Linear regression Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Linear regression questions. See exactly how to solve problems on least-squares, gradient, summary-statistics, intercept.

least-squaresgradientsummary-statisticsinterceptmean-pointresidual
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
A materials laboratory hangs a series of loads on a steel spring and measures the extension produced. For each observation, xx denotes the load (kg) and yy denotes the extension (mm). From the 55 paired observations it is found that Sxx=40S_{xx}=40 and Sxy=78S_{xy}=78. The least-squares regression line of yy on xx is the line y=a+bxy=a+bx that minimises the residual sum of squares ∑i=1n(yi−a−bxi)2\sum_{i=1}^{n}\left(y_i-a-bx_i\right)^2. Find the gradient bb of the least-squares regression line of yy on xx, giving your answer exactly.

Worked solution

  1. Recall the least-squares gradient

    b=SxySxxb=\frac{S_{xy}}{S_{xx}}

    Setting the partial derivatives of the residual sum of squares to zero gives this value of bb.

  2. Substitute the two summary statistics

    b=SxySxx=7840b=\frac{S_{xy}}{S_{xx}}=\frac{78}{40}

    The gradient is the ratio of the two sums of squares, nothing else is needed.

  3. Evaluate the quotient

    b=7840=1.95b=\frac{78}{40}=1.95

    The sign of bb is the sign of SxyS_{xy}, since Sxx>0S_{xx}>0 always.

  4. State the gradient

    b=1.95b=1.95

    This is the least-squares estimate of the change in yy per unit increase in xx.

Answer
b=1.95b=1.95
Question 2
2 markseasy
A chemist measures the initial rate of a reaction at several catalyst concentrations. For each observation, xx denotes the catalyst concentration (mmol per litre) and yy denotes the initial rate (micromol per second). From the 55 paired observations it is found that Sxx=20S_{xx}=20 and Sxy=43S_{xy}=43. The least-squares regression line of yy on xx is the line y=a+bxy=a+bx that minimises the residual sum of squares ∑i=1n(yi−a−bxi)2\sum_{i=1}^{n}\left(y_i-a-bx_i\right)^2. Find the gradient bb of the least-squares regression line of yy on xx, giving your answer exactly.

Worked solution

  1. Recall the least-squares gradient

    b=SxySxxb=\frac{S_{xy}}{S_{xx}}

    Setting the partial derivatives of the residual sum of squares to zero gives this value of bb.

  2. Substitute the two summary statistics

    b=SxySxx=4320b=\frac{S_{xy}}{S_{xx}}=\frac{43}{20}

    The gradient is the ratio of the two sums of squares, nothing else is needed.

  3. Evaluate the quotient

    b=4320=2.15b=\frac{43}{20}=2.15

    The sign of bb is the sign of SxyS_{xy}, since Sxx>0S_{xx}>0 always.

  4. State the gradient

    b=2.15b=2.15

    This is the least-squares estimate of the change in yy per unit increase in xx.

Answer
b=2.15b=2.15
Question 3
2 markseasy
A civil engineer crushes concrete cubes that have been cured for different numbers of days. For each observation, xx denotes the curing time (days) and yy denotes the compressive strength (newtons per square millimetre). From the 77 paired observations it is found that Sxx=100S_{xx}=100 and Sxy=120S_{xy}=120. The least-squares regression line of yy on xx is the line y=a+bxy=a+bx that minimises the residual sum of squares ∑i=1n(yi−a−bxi)2\sum_{i=1}^{n}\left(y_i-a-bx_i\right)^2. Find the gradient bb of the least-squares regression line of yy on xx, giving your answer exactly.

Worked solution

  1. Recall the least-squares gradient

    b=SxySxxb=\frac{S_{xy}}{S_{xx}}

    Setting the partial derivatives of the residual sum of squares to zero gives this value of bb.

  2. Substitute the two summary statistics

    b=SxySxx=120100b=\frac{S_{xy}}{S_{xx}}=\frac{120}{100}

    The gradient is the ratio of the two sums of squares, nothing else is needed.

  3. Evaluate the quotient

    b=120100=1.2b=\frac{120}{100}=1.2

    The sign of bb is the sign of SxyS_{xy}, since Sxx>0S_{xx}>0 always.

  4. State the gradient

    b=1.2b=1.2

    This is the least-squares estimate of the change in yy per unit increase in xx.

Answer
b=1.2b=1.2
Question 4
2 markseasy
An electronics technician measures the current drawn by a component at several applied voltages. For each observation, xx denotes the applied voltage (volts) and yy denotes the current (mA). From the 55 paired observations it is found that Sxx=250S_{xx}=250 and Sxy=215S_{xy}=215. The least-squares regression line of yy on xx is the line y=a+bxy=a+bx that minimises the residual sum of squares ∑i=1n(yi−a−bxi)2\sum_{i=1}^{n}\left(y_i-a-bx_i\right)^2. Find the gradient bb of the least-squares regression line of yy on xx, giving your answer exactly.

Worked solution

  1. Recall the least-squares gradient

    b=SxySxxb=\frac{S_{xy}}{S_{xx}}

    Setting the partial derivatives of the residual sum of squares to zero gives this value of bb.

  2. Substitute the two summary statistics

    b=SxySxx=215250b=\frac{S_{xy}}{S_{xx}}=\frac{215}{250}

    The gradient is the ratio of the two sums of squares, nothing else is needed.

  3. Evaluate the quotient

    b=215250=0.86b=\frac{215}{250}=0.86

    The sign of bb is the sign of SxyS_{xy}, since Sxx>0S_{xx}>0 always.

  4. State the gradient

    b=0.86b=0.86

    This is the least-squares estimate of the change in yy per unit increase in xx.

Answer
b=0.86b=0.86
Question 5
2 markseasy
A geologist measures the porosity of core samples taken from different depths in a borehole. For each observation, xx denotes the depth (tens of metres) and yy denotes the porosity (%). From the 66 paired observations it is found that b=−0.95b=-0.95, xˉ=10\bar{x}=10 and yˉ=60\bar{y}=60. Find the intercept aa of the least-squares regression line of yy on xx, giving your answer exactly.

Worked solution

  1. Recall the least-squares intercept

    a=yˉ−bxˉa=\bar{y}-b\bar{x}

    The fitted line always passes through the mean point (xˉ,yˉ)\left(\bar{x},\bar{y}\right).

  2. Substitute the gradient and the two means

    a=60−(−0.95)(10)a=60-\left(-0.95\right)\left(10\right)

    The line is forced through the mean point, which fixes aa once bb is known.

  3. Evaluate

    a=60−(−9.5)=69.5a=60-\left(-9.5\right)=69.5

    Careful with the sign: a negative gradient raises the intercept.

  4. State the intercept

    a=69.5a=69.5

    This is the value the model predicts for yy when x=0x=0.

Answer
a=69.5a=69.5

Unlock 65 more Linear regression questions

Create a free account to work through every Further Maths Linear regression question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Linear regression practice

Hard QuestionsPractice Questions

Related Statistics topics

Central limit theoremChi-squared testsCombinations of random variablesContinuous random variablesThe continuous uniform distributionDiscrete random variables