Question

x | 50 | 60 | 67 | 80 | 90 | 130 | 160
y | 60 | 40 | 69 | 60 | 80 | 83 | 120
residuals | 9.956 | -15.678 | 9.379 | -6.946 | 7.421 | -12.115 | 7.983
determine the strength and direction of the correlation between the variables in the context of the scenario.
there is a strong, negative correlation between the hours spent playing video games and the hours spent studying.
there is a strong, positive correlation between the hours spent playing video games and the hours spent studying
there is a weak, negative correlation between the hours spent playing video games and the hours spent studying
there is a weak, positive correlation between the hours spent playing video games and the hours spent studying

Explanation:

Step1: Analyze the trend of \(x\) (hours playing video games) and \(y\) (hours studying)

As \(x\) increases from \(50\) to \(160\), \(y\) has an overall increasing trend (from \(60\) to \(120\) with some fluctuations). Positive correlation means as one variable increases, the other variable also has a tendency to increase.

Step2: Consider the residuals

Residuals measure the deviation of the observed \(y\) - values from the predicted \(y\) - values. The presence of both positive and negative residuals indicates that the relationship is not perfectly linear. But looking at the general movement of \(x\) and \(y\) ( \(x\) up, \(y\) up), and the fact that the residuals are not extremely large in magnitude (ranging from \(- 15.678\) to \(9.956\) etc., not close to \(y\) - values like \(60,40,\cdots,120\)), we can say the correlation is not extremely strong.

Answer:

There is a weak positive correlation between the hours spent playing video games and the hours spent studying.