What is the correlation coefficient r and what does it mean

Question

1. For each correlation coefficient below, calculate what proportion of variance is shared by the two correlated variables:

r = 0.25
r = 0.33
r = 0.90
r = 0.14

2. For each coefficient of determination below, calculate the value of the correlation coefficient:

r₂ = 0.54
r₂ = 0.13
r₂ = 0.29
r₂ = 0.07

3. Suppose a researcher regressed surgical patients' length of stay (dependent variable) in the hospital on a scale of functional ability measured 24 hours after surgery. Given the following, solve for the value of the intercept constant and write out the full regression equation:

Mean length of stay = 6.5days; mean score on scale = 33; slope = -0.10

4. Using the regression equation calculated in Exercise 3, compute the predicted value of Y (length of hospital stay) for patients with the following functional ability scores:

X = 42
X = 68
X = 23
X = 10

5. Use the regression equation below for predicting graduate GPA for the three presented cases.

Y′ = -1.636 + 0.793(undergrad GPA) + 0.004(GREverbal) - 0.0009(GREquant) +0.009(Motivation)

Subject undergrad GPA GREverbal GREquant Motivation

1 2.9 560 540 55

2 3.2 550 590 65

3 3.4 600 550 70

6. Using the following information for R2, k, and N, calculate the value of the F statistic for testing the overall regression equation and determine whether F is statistically significant at the 0.05 level:

R² = 0.13, k = 5, N = 120
R² = 0.53, k = 5, N = 30
R² = 0.28, k = 4, N = 64
R² = 0.14, k = 4, N = 64

7. According to the University of Chicago, as men age, their cholesterol level goes up. A new drug (XAB) is being tested to determine if it can lower cholesterol in aging males and at what dose. The data for the first test subject is below:

Dose (mg) 2 3 5 6 8 10

Cholesterol level (mg/dL) 310 124 201 110 52 20

Plot the data and include a regression line in StatCrunch. Copy and paste your graph into your Word document for full credit.

What is the correlation coefficient r and what does it mean in this case?

What is the coefficient of determination and what does it mean in this case?

Is there a statistically significant correlation between dose and cholesterol level in this case?

What is the predicted cholesterol level for a person taking a dose of 4 mg? What about if they are not taking the drug at all (0 mg)?