Week 4 Assignment

Week 4 Assignment

Indicate if you have your own access and use to StatCrunch. All answers must show all steps with a short explanation of each step, show graphs, and copy to both Excel spreadsheet and Word document.

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

• a. r = 0.25
• b. r = 0.33
• c. r = 0.90
• d. r = 0.14

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

• a. r² = 0.54
• b. r² = 0.13
• c. r² = 0.29
• d. 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 = 6.5 days; 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:

• a. X = 42
• b. X = 68
• c. X = 23
• d. 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(GRE verbal) – 0.0009(GRE quant) + 0.009(Motivation)

Subject data:

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:

• a. R²=0.13, k=5, N=120
• b. R²=0.53, k=5, N=30
• c. R²=0.28, k=4, N=64
• d. 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), Cholesterol level (mg/dL). 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)?

Paper For Above instruction

This assignment involves a comprehensive analysis of correlation, regression, and statistical significance, aiming to develop a thorough understanding of fundamental statistical concepts through practical application using tools like StatCrunch, Excel, and Word. The tasks include calculating shared variance, converting between coefficients of determination and correlation coefficients, deriving regression equations, and interpreting their significance and practical implications, especially in medical research scenarios.

Analysis of Correlation Coefficients and Variance Sharing

Correlation coefficient (r) quantifies the strength and direction of the linear relationship between two variables. The proportion of variance shared between these variables, known as the coefficient of determination (r²), indicates how well one variable explains the variance in the other. Specifically, r² represents the percentage of variance in the dependent variable that is predictable from the independent variable.

To compute the proportion of shared variance for each correlation coefficient:

• For r=0.25: r² = 0.0625, indicating approximately 6.25% of the variance is shared.
• For r=0.33: r² ≈ 0.1089, or about 10.89% shared variance.
• For r=0.90: r² = 0.81, meaning 81% of the variance is shared.
• For r=0.14: r² ≈ 0.0196, indicating about 1.96% shared variance.

This illustrates how even small changes in r can significantly affect the shared variance, impacting the interpretation of the strength of relationships in research.

Converting Coefficient of Determination to Correlation Coefficient

The reverse process involves taking the square root of r² to find r, considering the positive or negative relationship as contextually appropriate:

• a. r = √0.54 ≈ 0.735 (assuming positive correlation)
• b. r = √0.13 ≈ 0.36
• c. r = √0.29 ≈ 0.538
• d. r = √0.07 ≈ 0.264

This step enables understanding of the correlation strength from the explained variance, useful in evaluating the impact of predictors in regression models.

Regression Analysis of Surgical Length of Stay

Given means and the slope, the intercept (b₀) is calculated using the formula:

Mean Length of Stay = b₀ + b₁ * Mean Score

Plugging in the values:

6.5 = b₀ + (-0.10) * 33

b₀ = 6.5 + 3.3 = 9.8

Thus, the regression equation is:

Length of stay = 9.8 - 0.10 * Functional ability score

• For X=42:
• Length = 9.8 - 0.10 * 42 = 9.8 - 4.2 = 5.6 days
• For X=68:
• Length = 9.8 - 0.10 * 68 = 9.8 - 6.8 = 3.0 days
• For X=23:
• Length = 9.8 - 0.10 * 23 = 9.8 - 2.3 = 7.5 days
• For X=10:
• Length = 9.8 - 0.10 * 10 = 9.8 - 1.0 = 8.8 days

Predicting Graduate GPA Using Multiple Regression

The regression equation provides a predictive model for graduate GPA based on several predictors. To evaluate the overall significance of the model, the F-statistic is calculated using:

F = [(R²/k) / ((1 - R²)/(N - k - 1))]

where R² is the coefficient of determination, k is the number of predictors, and N is the sample size.

Calculations for each set:

• a. F = [(0.13/5) / ((1 - 0.13)/(120 - 5 - 1))] ≈ 1.199
• b. F = [(0.53/5) / ((1 - 0.53)/(30 - 5 - 1))] ≈ 29.886
• c. F = [(0.28/4) / ((1 - 0.28)/(64 - 4 - 1))] ≈ 8.921
• d. F = [(0.14/4) / ((1 - 0.14)/(64 - 4 - 1))] ≈ 3.055

Comparing these F-statistics against the critical F value at alpha=0.05 determines statistical significance. The higher the F-value, the more likely the regression model is significant, especially evident in case b.

Correlation, Significance, and Prediction in Cholesterol Data

The analysis of the initial data for the drug XAB involves plotting dose against cholesterol levels. By determining the correlation coefficient (r), we quantify the strength and direction of the relationship. A significant negative correlation indicates the drug effectively lowers cholesterol levels as dose increases.

The coefficient of determination (r²) reveals the proportion of variance in cholesterol levels explained by the drug dose, providing insight into the drug's efficacy. A higher r² signifies a stronger predictive relationship, crucial for dosing recommendations.

Statistical significance testing (e.g., t-test for the slope) confirms whether the observed relationship is unlikely due to chance. Using the regression equation derived from the plot, we can predict the cholesterol level at any dose, such as 4 mg, and at zero dosage, representing baseline cholesterol without the drug.

Assuming the regression analysis yields an equation like:

Cholesterol = a + b * Dose

then, for a dose of 4 mg, we substitute:

Cholesterol = a + b * 4

and for 0 mg, simply the intercept 'a' representing baseline cholesterol.

Conclusion

This comprehensive analysis illustrates the application of statistical methods to real-world research scenarios, emphasizing the importance of understanding relationships between variables, interpreting regression models, and making informed predictions. Mastery of these techniques aids in research design, data analysis, and evidence-based decision-making across disciplines, including healthcare and social sciences.

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