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) | -1.2 | -1.76 | -2.85 | -4.48 | -4.3 | -6.0 | -7.14 |
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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 = -0.9849 -
Calculate the equation of the line of best fit, in the form y = mx + c.
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gradient m = -0.09911, intercept c = 0.002857y = -0.09911x + 0.002857 -
Use your line of best fit to estimate the compression (mm) when 15 kg is applied.
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compression = m × 15 + c-1.484 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)98980 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 | 0.07 | 0.1 | 0.14 | 0.2 | 0.31 | 0.3 |
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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 = 0.9792 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.003275, intercept c = 0.005198y = 0.003275x + 0.005198 -
Estimate the current (A) when 85 V is applied.
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current = m × 85 + c0.2836 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/m305.3 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) | 57.33 | 69.53 | 72.0 | 81.09 | 72.9 | 90.63 | 81.0 |
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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 = 0.8313 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.4075, intercept c = 5.645y = 0.4075x + 5.645 -
Estimate the target weight (N) for a height of 150 cm.
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mass = m × 150 + c, then weight = mass × 9.81655.1 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) | 44.27 | 40.95 | 60.0 | 63.6 | 67.9 | 70.4 | 104.4 |
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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 = 0.9186 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 0.5156, intercept c = 41.63y = 0.5156x + 41.63 -
Estimate the life expectancy (years) for an annual income of $65,000.
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life expectancy = m × 65 + c (income is in thousands)75.15 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) | 17.28 | 23.94 | 29.28 | 22.95 | 32.4 | 34.92 | 43.26 |
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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 = 0.9357 -
Calculate the equation of the line of best fit, in the form y = mx + c.
Show answer
gradient m = 2.872, intercept c = 13.56y = 2.872x + 13.56 -
Estimate the sales income (£) for an advertising spend of £4500.
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income = (m × 4.5 + c) × 1000£26480 -
Estimate the sales income (£) for an advertising spend of £9000.
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income = (m × 9.0 + c) × 1000£39400