Week 4 Assignment
1. For each correlation coefficient below, calculate what proportion of variance is shared by the two correlated variables:
2. For each coefficient of determination below, calculate the value of the correlation coefficient:
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
Y = a + bX Y = mean length of stay X = mean score on scale b = slope
6.5 = a + (-0.10)(33)
a = 9.8
Y = 9.8 -0.1X
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:
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 |
Subject 1: Y′ = -1.636 + 0.793(2.9) + 0.004(560) – 0.0009(540) + 0.009(55) = 2.9127
Subject 2: Y′ = -1.636 + 0.793(3.2) + 0.004(550) – 0.0009(590) + 0.009(65) = 3.1556
Subject 3: Y′ = -1.636 + 0.793(3.4) + 0.004(600) – 0.0009(550) + 0.009(70) = 3.5952
6. Using the following information for R^{2}, 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:
F = (R^{2}/k)/[(1-R^{2})/(N-k-1)]
R^{2} = 0.13, k = 5, N = 120 F = (0.13/5) / [(1-0.13)/(120-5-1)] = 4
3.4 > tabled F = 2.45; significant – reject the null hypothesis.
R^{2} = 0.53, k = 5, N = 30 F = (0.53/5) / [(1- 0.53)/(30-5-1)] = 6
5.6 > tabled F = 2.84; significant- reject the null hypothesis.
R^{2} = 0.28, k = 4, N = 64 F = (0.53/5) / [(1- 0.53)/(30-5-1)] = 6
5.6 > tabled F = 2.76; significant- reject the null hypothesis.
R^{2} = 0.14, k = 4, N = 64 F = (0.14/ 4) / [(1-0.14)/(64-4-1)] = 3
2.3< tabled F = 2.76; not significant- retain the null hypothesis.
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
a. Plot the data and include a regression line in StatCrunch. Copy and paste your graph into your Word document for full credit.
Simple linear regression results:
Dependent Variable: Cholesterol level (mg/dL)
Independent Variable: Dose (mg)
Cholesterol level (mg/dL) = 305.75 - 29.926471 Dose (mg)
Sample size: 6
R (correlation coefficient) = -0.85274561
R-sq = 0.72717508
Estimate of error standard deviation: 61.710186
Parameter estimates:
Parameter |
Estimate |
Std. Err. |
Alternative |
DF |
T-Stat |
P-value |
Intercept |
305.75 |
57.724593 |
≠ 0 |
4 |
5.2967025 |
0.0061 |
Slope |
-29.926471 |
9.1653284 |
≠ 0 |
4 |
-3.2651826 |
0.0309 |
Analysis of variance table for regression model:
Source |
DF |
SS |
MS |
F-stat |
P-value |
Model |
1 |
40600.245 |
40600.245 |
10.661417 |
0.0309 |
Error |
4 |
15232.588 |
3808.1471 |
||
Total |
5 |
55832.833 |
b. What is the correlation coefficient r and what does it mean in this case? The correlation coefficient r is -0.8527 and this represents the magnitude and nature of the relationship between variables. Because this value is a negative, it indicates that high values of one variable is associated with low values on the other variable. The higher the value, the stronger the relationship. In this case, we have a moderately strong negative relationship.
c. What is the coefficient of determination and what does it mean in this case? The coefficient of determination is 0.7272 and it means that 72.72% of the variation is explained by the linear correlation.
d. Is there a statistically significant correlation between dose and cholesterol level in this case? Because the F statistic is 10.66 is greater than its critical value of 6.61 and the p value is less than 0.05, there is a statistically significant correlation between dose and cholesterol level in this case.
e. 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)?
Cholesterol level (mg/dL) = 305.75 - 29.926471 Dose (mg)
Y = 305.75 – 29.93(4) = 186.03
The predicted cholesterol level for a person taking a dose of 4 mg is 186.04 mg/dL. For a person not taking the drug, the predicted cholesterol level would be 305.75 mg/dL.
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