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A-Level Correlation and linear regression
What the A-Level syllabus expects for Correlation and linear regression, and how to practise it.
What the syllabus expects
- Using a scatter diagram to decide whether a linear relationship between two variables is plausible
- The correlation coefficient as an indicator of how well a linear model fits the scatter
- Reading the product moment correlation coefficient, especially values near −1, 0 and 1
- Linear regression and the method of least squares for obtaining the regression line
- The ideas of interpolation and extrapolation
- Choosing the right regression line to predict or estimate a value in practice, and judging how well the linear model fits the situation
- Applying a square, reciprocal or logarithmic transformation to make the data linear
- Deriving the formulae, the relation r² = b₁b₂ between the two regression coefficients, and hypothesis tests
How it's examined
Questions on this topic most often ask you to compare, find, explain. About 5% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (3 marks)
Compare Model A: h=c+d√t with Model B: h=c+d ln t. Explain which fits the data better and give that model's least-squares regression line to 3 decimal places.
Show the worked answer
Neither model is linear in t, so transform the variable first so that each becomes a straight line, then compare how straight the transformed data is. Model A: h = c + d√t is linear in √t, so form the pairs (√t, h) and find the product moment correlation coefficient r. Model B: h = c + d ln t is linear in ln t, so form the pairs (ln t, h) and find r. From the data: Model A gives r = 0.974 Model B gives r = 0.997 The closer |r| is to 1, the closer the transformed points lie to a straight line, and therefore the better that model describes the data. Since 0.997 is closer to 1 than 0.974 is, Model B, h = c + d ln t, is the better fit. For Model B, the least-squares regression line of h on ln t has c = 3.890 and d = 5.311 (3 d.p.), so the line is h = 3.890 + 5.311 ln t.
More A-Level H2 Maths topics
Functions · Graphs and their transformations · Equations and inequalities · Sequences and series · Vectors in two and three dimensions: basic properties · Scalar and vector products · all of A-Level H2 Maths