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Date	May 2018	Marks available	7	Reference code	18M.2.SL.TZ1.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_8	Adapted from	N/A

Question

The following table shows values of ln x and ln y.

The relationship between ln x and ln y can be modelled by the regression equation ln y = a ln x + b.

Find the value of a and of b.

[3]

Use the regression equation to estimate the value of y when x = 3.57.

[3]

The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.

[7]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach (M1)

eg one correct value

−0.453620, 6.14210

a = −0.454, b = 6.14 A1A1 N3

[3 marks]

correct substitution (A1)

eg −0.454 ln 3.57 + 6.14

correct working (A1)

eg ln y = 5.56484

261.083 (260.409 from 3 sf)

y = 261, (y = 260 from 3sf) A1 N3

Note: If no working shown, award N1 for 5.56484.
If no working shown, award N2 for ln y = 5.56484.

[3 marks]

METHOD 1

valid approach for expressing ln y in terms of ln x (M1)

eg ${\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b$

correct application of addition rule for logs (A1)

eg ${\text{ln}}\,k + {\text{ln}}\,\left( {{x^n}} \right)$

correct application of exponent rule for logs A1

eg ${\text{ln}}\,k + n\,{\text{ln}}\,x$

comparing one term with regression equation (check FT) (M1)

eg $n = a,\,\,b = {\text{ln}}\,k$

correct working for k (A1)

eg ${\text{ln}}\,k = 6.14210,\,\,\,k = {e^{6.14210}}$

465.030

$n = - 0.454,\,\,k = 465$ (464 from 3sf) A1A1 N2N2

METHOD 2

valid approach (M1)

eg ${e^{{\text{ln}}\,y}} = {e^{a\,{\text{ln}}\,x + b}}$

correct use of exponent laws for ${e^{a\,{\text{ln}}\,x + b}}$ (A1)

eg ${e^{a\,{\text{ln}}\,x}} \times {e^b}$

correct application of exponent rule for $a\,{\text{ln}}\,x$ (A1)

eg ${\text{ln}}\,{x^a}$

correct equation in y A1

eg $y = {x^a} \times {e^b}$

comparing one term with equation of model (check FT) (M1)

eg $k = {e^b},\,\,n = a$

465.030

$n = - 0.454,\,\,k = 465$ (464 from 3sf) A1A1 N2N2

METHOD 3

valid approach for expressing ln y in terms of ln x (seen anywhere) (M1)

eg ${\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b$

correct application of exponent rule for logs (seen anywhere) (A1)

eg ${\text{ln}}\,\left( {{x^a}} \right) + b$

correct working for b (seen anywhere) (A1)

eg $b = {\text{ln}}\,\left( {{e^b}} \right)$

correct application of addition rule for logs A1

eg ${\text{ln}}\,\left( {{e^b}{x^a}} \right)$

comparing one term with equation of model (check FT) (M1)

eg $k = {e^b},\,\,n = a$

465.030

$n = - 0.454,\,\,k = 465$ (464 from 3sf) A1A1 N2N2

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.4—Pearsons, scatter diagrams, eqn of y on x

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