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)-0.87-1.82-2.49-3.96-5.6-6.12-7.84
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = -0.9927
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = -0.1165, intercept c = 0.5600
    y = -0.1165x + 0.5600
  3. Use your line of best fit to estimate the compression (mm) when 55 kg is applied.
    Show answer
    compression = m × 55 + c
    -5.847 mm
  4. What is the spring rate in N/m?
    Show answer
    spring rate = 9810 / m (1 kg ≈ 9.81 N, and mm → m)
    84210 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)0.030.080.070.150.180.280.31
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.9837
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.003203, intercept c = 0.001586
    y = 0.003203x + 0.001586
  3. Estimate the current (A) when 45 V is applied.
    Show answer
    current = m × 45 + c
    0.1457 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
    312.2 Ω
3. The data below shows figures for height (cm) and mass (kg) for UK adults.
Height (cm)140150160170180190200
Mass (kg)68.6767.573.4473.4477.7676.9588.2
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.9131
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.2922, intercept c = 25.47
    y = 0.2922x + 25.47
  3. Estimate the target weight (N) for a height of 190 cm.
    Show answer
    mass = m × 190 + c, then weight = mass × 9.81
    794.4 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)41.4644.5548.049.258.190.472.0
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.8612
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 0.4089, intercept c = 39.54
    y = 0.4089x + 39.54
  3. Estimate the life expectancy (years) for an annual income of $25,000.
    Show answer
    life expectancy = m × 25 + c (income is in thousands)
    49.76 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)22.3217.2222.5625.9229.140.6834.86
  1. Calculate the Pearson correlation coefficient (r) to 4 significant figures.
    Show answer
    Use your calculator's statistics mode, or a spreadsheet.
    r = 0.8719
  2. Calculate the equation of the line of best fit, in the form y = mx + c.
    Show answer
    gradient m = 2.492, intercept c = 13.99
    y = 2.492x + 13.99
  3. Estimate the sales income (£) for an advertising spend of £5500.
    Show answer
    income = (m × 5.5 + c) × 1000
    £27700
  4. Estimate the sales income (£) for an advertising spend of £9000.
    Show answer
    income = (m × 9.0 + c) × 1000
    £36420
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