Notes & theory for this topic

Regression

Write your working on paper. Reveal each answer only after you've had a go.

1. The table shows the compression of a spring when different masses are applied.
Mass (kg)10203040506070
Compression (mm)-0.91-2.28-3.57-4.4-5.7-6.72-5.88
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = -0.9539
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = -0.09257, intercept c = -0.5057
    y = -0.09257x − 0.5057
  3. Use your line of best fit to estimate the compression (mm) when 35 kg is applied.
    Show answer
    compression = m × 35 + c
    -3.746 mm
  4. What is the spring rate in N/m?
    Show answer
    spring rate = 9810 / m (1 kg ≈ 9.81 N, and mm → m)
    106000 N/m
2. To find the electrical resistance of a test piece of silicon a range of voltages (V) was applied and the resulting current (A) measured.
Voltage (V)102030406080100
Current (A)0.040.080.10.150.210.210.3
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.9795
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.002692, intercept c = 0.02498
    y = 0.002692x + 0.02498
  3. Estimate the current (A) when 15 V is applied.
    Show answer
    current = m × 15 + c
    0.06535 A
  4. What is the resistance of the silicon, in ohms?
    Show answer
    from I = V/R, the gradient is 1/R, so R = 1/m
    371.5 Ω
3. The data below shows figures for height (cm) and mass (kg) for UK adults.
Height (cm)140150160170180190200
Mass (kg)59.2264.865.5284.1588.2977.8187.3
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.8564
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.4751, intercept c = -5.470
    y = 0.4751x − 5.470
  3. Estimate the target weight (N) for a height of 180 cm.
    Show answer
    mass = m × 180 + c, then weight = mass × 9.81
    785.3 N
  4. What are the units of the gradient?
    Show answer
    gradient = change in mass (kg) / change in height (cm)
    kg/cm
4. The table shows annual income (in thousands of dollars) versus life expectancy in years.
Income ($ ×1000)0.51020406080100
Life expectancy (years)32.641.8553.561.881.270.494.5
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.9458
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.5521, intercept c = 37.77
    y = 0.5521x + 37.77
  3. Estimate the life expectancy (years) for an annual income of $15,000.
    Show answer
    life expectancy = m × 15 + c (income is in thousands)
    46.06 years
  4. What are the units of the gradient?
    Show answer
    gradient = change in years / change in income ($1000)
    years per $1000 of income
5. A company measures its advertising spend versus sales income (both in thousands of pounds).
Spend (£ ×1000)23456810
Income (£ ×1000)22.1419.5324.7224.324.026.6446.2
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.8309
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 2.606, intercept c = 12.64
    y = 2.606x + 12.64
  3. Estimate the sales income (£) for an advertising spend of £4500.
    Show answer
    income = (m × 4.5 + c) × 1000
    £24370
  4. Estimate the sales income (£) for an advertising spend of £9000.
    Show answer
    income = (m × 9.0 + c) × 1000
    £36100
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