Find the Least-Squares Regression Line and Use the Line to Make Predictions
-In order for a company's employees to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Find the equation of the regression line for the given data.
$\begin{array}{l|c|c|c|c|c|c|c|c|c|}
\text { Number of years, } \mathrm{x} & 3 & 4 & 4 & 5 & 3 & 6 & 2 & 7 & 3 \
\hline \text { Grades on test, } \mathrm{y} & 61 & 68 & 75 & 82 & 73 & 90 & 58 & 93 & 72
\end{array}$
A) $\hat {y} = 6.91 x + 46.26$
B) $\hat { y } = 6.91 \mathrm { x } - 46.26$
C) $\hat {y} = 46.26 x - 6.91$
D) $\hat { y } = 46.26 x + 6.91$

Question

Find the Least-Squares Regression Line and Use the Line to Make Predictions
-In order for a company's employees to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Find the equation of the regression line for the given data.
$\begin{array}{l|c|c|c|c|c|c|c|c|c|}
\text { Number of years, } \mathrm{x} & 3 & 4 & 4 & 5 & 3 & 6 & 2 & 7 & 3 \
\hline \text { Grades on test, } \mathrm{y} & 61 & 68 & 75 & 82 & 73 & 90 & 58 & 93 & 72
\end{array}$
A) $\hat {y} = 6.91 x + 46.26$
B) $\hat  { y } = 6.91 \mathrm { x } - 46.26$
C) $\hat {y} = 46.26 x - 6.91$
D) $\hat { y } = 46.26 x + 6.91$

Quizplus · Accepted Answer

To find the least-squares regression line, we calculate the slope ($m$) and the y-intercept ($b$) using the formulas derived from the given data points. The correct equation, based on the calculation from the provided data, is $\hat {y} = 6.91x + 46.26$, which matches choice A. This equation represents the best fit line that minimizes the squared differences between the observed values and the values predicted by the line.