taiwan-gsat 2025 Q5

taiwan-gsat · Other · ast__math-b 5 marks Linear regression
A company collected data on the number of customers $x$ (in units of 100 people) and sales revenue $y$ (in units of 10,000 yuan) from 8 branches last week, obtaining 8 data points $(x, y)$ as follows: $(3,3), (3,5), (3,2), (4,4), (5,8), (6,7), (8,12), (8,7)$. These 8 points are plotted on the coordinate plane, and the regression line equation for $y$ with respect to $x$ is determined to be $y = \frac { 5 } { 4 } x - \frac { 1 } { 4 }$.
The company wants to analyze from another perspective. The 8 data points are sorted separately from smallest to largest for the number of customers and sales revenue, resulting in new 8 data points $(x, y)$ as follows: $(3,2), (3,3), (3,4), (4,5), (5,7), (6,7), (8,8), (8,12)$. Let the regression line equation for $y$ with respect to $x$ for the new 8 data points be $y = m x + b$, where $m, b$ are real numbers. Based on the above, select the correct option.
(1) $m = \frac { 5 } { 4 }$ and $b = - \frac { 1 } { 4 }$
(2) $m > \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$
(3) $m > \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$
(4) $m < \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$
(5) $m < \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$
A company collected data on the number of customers $x$ (in units of 100 people) and sales revenue $y$ (in units of 10,000 yuan) from 8 branches last week, obtaining 8 data points $(x, y)$ as follows:\\
$(3,3), (3,5), (3,2), (4,4), (5,8), (6,7), (8,12), (8,7)$.\\
These 8 points are plotted on the coordinate plane, and the regression line equation for $y$ with respect to $x$ is determined to be $y = \frac { 5 } { 4 } x - \frac { 1 } { 4 }$.

The company wants to analyze from another perspective. The 8 data points are sorted separately from smallest to largest for the number of customers and sales revenue, resulting in new 8 data points $(x, y)$ as follows:\\
$(3,2), (3,3), (3,4), (4,5), (5,7), (6,7), (8,8), (8,12)$.\\
Let the regression line equation for $y$ with respect to $x$ for the new 8 data points be $y = m x + b$, where $m, b$ are real numbers. Based on the above, select the correct option.\\
(1) $m = \frac { 5 } { 4 }$ and $b = - \frac { 1 } { 4 }$\\
(2) $m > \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$\\
(3) $m > \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$\\
(4) $m < \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$\\
(5) $m < \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$