A high school counselor want to look at the relationship between the grade point average (GPA) and the number of absences for students in the senior class this past year. The data show a linear pattern with the summary statistics shown below:

0

The equation for the regression line is \widehat{y}= a+bx

y

=a+bx, where a is the y-intercept and b is the slope.

We can determine the slope from b=b=r\left(\ \frac{s_y}{s_x}\right)r(

s

x

s

y

)

Plugging in the values of rr, s_{y}s

y

and s_{x}s

x

from the table

b = 0.83\Big(\frac{0.7}{2.5}\Big) = 0.2324b=0.83(

2.5

0.7

)=0.2324

a = \overline{y} - b\overline{x} a=

y

−b

x

a = 3.2 - (0.2324) (4)a=3.2−(0.2324)(4)

a = 3.2 - 0.9296 = 2.2704a=3.2−0.9296=2.2704

Plugging the values of a and b in \widehat{y} = a + bx

y

=a+bx and rounding the values of a and b to the nearest hundredths, we get

\widehat{y} = 2.27 + 0.23 x

y

=2.27+0.23x

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

the answer is 21%.