Regression and correlation Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Regression and correlation questions. See exactly how to solve problems on pmcc, summary-statistics, regression-line, prediction.

pmccsummary-statisticsregression-linepredictionexponential-modellogs
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
For a set of bivariate data, Sxx=40S_{xx} = 40, Syy=30S_{yy} = 30 and Sxy=32S_{xy} = 32. Calculate the product moment correlation coefficient rr, giving your answer to 3 decimal places.

Worked solution

  1. Recall the formula for the PMCC

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    The product moment correlation coefficient uses the three summary statistics.

  2. Write down the given summary statistics

    Sxx=40, Syy=30, Sxy=32S_{xx}=40,\ S_{yy}=30,\ S_{xy}=32

    These values are provided in the question.

  3. State the PMCC to 3 decimal places

    r=3240×30=0.924r=\frac{32}{\sqrt{40\times 30}}=0.924

    This is the required correlation coefficient.

Answer
0.9240.924
Question 2
2 markseasy
For a set of bivariate data, Sxx=50S_{xx} = 50, Syy=45S_{yy} = 45 and Sxy=−30S_{xy} = -30. Calculate the product moment correlation coefficient rr, giving your answer to 3 decimal places.

Worked solution

  1. Recall the formula for the PMCC

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    The product moment correlation coefficient uses the three summary statistics.

  2. Write down the given summary statistics

    Sxx=50, Syy=45, Sxy=−30S_{xx}=50,\ S_{yy}=45,\ S_{xy}=-30

    These values are provided in the question.

  3. State the PMCC to 3 decimal places

    r=−3050×45=−0.632r=\frac{-30}{\sqrt{50\times 45}}=-0.632

    This is the required correlation coefficient.

Answer
−0.632-0.632
Question 3
2 markseasy
For a set of bivariate data, Sxx=18S_{xx} = 18, Syy=50S_{yy} = 50 and Sxy=24S_{xy} = 24. Calculate the product moment correlation coefficient rr, giving your answer to 3 decimal places.

Worked solution

  1. Recall the formula for the PMCC

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    The product moment correlation coefficient uses the three summary statistics.

  2. Write down the given summary statistics

    Sxx=18, Syy=50, Sxy=24S_{xx}=18,\ S_{yy}=50,\ S_{xy}=24

    These values are provided in the question.

  3. State the PMCC to 3 decimal places

    r=2418×50=0.800r=\frac{24}{\sqrt{18\times 50}}=0.800

    This is the required correlation coefficient.

Answer
0.8000.800
Question 4
2 markseasy
For a set of bivariate data, Sxx=64S_{xx} = 64, Syy=81S_{yy} = 81 and Sxy=60S_{xy} = 60. Calculate the product moment correlation coefficient rr, giving your answer to 3 decimal places.

Worked solution

  1. Recall the formula for the PMCC

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    The product moment correlation coefficient uses the three summary statistics.

  2. Write down the given summary statistics

    Sxx=64, Syy=81, Sxy=60S_{xx}=64,\ S_{yy}=81,\ S_{xy}=60

    These values are provided in the question.

  3. State the PMCC to 3 decimal places

    r=6064×81=0.833r=\frac{60}{\sqrt{64\times 81}}=0.833

    This is the required correlation coefficient.

Answer
0.8330.833
Question 5
2 markseasy
A shop models weekly sales. A regression line of yy on xx is y=2.5+1.4xy = 2.5 + 1.4x. Use it to estimate yy when x=10x = 10, giving your answer to 2 decimal places.

Worked solution

  1. Write down the regression equation

    y=2.5+1.4xy = 2.5 + 1.4x

    This is the fitted line of y on x.

  2. Substitute the given value of x

    y=2.5+1.4×10y = 2.5 + 1.4\times 10

    Replace x with the value we want a prediction for.

  3. State the predicted value to 2 decimal places

    y≈16.50y \approx 16.50

    This is the estimate given by the regression line.

Answer
16.5016.50

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