Mathematics · Oct / Nov 2021 · Paper 61 · Question 2

YMA01

As printed in the paper

The question as text

2. A large company is analysing how much money it spends on paper in its offices each year. The number of employees in the office, x, and the amount spent on paper in a year, p ($ hundreds), in each of 12 randomly selected offices were recorded. The results are summarised in the following statistics. ∑ x = 93 Sxx = 148.25 ∑ p = 273 ∑ p2 = 6602.72 ∑ xp = 2347

Read automatically from the paper; the printed page above is exact.

Show the answer and mark scheme

2(a)[1]
S
=
2347
−
or 2347
−
=
231.25
(*)
xp
12
12
273
2
2(b)[3]
[
]
S
=
6602.72
−
=
391.97
pp
12
231.25
[
]
r =
148.25 "391.97"
×
= 0.959307… awrt 0.959
S
231.25
2(c)[4]
[
]
b
=
xp
=
=
1.559865...
S
148.25
xx
273
93
[
]
a =
−
"1.56"
×
or 22.75 "1.56" 7.75
−
×
=
10.66...
M1
12
12
b = awrt 1.6 or a = awrt 11
p = 10.7 + 1.56x
(d) e.g. each extra employee costs the company (on average)[$"]156" a year in paper
"1.559..."
93
(e) [New p = ]
0.8 "10.66..."
×
+
×
[=14.573…]
2
12
22.75 14.573...
−
[compared with
[
]
p =
22.75
] so percentage saving is
×
100
22.75
= 35.94… awrt 36[%]
Notes
(a) B1 for either correct expression [don’t need = 231.25]
(b)
1st M1 for attempt at correct expression for Spp Allow one transcription error e.g. 6620.72 May be
seen in part (a)
If no correct expression seen allowSpp = awrt 392 or correctly placed in formula for r
2nd M1 for a correct expression for r, ft their Spp
A1 for awrt 0.959
(c) 1st M1 for a correct expression for b (may be implied by a correct value of awrt 1.56)
2nd M1 for a correct expression for a ft their value for b. May be implied by awrt 10.7
1st A1 for b = awrt 1.6 or a = awrt 11
2nd A1 for correct equation in p and x with b = awrt 1.56 and a = awrt 10.7
(d) B1 for a suitable contextual comment that mentions their value of b
Allow multiples eg every extra 100 employees costs the company "$15600". Condone missing $
sign or use of £. Do not allow " $1.56" for 1 person unless indicates in 100's
(e) 1st M1 for a correct expression for average value of p using new model [ft their a and b]
14.573...
[
]
2nd M1 for correct percentage saving calculation using 22.75 (e.g.
×
100
) Allow use of
22.75
93
"10.7" 1.56"
+
×
[
≈
22.79]
for 22.75 May be implied by correct answer.
12
A1 for awrt 36
SC use of 93 throughout part (e) rather than 7.75 leading to awrt 48 or 0.48 (they will need to use the
regression line from part(c) to calculate the original value) gains M0M1A0
SC use of 93 in part(c) Answer of 36% gains M1M1A1, 64% or 0.64 gains M1M1A0.
3. (a) [Median =] 53
(b)
Q
=
45
Q
=
61
1
3
[IQR =] 61 – 45 = 16 (*)
(c)
Q −
1.5 (IQR)
×
=
45 1.5 16
−
×
[= 21] or
Q +
1.5 (IQR)
×
=
61 1.5 16
+
×
[=85]
1
3
Outliers are < 21 or > 85
So there are three outliers at 13, 87 and 88
2(d)[1]
(e) e.g. the females are generally older than the men as median is higher (67 > 53)
(f)(i) No change to box plot means one in each section so granddaughter [34~56]
(ii) Eldest daughter in range [67~72] or Anja’s age [72~93]
Since Anja 23 years older than eldest daughter Anja in range [90~93]
Notes [14]
(a) B1 for 53
(b) M1 for an attempt at both and at least one correct. No need to be labelled.
A1cso for both correct quartiles seen and 61 – 45 leading to 16
(c) M1 for attempting at least one of the limits. Can ft their quartiles. May be implied by 85 or 21
1st A1ft for both outlier limits correct or correct ft using their quartiles
2nd A1 for identifying the three outliers at 13, 87, 88 (dep on seeing both correct limits)
(d) M1 for drawing a box with only two whiskers one at each end
1st A1ft for
Q Q
,
and
Q as a correctly drawn box (ft their values for
Q Q
,
and
Q )
1
2
3
1
2
2nd A1 for upper whisker ending at 76 (or 85) and lower whisker ending at 27 (or 21) NB
Must be correctly paired.
3rd A1 for the 3 outliers correctly shown (accuracy half a small square throughout)
SC fully correct but with both whiskers correct on each side. M0A1A0A1
(e) B1 for a correct comment, referring to ages, with reference to a correctly named statistic. Must
include the figures compared.
eg Females older than men and comparison of median, upper quartile or lower quartile, allow
Q Q
,
and
Q with their figures which must agree with the statement .
1
2
3
eg Males ages more spread out than female and comparison of ranges with males = 75 and
females = 73
eg Females older than males since Males are symmetrical [Q3 – Q2 : Q2 – Q1] 8 : 8 Female
are negative skew 5 : 11
NB use of mean/ IQR/ minimum/ maximum is B0. Ignore incorrect comments.
(f)(i) B1 for deducing granddaughter is at or below lower quartile but not below 34
Allow any reasonable adjustment for her mother’s age, {34 to x} where 35 x 56
(ii) M1 Suitable range for eldest daughter or Anja above upper quartile. Ignore any incorrect upper limit.
May be implied by a correct range.
A1 for a range of [90~93] for Anja’s age
2(e)[3]
S
=
2347
−
or 2347
−
=
231.25
(*)
xp
12
12
273
2
[
]
S
=
6602.72
−
=
391.97
pp
12
231.25
[
]
r =
148.25 "391.97"
×
= 0.959307… awrt 0.959
S
231.25
[
]
b
=
xp
=
=
1.559865...
S
148.25
xx
273
93
[
]
a =
−
"1.56"
×
or 22.75 "1.56" 7.75
−
×
=
10.66...
M1
12
12
b = awrt 1.6 or a = awrt 11
p = 10.7 + 1.56x
(d) e.g. each extra employee costs the company (on average)[$"]156" a year in paper
"1.559..."
93
(e) [New p = ]
0.8 "10.66..."
×
+
×
[=14.573…]
2
12
22.75 14.573...
−
[compared with
[
]
p =
22.75
] so percentage saving is
×
100
22.75
= 35.94… awrt 36[%]
Notes
(a) B1 for either correct expression [don’t need = 231.25]
(b)
1st M1 for attempt at correct expression for Spp Allow one transcription error e.g. 6620.72 May be
seen in part (a)
If no correct expression seen allowSpp = awrt 392 or correctly placed in formula for r
2nd M1 for a correct expression for r, ft their Spp
A1 for awrt 0.959
(c) 1st M1 for a correct expression for b (may be implied by a correct value of awrt 1.56)
2nd M1 for a correct expression for a ft their value for b. May be implied by awrt 10.7
1st A1 for b = awrt 1.6 or a = awrt 11
2nd A1 for correct equation in p and x with b = awrt 1.56 and a = awrt 10.7
(d) B1 for a suitable contextual comment that mentions their value of b
Allow multiples eg every extra 100 employees costs the company "$15600". Condone missing $
sign or use of £. Do not allow " $1.56" for 1 person unless indicates in 100's
(e) 1st M1 for a correct expression for average value of p using new model [ft their a and b]
14.573...
[
]
2nd M1 for correct percentage saving calculation using 22.75 (e.g.
×
100
) Allow use of
22.75
93
"10.7" 1.56"
+
×
[
≈
22.79]
for 22.75 May be implied by correct answer.
12
A1 for awrt 36
SC use of 93 throughout part (e) rather than 7.75 leading to awrt 48 or 0.48 (they will need to use the
regression line from part(c) to calculate the original value) gains M0M1A0
SC use of 93 in part(c) Answer of 36% gains M1M1A1, 64% or 0.64 gains M1M1A0.
3. (a) [Median =] 53
(b)
Q
=
45
Q
=
61
1
3
[IQR =] 61 – 45 = 16 (*)
(c)
Q −
1.5 (IQR)
×
=
45 1.5 16
−
×
[= 21] or
Q +
1.5 (IQR)
×
=
61 1.5 16
+
×
[=85]
1
3
Outliers are < 21 or > 85
So there are three outliers at 13, 87 and 88
(e) e.g. the females are generally older than the men as median is higher (67 > 53)
(f)(i) No change to box plot means one in each section so granddaughter [34~56]
(ii) Eldest daughter in range [67~72] or Anja’s age [72~93]
Since Anja 23 years older than eldest daughter Anja in range [90~93]
Notes [14]
(a) B1 for 53
(b) M1 for an attempt at both and at least one correct. No need to be labelled.
A1cso for both correct quartiles seen and 61 – 45 leading to 16
(c) M1 for attempting at least one of the limits. Can ft their quartiles. May be implied by 85 or 21
1st A1ft for both outlier limits correct or correct ft using their quartiles
2nd A1 for identifying the three outliers at 13, 87, 88 (dep on seeing both correct limits)
(d) M1 for drawing a box with only two whiskers one at each end
1st A1ft for
Q Q
,
and
Q as a correctly drawn box (ft their values for
Q Q
,
and
Q )
1
2
3
1
2
2nd A1 for upper whisker ending at 76 (or 85) and lower whisker ending at 27 (or 21) NB
Must be correctly paired.
3rd A1 for the 3 outliers correctly shown (accuracy half a small square throughout)
SC fully correct but with both whiskers correct on each side. M0A1A0A1
(e) B1 for a correct comment, referring to ages, with reference to a correctly named statistic. Must
include the figures compared.
eg Females older than men and comparison of median, upper quartile or lower quartile, allow
Q Q
,
and
Q with their figures which must agree with the statement .
1
2
3
eg Males ages more spread out than female and comparison of ranges with males = 75 and
females = 73
eg Females older than males since Males are symmetrical [Q3 – Q2 : Q2 – Q1] 8 : 8 Female
are negative skew 5 : 11
NB use of mean/ IQR/ minimum/ maximum is B0. Ignore incorrect comments.
(f)(i) B1 for deducing granddaughter is at or below lower quartile but not below 34
Allow any reasonable adjustment for her mother’s age, {34 to x} where 35 x 56
(ii) M1 Suitable range for eldest daughter or Anja above upper quartile. Ignore any incorrect upper limit.
May be implied by a correct range.
A1 for a range of [90~93] for Anja’s age

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