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) | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|---|
| Compression (mm) | -0.87 | -1.82 | -2.49 | -3.96 | -5.6 | -6.12 | -7.84 |
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Calculate the Pearson correlation coefficient (r) to 4 significant figures.
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Use your calculator's statistics mode, or a spreadsheet.r = -0.9927 -
Calculate the equation of the line of best fit, in the form y = mx + c.
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gradient m = -0.1165, intercept c = 0.5600y = -0.1165x + 0.5600 -
Use your line of best fit to estimate the compression (mm) when 55 kg is applied.
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compression = m × 55 + c-5.847 mm -
What is the spring rate in N/m?
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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) | 10 | 20 | 30 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|---|---|
| Current (A) | 0.03 | 0.08 | 0.07 | 0.15 | 0.18 | 0.28 | 0.31 |
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Calculate the Pearson correlation coefficient (r) to 4 significant figures.
Show answer
Use your calculator's statistics mode, or a spreadsheet.r = 0.9837 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.003203, intercept c = 0.001586y = 0.003203x + 0.001586 -
Estimate the current (A) when 45 V is applied.
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current = m × 45 + c0.1457 A -
What is the resistance of the silicon, in ohms?
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from I = V/R, the gradient is 1/R, so R = 1/m312.2 Ω
3. The data below shows figures for height (cm) and mass (kg) for UK adults.
| Height (cm) | 140 | 150 | 160 | 170 | 180 | 190 | 200 |
|---|---|---|---|---|---|---|---|
| Mass (kg) | 68.67 | 67.5 | 73.44 | 73.44 | 77.76 | 76.95 | 88.2 |
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Calculate the Pearson correlation coefficient (r) to 4 significant figures.
Show answer
Use your calculator's statistics mode, or a spreadsheet.r = 0.9131 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.2922, intercept c = 25.47y = 0.2922x + 25.47 -
Estimate the target weight (N) for a height of 190 cm.
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mass = m × 190 + c, then weight = mass × 9.81794.4 N -
What are the units of the gradient?
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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) | 0.5 | 10 | 20 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|---|---|
| Life expectancy (years) | 41.46 | 44.55 | 48.0 | 49.2 | 58.1 | 90.4 | 72.0 |
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Calculate the Pearson correlation coefficient (r) to 4 significant figures.
Show answer
Use your calculator's statistics mode, or a spreadsheet.r = 0.8612 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.4089, intercept c = 39.54y = 0.4089x + 39.54 -
Estimate the life expectancy (years) for an annual income of $25,000.
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life expectancy = m × 25 + c (income is in thousands)49.76 years -
What are the units of the gradient?
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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) | 2 | 3 | 4 | 5 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|---|
| Income (£ ×1000) | 22.32 | 17.22 | 22.56 | 25.92 | 29.1 | 40.68 | 34.86 |
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Calculate the Pearson correlation coefficient (r) to 4 significant figures.
Show answer
Use your calculator's statistics mode, or a spreadsheet.r = 0.8719 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 2.492, intercept c = 13.99y = 2.492x + 13.99 -
Estimate the sales income (£) for an advertising spend of £5500.
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income = (m × 5.5 + c) × 1000£27700 -
Estimate the sales income (£) for an advertising spend of £9000.
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income = (m × 9.0 + c) × 1000£36420