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) | -1.18 | -1.64 | -3.54 | -4.36 | -5.8 | -5.46 | -7.77 |
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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 = -0.9759 -
Calculate the equation of the line of best fit, in the form y = mx + c.
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gradient m = -0.1060, intercept c = -0.01143y = -0.1060x − 0.01143 -
Use your line of best fit to estimate the compression (mm) when 45 kg is applied.
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compression = m × 45 + c-4.780 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)92580 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) | 0.04 | 0.08 | 0.09 | 0.1 | 0.21 | 0.28 | 0.25 |
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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 = 0.9493 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.002729, intercept c = 0.01744y = 0.002729x + 0.01744 -
Estimate the current (A) when 25 V is applied.
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current = m × 25 + c0.08567 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/m366.4 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) | 61.74 | 67.5 | 78.48 | 77.27 | 87.48 | 81.22 | 94.5 |
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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 = 0.9314 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.4811, intercept c = -3.481y = 0.4811x − 3.481 -
Estimate the target weight (N) for a height of 180 cm.
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mass = m × 180 + c, then weight = mass × 9.81815.4 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) | 47.9 | 41.4 | 56.5 | 60.0 | 74.2 | 88.0 | 73.8 |
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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 = 0.8849 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.3894, intercept c = 45.84y = 0.3894x + 45.84 -
Estimate the life expectancy (years) for an annual income of $45,000.
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life expectancy = m × 45 + c (income is in thousands)63.36 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) | 18.72 | 20.37 | 29.28 | 29.97 | 25.8 | 26.28 | 45.78 |
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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 = 0.8199 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 2.584, intercept c = 14.00y = 2.584x + 14.00 -
Estimate the sales income (£) for an advertising spend of £3500.
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income = (m × 3.5 + c) × 1000£23050 -
Estimate the sales income (£) for an advertising spend of £8000.
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income = (m × 8.0 + c) × 1000£34670