To find the equation of the least-squares regression line for the given data, we need to find the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the regression line equation:
[tex]\[ y = mx + b \][/tex]
Let's break down the steps to find this equation.
### Step 1: Calculate the Means
First, we calculate the mean of the first test scores ([tex]\( x \)[/tex]) and the mean of the second test scores ([tex]\( y \)[/tex]).
The given scores for the first test ([tex]\( x \)[/tex]) are: 60, 82, 46, 72, 43, 55, 77, 95, 72, 42, 79, 64. The mean ([tex]\( \overline{x} \)[/tex]) of these scores is:
[tex]\[ \overline{x} = 65.583 \][/tex]
The given scores for the second test ([tex]\( y \)[/tex]) are: 54, 74, 48, 55, 40, 61, 65, 75, 72, 29, 70, 59. The mean ([tex]\( \overline{y} \)[/tex]) of these scores is:
[tex]\[ \overline{y} = 58.5 \][/tex]
### Step 2: Calculate the Slope ( [tex]\( m \)[/tex] )
The slope ([tex]\( m \)[/tex]) of the regression line is calculated using the formula:
[tex]\[ m = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2} \][/tex]
After performing the necessary calculations:
[tex]\[ m = 0.752 \][/tex]
### Step 3: Calculate the Y-Intercept ( [tex]\( b \)[/tex] )
The y-intercept ([tex]\( b \)[/tex]) is calculated using the formula:
[tex]\[ b = \overline{y} - m \cdot \overline{x} \][/tex]
After substituting the values of [tex]\( \overline{x} \)[/tex], [tex]\( \overline{y} \)[/tex], and [tex]\( m \)[/tex]:
[tex]\[ b = 9.159 \][/tex]
### Step 4: Form the Equation
With the calculated values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex], we can write the equation of the least-squares regression line.
[tex]\[ y = 0.752x + 9.159 \][/tex]
So, the equation of the least-squares regression line, rounded to three decimal places, is:
[tex]\[ y = 0.752x + 9.159 \][/tex]
