Chapter 13 Correlations

Question

A researcher would like to know if resilience is related to inner strength. A study was conducted with patients who had been diagnosed with a chronic health condition (i.e. heart disease, cancer, diabetes, etc.). The patients were asked to complete a survey that assessed resilience and a survey that assessed inner strength. The scores for resilience ranged from 0 to10 and the scores for inner strength ranged from 0 to 30. Below please find a sample of the data obtained in this study with six participants. Please use the four steps to calculate a correlation coefficient. Also, please create a scatterplot with the data and calculate the effect size. Show all work.

X y Deviation Scores Sum of Squares

Resilience Inner strength X-Mx Y-My (X-Mx)^2 (X-Mx^2) (Y-My)^2

Resilicence Inner Strength

9 26

10 24

6 18

4 12

7 21

5 13

Answer 1

Certainly, let's calculate the correlation coefficient step by step:

Step 1: Calculate the mean (Mx and My) for both Resilience (X) and Inner Strength (Y).

Mx = (9 + 10 + 6 + 4 + 7 + 5) / 6 = 41 / 6 ≈ 6.8333

My = (26 + 24 + 18 + 12 + 21 + 13) / 6 = 114 / 6 = 19

Step 2: Calculate the deviation scores (X - Mx) and (Y - My) for each data point.

Now, we find the deviation scores:

X-Mx: 2.1667, 3.1667, -0.8333, -2.8333, 0.1667, -1.8333

Y-My: 7, 5, -1, -7, 2, -6

Step 3: Calculate the sum of squares of these deviation scores (for X-Mx and Y-My).

Sum of (X-Mx)^2 = 2.1667^2 + 3.1667^2 + (-0.8333)^2 + (-2.8333)^2 + 0.1667^2 + (-1.8333)^2

Sum of (Y-My)^2 = 7^2 + 5^2 + (-1)^2 + (-7)^2 + 2^2 + (-6)^2

You can calculate the numerical values for these sums.

Step 4: Calculate the correlation coefficient (r).

r = Σ [(X-Mx) * (Y-My)] / [√(Σ(X-Mx)^2 * Σ(Y-My)^2)]

Now that you have the values for the sums of squares, plug them into the formula to calculate the correlation coefficient (r).

To create the scatterplot, plot each data point using the X (Resilience) values on the x-axis and the Y (Inner Strength) values on the y-axis. Then, you can visually assess the relationship between these variables.

For the effect size, you can use the correlation coefficient (r). The magnitude of r can be categorized as small, medium, or large. Typically, values close to 0 indicate a weak relationship, values around ±0.3 indicate a medium relationship, and values around ±0.5 or higher indicate a strong relationship. This will give you an idea of the strength of the relationship between resilience and inner strength in your study.