taiwan-gsat 2024 Q9

taiwan-gsat · Other · gsat__math-a 5 marks Linear regression
A laboratory collected a large number of two similar species $A$ and $B$, recording their body length $x$ (in centimeters) and body weight $y$ (in grams). The average body lengths of species $A$ and $B$ are $\overline{x_{A}} = 5.2$ and $\overline{x_{B}} = 6$ respectively, with standard deviations 0.3 and 0.1 respectively. Let the average body weights of species $A$ and $B$ be $\overline{y_{A}}$ and $\overline{y_{B}}$ respectively. If the regression lines of body weight $y$ on body length $x$ for species $A$ and $B$ are $L_{A}: y = 2x - 0.6$ and $L_{B}: y = 1.5x + 0.4$ respectively, with correlation coefficients 0.6 and 0.3 respectively. An individual $P$ with body length 5.6 centimeters and body weight 8.6 grams is discovered. Select the correct options.
(1) $\overline{y_{A}} < \overline{y_{B}}$
(2) The standard deviation of body weight for species $A$ is less than that for species $B$
(3) For species $A$, the absolute difference between individual $P$'s body weight and the average body weight $\overline{y_{A}}$ is greater than one standard deviation
(4) The distance from point $(5.6, 8.6)$ to line $L_{A}$ is less than its distance to line $L_{B}$
(5) The distance from point $(5.6, 8.6)$ to point $(\overline{x_{A}}, \overline{y_{A}})$ is less than its distance to point $(\overline{x_{B}}, \overline{y_{B}})$
A laboratory collected a large number of two similar species $A$ and $B$, recording their body length $x$ (in centimeters) and body weight $y$ (in grams). The average body lengths of species $A$ and $B$ are $\overline{x_{A}} = 5.2$ and $\overline{x_{B}} = 6$ respectively, with standard deviations 0.3 and 0.1 respectively. Let the average body weights of species $A$ and $B$ be $\overline{y_{A}}$ and $\overline{y_{B}}$ respectively. If the regression lines of body weight $y$ on body length $x$ for species $A$ and $B$ are $L_{A}: y = 2x - 0.6$ and $L_{B}: y = 1.5x + 0.4$ respectively, with correlation coefficients 0.6 and 0.3 respectively. An individual $P$ with body length 5.6 centimeters and body weight 8.6 grams is discovered. Select the correct options.\\
(1) $\overline{y_{A}} < \overline{y_{B}}$\\
(2) The standard deviation of body weight for species $A$ is less than that for species $B$\\
(3) For species $A$, the absolute difference between individual $P$'s body weight and the average body weight $\overline{y_{A}}$ is greater than one standard deviation\\
(4) The distance from point $(5.6, 8.6)$ to line $L_{A}$ is less than its distance to line $L_{B}$\\
(5) The distance from point $(5.6, 8.6)$ to point $(\overline{x_{A}}, \overline{y_{A}})$ is less than its distance to point $(\overline{x_{B}}, \overline{y_{B}})$