Mathematics · May / June 2021 · Paper 61 · Question 6
YMA01
As printed in the paper
The question as text
6. Two economics students, Andi and Behrouz, are studying some data relating to unemployment, x %, and increase in wages, y %, for a European country. The least squares regression line of y on x has equation y = 3.684 − 0.3242x ∑ y = 23.7 ∑ y 2 = 42.63 ∑ x 2 = 756.81 n = 16 and
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Show the answer and mark scheme
6(a)[1]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
6(b)[4]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
6(c)[3]
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
6(d)[1]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
6(e)[3]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
6(f)[2]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
6(g)[2]
S
yy =
42.63
−
= [7.524375]
16
3.684
−
23.7
∑
(b) Use of
y
=
3.684
−
0.3242
x
; so
x
=
16
×
16
= 108.71067..
0.3242
(
)
2
"108.71.."
{
}
S
xx =
756.81
−
; = 18.18435… awrt 18.2
16
S
" 5.89536..."
−
(
)
b =
xy
⇒
S
=
"18.1843.."
× −
0.3242 [
= −
5.8953..]
;
r
=
S
xy
"18.184.." 7.524375
×
xx
= – 0.50399… = – 0.49 ~ –0.51
(d) Sub x = 2 in the regression line gives y = 3.0356
S
"18.184..."
(e) St.dev =
n =
xx
=
1.066...
16
So limits are: "108.71..."
+ 3×"1.066…" = 3.5965…~ 9.9929… = awrt 3.6~10 M1, A1
16
(f) The probability of x = 2 being in the range is very small;
so Behrouz’s estimate is unreliable
(g) Should use regression of x on y to estimate unemployment or equivalent
So Andi’s suggestion is not suitable or not to be recommended
Notes
(a) B1
Value given so must see sight of a correct expression – allow 561.69 for 23.72
∑
(b) 1st M1
for clear use of regression line with
y
or
y
1st A1 for
∑ = awrt 109
x
2nd M1 for a correct expression for Sxx ft their Σx
2nd A1 for awrt 18.2
(c) 1st M1 for use of gradient to find Sxy
2nd M1 for a correct expression for r ft their Sxy and Sxx
A1
for an answer in the range –0.49 ~ –0.51
(d) B1
for sight of y = 3.03… or better. Allow 3.04
756.81
"108.71..."
(e) 1st M1 for a correct attempt at st. dev. ft their Sxx or
−
16
16
2nd M1 for one correct calc…ft their values
A1
for a range awrt 3.6~10
(f) 1st B1ft for a correct reason ft their range in part (e) eg x = 2 is outside the range. Allow
extrapolation
2nd dB1ft dep on 1st B1 for stating a correct conclusion for their range
(g) 1st B1
for a suitable reason based on reg line, eg regression line (y on x) can only be used
to estimate wages. Allow x instead of unemployment and y instead of wages
2nd dB1 dep on 1st B1 for suggesting not suitable (or equivalent)
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
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