \div \sqrt{200}\) or \(450^2 \div 200^2\)

A1: BC cao

A1: Explicit comparison
Allow compare (any value \(\geqslant\) 0.65) with 0.975

Alternative method 2 for M1A1A1

Scheme	Marks
\(\dfrac{a - 3300}{450 \div \sqrt{200}} = 1.96\)	M1*
CR, \(a = 3362\) (NB 4 sf)	A1
\(3360 \lt 3362\)   or 3360 is in acceptance region	A1

M1*: May be implied, eg by 3362
Allow \(450^2 \div \sqrt{200}\) or \(450^2 \div 200^2\)

A1: Explicit comparison of their \(a\) with 3360

Alternative method 3 for M1A1A1

Scheme	Marks
\(\dfrac{3360 - 3300}{450 \div \sqrt{200}}\)	M1*
CV of \(z = 1.886\) or 1.89 (NB 3 sf)	A1
1.89 (or 1.90 or 1.9) \(\lt 1.96\)	A1

M1*: Allow 3359.5 or 3360.5 May be implied, eg by 1.89
Allow \(450^2 \div \sqrt{200}\) or \(450^2 \div 200^2\)

A1: cao

A1: Explicit comparison

Alternative scheme for incorrect method using 2-tails

Hypotheses	B1B0
Calculation: as above	M1A1
Compare 0.0125 oe	A1
Conclusion	M0A0

Exemplars for Q11

	Response	Marks
Hypotheses
A	\(\mathrm{H}_0: \mu = 3300\) \(\mathrm{H}_1: \mu \gt 3300\) where \(\mu =\) (pop) mean mass	B1B1
B	\(\mathrm{H}_0: \mu = 3300\) \(\mathrm{H}_1: \mu \gt 3300\)	B1B0
C	\(\mathrm{H}_0\): The (pop) mean mass is 3300 \(\mathrm{H}_1\): The (pop) mean mass is greater than 3300 See Specimen paper q10 MS "Must be in terms of parameter values"	B1B0
D	\(\mathrm{H}_0 = 3300\) \(\mathrm{H}_0 \gt 3300\)	B0B0
E	\(\mathrm{H}_0: \mu = 3300\) \(\mathrm{H}_1: \mu \ne 3300\) where \(\mu =\) (pop) mean mass	B1B0
F	\(\mathrm{H}_0: \mu = 3300\) \(\mathrm{H}_1: \mu \ne 3300\)	B0B0
Calculation, comparison and conclusion
G	No statement of distribution \(\mathrm{P}(\overline{X} = 3360) = 0.0297\) \(0.0297 \gt 0.025\) Don't reject \(\mathrm{H}_0\) There is no evidence that mean mass has increased	M1A1 A1 M1 A1
H	\(\mathrm{P}(\overline{X} = 3360.5) = 0.0286\) \(0.0286 \gt 0.025\) Accept \(\mathrm{H}_0\) There is evidence that mean mass hasn't increased	M1A0 A1 M1 A0
I	\(\mathrm{P}(\overline{X} \gt 3360.5) = 0.0286\)   Accept \(\mathrm{H}_0\) There is evidence that mean mass hasn't increased	M1A0 A0 M1 A0
J	\(\mathrm{P}(\overline{X} = 3359.5) = 0.024\) \(0.024 \lt 0.025\) Reject \(\mathrm{H}_0\) There is evidence that mean mass has increased	M1A0 A1 M1 A1ft
K	\(\mathrm{P}(\overline{X} \lt 3360) = 0.970\) \(0.970 \lt 0.975\) Reject \(\mathrm{H}_1\) Insufficient evidence that mean mass has changed	M1A1 A1 M1 A0
L	\(\mathrm{P}(\overline{X} \gt 3360) = 0.970\) \(0.970 \gt 0.025\) Insufficient evidence that mean mass has increased	M1A1 A0 M0A0
M	\(\overline{X} \sim \mathrm{N}(3300, 1012.5)\) \(\mathrm{P}(\overline{X} \gt 3360) = 0.297\) \(0.297 \gt 0.025\) Insufficient evidence that mean mass has increased	M1A0 A1 M1A1
N	\(\mu \pm 1.96\sigma = 3237\) to 3362 3360 lies within this range Can’t reject \(\mathrm{H}_0\) Mean mass hasn’t increased	M1A1 A1 M1 A0
O	CV = 3362 \(3360 \lt 3362\) Reject \(\mathrm{H}_0\). Evidence that level of poll't has increased.	M1A1 A1 M0A0
P	\((3360 - 3300) \div (450 \div \sqrt{200}) = 1.886\) \(1.866 \lt 1.96\) Don't reject \(\mathrm{H}_0\). Mean mass hasn't increased.	M1A1 A1 M1A0
2-tail
Q	\(\mathrm{H}_0: \mu = 3360\) \(\mathrm{H}_1: \mu \ne 3360\) \(0.0297 \gt 0.0125\) Don't reject \(\mathrm{H}_0\) There is no evidence that mean mass has changed	B0B0 A1 M0 A0
R	\(\mathrm{H}_0: \mu = 3360\) \(\mathrm{H}_1: \mu \ne 3360\) where \(\mu =\) (pop) mean mass \(0.0297 \gt 0.025\) Don't reject \(\mathrm{H}_0\) There is no evidence that mean mass has changed	B1B0 A0 M0 A0
S	\(\mathrm{H}_0\): The (pop) mean mass = 3360 \(\mathrm{H}_1\): The (pop) mean mass \(\ne 3360\) \(0.97 \lt 0.9875\) Accept \(\mathrm{H}_0\) There is no evidence that mean mass has changed	B0B0 A1 M0 A0