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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 ____.

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Answer:


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%.

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