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Inferential Statistics · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQRegression analysis10 marksPaper 3~15 min
A biostatistician investigates the relationship between enzyme concentration, xx (in U L−1\text{U\,L}^{-1}), and reaction rate, yy (in μmol min−1\mu\text{mol\,min}^{-1}), from n=8n = 8 independent experiments. xx — 2.02.0 — 3.53.5 — 5.05.0 — 6.06.0 — 7.57.5 — 9.09.0 — 10.510.5 — 12.012.0 yy — 1.81.8 — 3.23.2 — 4.54.5 — 5.15.1 — 6.86.8 — 7.97.9 — 9.39.3 — 10.610.6 A simple linear regression model y=α+βx+εy = \alpha + \beta x + \varepsilon is proposed, where ε\varepsilon represents random error.
(a)
Define the sum of squared residuals S=∑i=1n(yi−α−βxi)2S = \displaystyle\sum_{i=1}^{n}(y_i - \alpha - \beta x_i)^2. By minimising SS with respect to α\alpha and β\beta using partial differentiation, prove that the least squares estimates satisfy β^=SxySxx,α^=yˉ−β^ xˉ,\hat{\beta} = \frac{S_{xy}}{S_{xx}}, \qquad \hat{\alpha} = \bar{y} - \hat{\beta}\,\bar{x}, where Sxy=∑i=1n(xi−xˉ)(yi−yˉ)S_{xy} = \displaystyle\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) and Sxx=∑i=1n(xi−xˉ)2S_{xx} = \displaystyle\sum_{i=1}^{n}(x_i - \bar{x})^2. [5 marks]
(b)
Using the data above, calculate α^\hat{\alpha} and β^\hat{\beta}, giving your answers correct to three significant figures. [2 marks]
(c)
The eight residuals ei=yi−y^ie_i = y_i - \hat{y}_i, computed using your regression line from (b), are given below (rounded to two decimal places). y^i\hat{y}_i — 1.641.64 — 3.013.01 — 4.384.38 — 5.305.30 — 6.676.67 — 8.048.04 — 9.419.41 — 10.7810.78 eie_i — 0.160.16 — 0.190.19 — 0.120.12 — −0.20-0.20 — 0.130.13 — −0.14-0.14 — −0.11-0.11 — −0.18-0.18 By referring to the residuals above, evaluate the validity of the linear regression model with respect to linearity and homoscedasticity. Hence state, with a reason, whether a linear model or a transformed model is more appropriate for these data. [3 marks]
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