To find the equation of the least-squares regression line for the given data, we follow several steps:
1. List the Data: We start by listing the test scores for clarity.
- First test scores ([tex]\( x \)[/tex]): 86, 82, 47, 69, 73, 47, 63, 50, 59, 77, 88, 87
- Second test scores ([tex]\( y \)[/tex]): 81, 71, 51, 75, 69, 44, 48, 48, 58, 68, 83, 77
2. Calculate Means: Determine the mean of the first test scores ([tex]\( \bar{x} \)[/tex]) and the mean of the second test scores ([tex]\( \bar{y} \)[/tex]).
- Mean of [tex]\( x \)[/tex]: [tex]\( \bar{x} \)[/tex]
- Mean of [tex]\( y \)[/tex]: [tex]\( \bar{y} \)[/tex]
3. Compute the Slope: Use the formula for the slope ([tex]\( b \)[/tex]) of the regression line:
[tex]\[
b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\][/tex]
4. Compute the Intercept: Use the formula for the intercept ([tex]\( a \)[/tex]):
[tex]\[
a = \bar{y} - b \bar{x}
\][/tex]
Given the data and calculations, we get:
- Slope ([tex]\( b \)[/tex]): 0.823 (rounded to three decimal places)
- Intercept ([tex]\( a \)[/tex]): 7.610 (rounded to three decimal places)
5. Form the Equation: Combine the slope and intercept to form the equation of the least-squares regression line:
[tex]\[
y = bx + a
\][/tex]
Substituting the values of [tex]\( b \)[/tex] and [tex]\( a \)[/tex]:
[tex]\[
y = 0.823x + 7.610
\][/tex]
Hence, the equation of the least-squares regression line is:
[tex]\[
y = 0.823x + 7.610
\][/tex]
