The table shows the hours spent exercising and the hours she spent playing video games. What is the strength of the correlation between the hours spent exercising and the hours spent playing video games?
A. weak negative correlation
B. moderate negative correlation
C. weak positive correlation
D. moderate positive correlation
Answer (1)
Calculate the means of hours spent exercising and playing video games.
Calculate the standard deviations for both variables.
Apply the correlation coefficient formula.
Determine the correlation strength based on the calculated coefficient: m o d er a t e n e g a t i v e corre l a t i o n .
Explanation
Understand the problem and provided data We are given a table of data showing the hours spent exercising and the hours spent playing video games for 5 individuals. The data points are (2, 16), (4, 7), (3, 5), (4, 15), (5, 8). Our goal is to determine the strength of the correlation between the hours spent exercising and the hours spent playing video games.
Plan the solution To determine the strength of the correlation, we need to calculate the correlation coefficient (r) between the two variables. The correlation coefficient will tell us both the strength and direction of the linear relationship between the two variables.
Introduce the formula We will calculate the correlation coefficient using the formula: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where:
x i are the hours spent exercising
y i are the hours spent playing video games
x ˉ is the mean of the hours spent exercising
y ˉ is the mean of the hours spent playing video games
s x is the standard deviation of the hours spent exercising
s y is the standard deviation of the hours spent playing video games
n is the number of data points.
Calculate means and standard deviations First, we calculate the means of the two variables: x ˉ = 5 2 + 4 + 3 + 4 + 5 = 5 18 = 3.6 y ˉ = 5 16 + 7 + 5 + 15 + 8 = 5 51 = 10.2 Next, we calculate the standard deviations of the two variables: s x = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 = 5 − 1 ( 2 − 3.6 ) 2 + ( 4 − 3.6 ) 2 + ( 3 − 3.6 ) 2 + ( 4 − 3.6 ) 2 + ( 5 − 3.6 ) 2 = 4 2.56 + 0.16 + 0.36 + 0.16 + 1.96 = 4 5.2 = 1.3 ≈ 1.14 s y = n − 1 ∑ i = 1 n ( y i − y ˉ ) 2 = 5 − 1 ( 16 − 10.2 ) 2 + ( 7 − 10.2 ) 2 + ( 5 − 10.2 ) 2 + ( 15 − 10.2 ) 2 + ( 8 − 10.2 ) 2 = 4 33.64 + 10.24 + 27.04 + 23.04 + 4.84 = 4 98.8 = 24.7 ≈ 4.97
Calculate the correlation coefficient Now, we calculate the correlation coefficient: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) = ( 5 − 1 ) ( 1.14 ) ( 4.97 ) ( 2 − 3.6 ) ( 16 − 10.2 ) + ( 4 − 3.6 ) ( 7 − 10.2 ) + ( 3 − 3.6 ) ( 5 − 10.2 ) + ( 4 − 3.6 ) ( 15 − 10.2 ) + ( 5 − 3.6 ) ( 8 − 10.2 ) = 4 ( 1.14 ) ( 4.97 ) ( − 1.6 ) ( 5.8 ) + ( 0.4 ) ( − 3.2 ) + ( − 0.6 ) ( − 5.2 ) + ( 0.4 ) ( 4.8 ) + ( 1.4 ) ( − 2.2 ) = 22.66 − 9.28 − 1.28 + 3.12 + 1.92 − 3.08 = 22.66 − 8.6 ≈ − 0.38
Determine the strength of the correlation The correlation coefficient is approximately -0.38. Since − 0.3 < r < 0 , the correlation is moderate and negative.
Final Answer The strength of the correlation between the hours spent exercising and the hours spent playing video games is moderate negative correlation.
Examples
Understanding the correlation between exercise and video game habits can help health professionals design effective interventions. For instance, if a moderate negative correlation exists, encouraging more exercise might lead to a decrease in video game time, promoting a healthier lifestyle. This knowledge can also inform personalized health recommendations, balancing physical activity and digital entertainment for optimal well-being.
