Question

15. The data set below shows the GMAT scores for five MBA students and the students' grade point averages (GPA) upon graduation. GMAT GPA 660 3.7 580 3 480 3.2 710 4 600 3.5 (a) Calculate r , the correlation coefficient between these two variables. r = (b) Interpret the value of r : the association is Select an answer and Select an answer (c) Compute the regression line for predicting GPA from GMAT score. ˆ y = x + (d) Predict the GPA of a student who gets a score of 500 on the GMAT. (e) Does the student with a GMAT score of 580 have a higher or lower GPA than the one predicted by the regression line? Higher Lower

Answer 1

We have a table that relates GMAT with GPA.

We have to calculate the correlation coefficient (r).

The formular for r is:

`\begin{gathered} r_(xy)=(1)/(n-1)\sum ^{}_{}\frac{(x-\bar{x})}{s_x}\cdot\frac{(y-\bar{y})}{s_y} \\ r_(xy)=(1)/(n-1)(1)/(s_x\cdot s_y)\sum ^{}_{}(x-\bar{x})(y-\bar{y}) \end{gathered}`

sx will be the standard deviation of the sample of GMAT, while sy will be the standard deviation of the sample of GPA.

Using software, we can calculate the mean and standard deviation of both variables:

`\begin{gathered} \bar{x}=606 \\ \bar{y}=3.48 \\ s_x=87.063 \\ s_y=0.396 \\ n=5 \end{gathered}`

Then, we can calculate r as:

The correlation coefficient is r = 0.823, which indicates a strong positive correlation, as the value of r is within the interval 0.7 and 0.9.

Now, we have to find the linear regression model to relate the two variables.

`\hat{y}=\beta_0+\beta_1\cdot x`

Then we can calculate the coefficients as:

`\begin{gathered} {\displaystyle{\begin{aligned}{\beta_1} & ={\frac{\displaystyle\sum_(i=1)^(n)x_(i)y_(i)-n{\bar{x}}{\bar{y}}}{\displaystyle\sum_(i=1)^(n)x_(i)^(2)-{(1)/(n)}\left(\sum_(i=1)^(n)x_(i)\right)^(2)}}\end{aligned}}} \\ \beta_0={(\displaystyle\sum_(i=1)^(n)x_(i)^(2)\sum_(i=1)^(n)y_(i)-\sum_(i=1)^(n)x_(i)y_(i)\sum_(i=1)^(n)x_(i))/(\displaystyle n\sum_(i=1)^(n)x_(i)^(2)-\left(\sum_(i=1)^(n)x_(i)\right)^(2))}={\bar{y}}-{\beta_1\cdot}\bar{x} \end{gathered}`

We can calculate the coefficients as:

`\begin{gathered} {\displaystyle{\begin{aligned}{\beta_1} & ={\frac{\displaystyle\sum^n_(i=1)x_iy_i-n{\bar{x}}{\bar{y}}}{\displaystyle\sum^n_(i=1)x^2_i-{(1)/(n)}(\sum^n_(i=1)x_i)^2}}\end{aligned}}} \\ {\displaystyle{\begin{aligned}{\beta_1} & ={\frac{\lbrack\displaystyle\mleft(660\mright)\mleft(3.7\mright)+\mleft(580\mright)\mleft(3\mright)+\mleft(480\mright)\mleft(3.2\mright)+\mleft(710\mright)\mleft(4\mright)+\mleft(600\mright)\mleft(3.5\mright)\rbrack_{}-5\cdot606\cdot3.48}{(660^2+580^2+480^2+710^2+600^2)-{(1)/(5)}(660+580+480+710+600)^2}}\end{aligned}}} \\ \beta_1=(\lbrack2442+1740+1536+2840+2100\rbrack-10544.4)/((435600+336400+230400+504100+360000)-(1)/(5)(3030)^2) \\ \beta_1=(10658-10544.4)/(1866500-(1)/(5)\cdot9180900) \\ \beta_1=(113.6)/(1866500-1836180) \\ \beta_1=(113.6)/(30320) \\ \beta_1\approx0.00375 \end{gathered}`

Now we can calculate the y-intercept of the regression model as:

`\begin{gathered} \beta_0=\bar{y}-\beta_1\cdot\bar{x} \\ \beta_0=3.48-0.00375\cdot606 \\ \beta_0=3.48-2.27 \\ \beta_0=1.21 \end{gathered}`

Then, we can write the regression model as:

`\hat{y}=0.00375x+1.21`

We can graph this with points as:

We can use the model to predict values of GPA for an specific value of GMAT.

If GMAT = 500, then we predict a value of GPA of:

`\begin{gathered} \hat{y}(500)=0.00375(500)+1.21 \\ \hat{y}(500)=1.875+1.21 \\ \hat{y}(500)\approx3.09 \end{gathered}`

We can use this model to predict the GPA for a GMAT score of 580 as:

`\begin{gathered} \hat{y}(580)=0.00375(580)+1.21 \\ \hat{y}(580)=2.18+1.21 \\ \hat{y}(580)=3.39 \end{gathered}`

The data point for GMAT = 580 indicates a GPA of 3, so the regression model estimates a higher value of GPA in this case (the actual value is lower than the predicted value).

Answer:

a) r = 0.823

b) The correlation is strong and positive.

c) y = 0.00375x + 1.21

d) GPA = 3.09

e) Lower