Question

A high school counselor wants to look at the relationship between GPA and the numberof absences for students in the senior class this year. That data shows a linear patternwith the summary statistics shown below.Please help with all parts

Answer 1

Let x be the number of absences and y be the GPA

Given;

`\begin{gathered} \bar{x}=5.0,\bar{y}=2.9 \\ S_x=1.2,S_y=0.3,r=-0.65 \end{gathered}`

(a) The slope b of the lease squares regression line is;

`\begin{gathered} b=r((S_y)/(S_x)) \\ \text{Where r is the correlation between variable x and y}, \\ S_y\text{ is the standard deviation of y-scores} \\ S_x\text{ is the standard deviation of x-scores} \end{gathered}`
`\begin{gathered} b=-0.65((0.3)/(1.2)) \\ b=-0.65(0.25) \\ b=-0.1625 \end{gathered}`

Hence, the slope is -0.1625

(b) The y-intercept of the line is;

`a=\bar{y}-b(\bar{x})`
`\begin{gathered} a=2.9-(-0.1625)(5.0) \\ a=2.9+0.8125 \\ a=3.7125 \end{gathered}`

Hence, the y-intercept is 3.7125

(c) The equation of the line is;

`\begin{gathered} \hat{y}=a+bx \\ \text{Where;} \\ a=3.7125,b=-0.1625 \end{gathered}`
`\begin{gathered} \hat{y}=3.7125+(-0.1625)x \\ \hat{y}=3.7125-0.1625x \end{gathered}`

(d) The slope of the least-squares regression line is the average change in the predicted values of the response variable when the explanatory variable increases by 1 unit.

Hence, the slope of the least squares regression lines shows the change in the GPA due to number of absences.

(e) The estimate GPA for a student with three absences is;

`\begin{gathered} \text{at x=3;} \\ \hat{y}=3.7125-0.1625x \\ \hat{y}=3.7125-0.1625(3) \\ \hat{y}=3.7125-0.4875 \\ \hat{y}=3.2250 \end{gathered}`

(f) The r-squared value of the data is;

`\begin{gathered} r^2=(-0.65)^2 \\ r^2=0.4225 \end{gathered}`

An r-squared of 42% shows that 42% of the data fit the regression model.