(a) Writing the Least Squares Regression Equation:
To find the least squares regression line, we need to determine the slope and the intercept. Let [tex]\( x \)[/tex] be the average sleep in hours and [tex]\( y \)[/tex] be the test score in percentage.
The regression equation is of the form:
[tex]\[ y = mx + b \][/tex]
Where:
- [tex]\( m \)[/tex] is the slope,
- [tex]\( b \)[/tex] is the y-intercept.
After performing the calculations, we find:
- Slope ([tex]\( m \)[/tex]): 6.3333 (rounded to four decimal places)
- Intercept ([tex]\( b \)[/tex]): 38.6875 (rounded to four decimal places)
Thus, the least squares regression equation is:
[tex]\[ y = 6.3333x + 38.6875 \][/tex]
(b) Determining the Approximate Test Score for a Student Who Sleeps 8 Hours a Night:
Now, we use the regression equation to determine the test score for a student who averages 8 hours of sleep each night. Substitute [tex]\( x = 8 \)[/tex] into the regression equation:
[tex]\[ y = 6.3333 \times 8 + 38.6875 \][/tex]
Step-by-step calculation:
1. Multiply the slope by the number of sleep hours:
[tex]\[ 6.3333 \times 8 = 50.6664 \][/tex]
2. Add the intercept to this value:
[tex]\[ 50.6664 + 38.6875 = 89.354 \][/tex]
So, the approximate test score for a student who sleeps an average of 8 hours a night is:
[tex]\[ y \approx 89.354 \][/tex]
Thus, the student's predicted test score is approximately 89.35.
