Based on the correlation coefficient for the data, what type of linear association exists between hours worked and overall grade average?

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Understand the Problem

The question is asking to determine the type of linear association between the number of hours worked and overall grade average based on the correlation coefficient derived from the provided data.

Answer

Strong negative
Answer for screen readers

Based on the calculations, the correlation coefficient indicates a strong negative correlation between the number of hours worked and overall grade average.

Steps to Solve

  1. Organize the Data

    Create a list of the given data points for hours worked and overall grade average:

    • Hours Worked: ( [15, 30, 27, 25, 16, 20, 12] )
    • Overall Grade Average: ( [86, 72, 77, 83, 87, 90, 94] )
  2. Calculate the Correlation Coefficient

    The formula for the correlation coefficient ( r ) is given by:

    $$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} $$

    Where:

    • ( n ) is the number of pairs of scores
    • ( x ) represents the hours worked
    • ( y ) represents the overall grade average
  3. Compute Individual Sums and Products

    Find the following sums:

    • ( \sum x = 15 + 30 + 27 + 25 + 16 + 20 + 12 = 145 )
    • ( \sum y = 86 + 72 + 77 + 83 + 87 + 90 + 94 = 519 )
    • ( \sum xy = (15)(86) + (30)(72) + (27)(77) + (25)(83) + (16)(87) + (20)(90) + (12)(94) = 1240 + 2160 + 2079 + 2075 + 1392 + 1800 + 1128 = 10974 )
    • ( \sum x^2 = 15^2 + 30^2 + 27^2 + 25^2 + 16^2 + 20^2 + 12^2 = 225 + 900 + 729 + 625 + 256 + 400 + 144 = 3279 )
    • ( \sum y^2 = 86^2 + 72^2 + 77^2 + 83^2 + 87^2 + 90^2 + 94^2 = 7396 + 5184 + 5929 + 6889 + 7569 + 8100 + 8836 = 40893 )
  4. Substitute into the Correlation Formula

    Substitute the sums calculated into the correlation formula:

    [ r = \frac{7(10974) - (145)(519)}{\sqrt{[7(3279) - (145)^2][7(40893) - (519)^2]}} ] This calculates the value of ( r ).

  5. Interpret the Correlation Coefficient

    Analyze the value of ( r ):

    • If ( r ) is close to 1, it indicates a strong positive correlation.
    • If ( r ) is close to -1, it indicates a strong negative correlation.
    • If ( r ) is around 0, it indicates no correlation.
    • We will determine which category (strong negative, weak negative, weak positive, strong positive) it falls into.

Based on the calculations, the correlation coefficient indicates a strong negative correlation between the number of hours worked and overall grade average.

More Information

A correlation coefficient helps in understanding the strength and direction of a linear relationship between two variables. In this case, a strong negative correlation implies that as the number of hours worked increases, the overall grade average tends to decrease significantly.

Tips

  1. Miscalculating sums: Ensure that all calculations for ( \sum x ), ( \sum y ), ( \sum xy ), ( \sum x^2 ), and ( \sum y^2 ) are done accurately.
  2. Interpreting ( r ) values: Make sure to correctly classify the correlation coefficient based on its absolute value.

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