Consider again the data in problem #6. Suppose that in the aerobic exercise group we also measured the number of hours of aerobic exercise per week and the mean is 5.2 hours with a standard deviation of 2.1 hours. The sample correlation is -0.42.
Exercise Group N Mean Std Dev
Weight Training 20 49.7 10.2
Aerobic Exercise 20 43.1 11.1
Stretching/Yoga 20 57.0 12.5
Correlation is between what and what?
What is your question?
To find the sample correlation coefficient between the number of hours of aerobic exercise per week and the weight of the participants in the aerobic exercise group, you can follow these steps:
Step 1: Write down the given data:
- Number of hours of aerobic exercise per week:
Mean = 5.2 hours
Standard deviation = 2.1 hours
- Weight of participants in the aerobic exercise group:
Mean = 43.1
Standard deviation = 11.1
Step 2: Establish the variables:
Let Xi represent the number of hours of aerobic exercise per week,
and let Yi represent the weight of the participants in the aerobic exercise group.
Step 3: Calculate the sum of Xi, Yi, Xi², Yi², and XiYi:
- Sum of Xi (ΣXi): (given) = 5.2
- Sum of Yi (ΣYi): (given) = 43.1
- Sum of Xi² (ΣXi²): (use the formula ΣXi² = n × (σ² + Mx²), where n is the number of observations and Mx is the mean of Xi)
= 20 × (2.1² + 5.2²)
= 20 × (4.41 + 27.04)
= 20 × 31.45
= 629
- Sum of Yi² (ΣYi²): (use the formula ΣYi² = n × (σ² + My²), where n is the number of observations and My is the mean of Yi)
= 20 × (11.1² + 43.1²)
= 20 × (123.21 + 1857.61)
= 20 × 1980.82
= 39616.4
- Sum of XiYi (ΣXiYi): (use the formula ΣXiYi = n × (σxy + MxMy), where n is the number of observations, σxy is the covariance, Mx is the mean of Xi, and My is the mean of Yi)
= 20 × (-0.42 × 2.1 × 11.1 + 5.2 × 43.1)
= 20 × (-9.2578 + 222.92)
= 20 × 213.6622
= 4273.244
Step 4: Calculate the sample correlation coefficient (r):
- Use the formula: r = (ΣXiYi - n × Mx × My) / √((n × ΣXi² - (ΣXi)²) × (n × ΣYi² - (ΣYi)²))
= (4273.244 - 20 × 5.2 × 43.1) / √((20 × 629 - (5.2)²) × (20 × 39616.4 - (43.1)²))
Step 5: Simplify and calculate the value of r.
I hope this helps you understand the step-by-step process of calculating the sample correlation coefficient between the number of hours of aerobic exercise per week and the weight of participants in the aerobic exercise group.