Four research participants take a test of manual dexterity (high scores mean better dexterity) and anxiety test (high scores mean more anxiety). The scores are as follows
Person Dexterity Anxiety
1 1 10
2 1 8
3 2 4
4 4 -2
What is your question?
To analyze the relationship between manual dexterity and anxiety for the four research participants, we can calculate the correlation coefficient (r) between the two variables. The correlation coefficient measures the strength and direction of the linear relationship between two variables, ranging from -1 to +1.
To calculate the correlation coefficient, we can use the formula:
r = Σ((X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)^2 * Σ(Y - Ȳ)^2)
Where:
- X represents the scores for manual dexterity
- X̄ represents the mean of the manual dexterity scores
- Y represents the scores for anxiety
- Ȳ represents the mean of the anxiety scores
- Σ represents the sum of the values
First, let's calculate the means (X̄ and Ȳ):
X̄ = (1 + 1 + 2 + 4) / 4 = 2
Ȳ = (10 + 8 + 4 - 2) / 4 = 5
Next, let's calculate the individual components needed for the correlation coefficient formula:
For Person 1:
(X - X̄) = 1 - 2 = -1
(Y - Ȳ) = 10 - 5 = 5
(-1 * 5) = -5
For Person 2:
(X - X̄) = 1 - 2 = -1
(Y - Ȳ) = 8 - 5 = 3
(-1 * 3) = -3
For Person 3:
(X - X̄) = 2 - 2 = 0
(Y - Ȳ) = 4 - 5 = -1
(0 * -1) = 0
For Person 4:
(X - X̄) = 4 - 2 = 2
(Y - Ȳ) = -2 - 5 = -7
(2 * -7) = -14
Now, let's sum these components:
Σ((X - X̄)(Y - Ȳ)) = -5 + (-3) + 0 + (-14) = -22
Next, let's calculate the sum of squares for X and Y:
Σ(X - X̄)^2 = (1 - 2)^2 + (1 - 2)^2 + (2 - 2)^2 + (4 - 2)^2 = 2
Σ(Y - Ȳ)^2 = (10 - 5)^2 + (8 - 5)^2 + (4 - 5)^2 + (-2 - 5)^2 = 114
Now, let's calculate the correlation coefficient (r):
r = Σ((X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)^2 * Σ(Y - Ȳ)^2) = -22 / √(2 * 114)
r ≈ -0.422
Therefore, the correlation coefficient (r) between manual dexterity and anxiety for these four research participants is approximately -0.422. This negative value suggests a weak negative correlation, indicating that higher manual dexterity scores tend to be associated with lower anxiety scores, and vice versa. However, the correlation is not very strong since the absolute value of r is less than 0.5.