Question

A group of researchers conducted a study to determine whether the final grade in an honors section of introductory psychology was related to a student’s performance on a test of math ability administered for college entrance. The researchers looked at the test scores of 200 students (n= 200) and found a correlation of r= .45 between math ability scores and final course grade. The proportion of the variability seen in final grade performance that can be predicted by math ability scores is ____.

Answer 1

`r=(n(\sum xy)-(\sum x)(\sum y))/(√([n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]))`

And for this case
`r =0.45`

The % of variation is given by the determination coefficient given by
`r^2` and on this case
`0.45^2 =0.2025`, so then the % of variation explained is 20.25%.

The proportion of the variability seen in final grade performance that can be predicted by math ability scores is 20.25%.

Explanation:

For this case we asume that we fit a linear model:

`y = mx+b`

Where y represent the final grade and x the math ability scores

`m=(S_(xy))/(S_(xx))`

Where:

`S_(xy)=\sum_(i=1)^n x_i y_i -((\sum_(i=1)^n x_i)(\sum_(i=1)^n y_i))/(n)`

`S_(xx)=\sum_(i=1)^n x^2_i -((\sum_(i=1)^n x_i)^2)/(n)`

`\bar x= (\sum x_i)/(n)`

`\bar y= (\sum y_i)/(n)`

And we can find the intercept using this:

`b=\bar y -m \bar x`

The correlation coeffcient is given by:

`r=(n(\sum xy)-(\sum x)(\sum y))/(√([n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]))`

And for this case
`r =0.45`

The % of variation is given by the determination coefficient given by
`r^2` and on this case
`0.45^2 =0.2025`, so then the % of variation explained is 20.25%.

The proportion of the variability seen in final grade performance that can be predicted by math ability scores is 20.25%.