Using the correlation coefficient results from Problem 34, calculate the statistical significance of the one correlation between X and Y and then between X and Y holding constant Z. How does controlling for Z change the significance of the correlation between X and Y?

Data From Problem 34

With regard to the earlier problems, many sociologists and developmental psychologists have noted that it might not be listening to Mozart per se that increases a child’s neural development. Instead, they argue, increased intelligence may be attributable to having the type of parents who would intentionally play music to their infant because it reputedly stimulates child development. This leads to a different hypothesis: There is a positive correlation between children’s intelligence and parents’ intervention in general, and not the specific nature of the intervention. To test this alternative explanation, calculate and interpret the strength and direction of a partial correlation between hours of Mozart (X) and IQ score at age 16 (Y) holding constant the amount of time (in hours per week) parents spent on interventions with their child (Z). Make sure to construct a correlation matrix, it will aid you in solving for r_XY.Z.

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IQ Score 128 104 96 107 106 99 91 111 131 117 Hours of Mozart/Year 68 51 18 26 36 47 10 72 94 48 Parental Intervention Hours 14 6 4 7 7 5 2 9 14 12