Week 4 Assignment For Each Correlation Coefficient Below Cal

Week 4 Assignment1for Each Correlation Coefficient Below Calculate

Analyze multiple statistical problems involving correlation coefficients, coefficients of determination, regression equations, prediction calculations, regression analysis, ANOVA testing, and interpretation of statistical output related to a research study on job stress and other variables.

Paper For Above instruction

Correlation coefficients are fundamental in understanding the degree and direction of the relationship between two variables. The coefficient of determination (r²) indicates the proportion of variance shared between the variables. In this assessment, we will explore the relationship between these statistical measures, regression analysis, and their application in real-world research, particularly focusing on a study examining job stress factors among employees.

Part 1: Computing the Proportion of Shared Variance from Correlation Coefficients

Given correlation coefficients (r), the proportion of shared variance between two variables is calculated by squaring the correlation coefficient, i.e., r². This value represents the percentage of variance in one variable that can be explained by the other.

For r = 0.25, r² = 0.0625, meaning approximately 6.25% of the variance is shared. When r = 0.33, r² ≈ 0.1089, or about 10.89%. For a high correlation, r = 0.90, r² = 0.81, indicating 81% of the variance is shared. If an r were 0, the shared variance would be zero, signifying no relationship.

Part 2: Computing the Correlation Coefficient from the Coefficient of Determination

Using r² values, we take the square root to find r. For r² = 0.54, r ≈ √0.54 ≈ 0.735. For r² = 0.13, r ≈ √0.13 ≈ 0.36. When r² = 0.29, r ≈ √0.29 ≈ 0.538. When r² = 0, r = 0, indicating no linear relationship.

Part 3: Regression Equation Derivation

Given the mean length of stay (Ȳ) = 6.5 days, mean functional ability score (X̄) = 33, and slope (b) = -0.1, the intercept (a) is calculated using the formula: a = Ȳ - b X̄. Substituting the values: a = 6.5 - (-0.1 33) = 6.5 + 3.3 = 9.8. Therefore, the full regression equation is:

Y = 9.8 - 0.1X

Part 4: Predictions Using the Regression Equation

Using the regression equation Y = 9.8 - 0.1X, predictions for specified functional ability scores are:

• X = 42: Y = 9.8 - 0.1 * 42 = 9.8 - 4.2 = 5.6 days
• X = 68: Y = 9.8 - 0.1 * 68 = 9.8 - 6.8 = 3.0 days
• X = 23: Y = 9.8 - 0.1 * 23 = 9.8 - 2.3 = 7.5 days

Part 5: Multiple Regression Model and Interpretation

The regression model predicting graduate GPA includes predictors: undergraduate GPA, GRE verbal, GRE quant, and motivation. The regression equation is:

Y' = -1.636 + 0.793(UGPA) + 0.004(GRE Verbal) - 0.0009(GRE Quant) + 0.009(Motivation)

The coefficient of determination (R²) indicates the proportion of variance in GPA explained by all predictors combined. For this model, R² can be calculated from the sum of squares: R² = Regression Sum of Squares / Total Sum of Squares. Based on the ANOVA output, R² is approximately 0.13 or 13%, meaning the predictors collectively explain about 13% of the variance in GPA.

The F-statistic assesses the overall significance of the regression model. Using the formula:

F = (R² / k) / ((1 - R²) / (N - k - 1))

and plugging in R² ≈ 0.13, k = 4, and N = 30, we obtain F ≈ 2.6, which can be compared against the critical F-value at alpha = 0.05 to determine significance. In this case, since the p-value associated with the F-test exceeds 0.05, the model is not statistically significant, indicating predictors do not reliably explain GPA variance.

Part 6: Testing Regression Significance via F-Statistic

Given R², sample size (N), and number of predictors (k), the F-statistic is calculated to test the model's overall significance. The null hypothesis posits that all regression coefficients are zero, implying no relationship between predictors and the dependent variable. The alternative hypothesis suggests at least one predictor significantly predicts the outcome.

Part 7: Regression Analysis on Cholesterol and Drug Dose Data

The data on dose and cholesterol levels were analyzed to understand the drug's effect. Plotting the data with a regression line helps visualize the relationship. The correlation coefficient (r) quantifies the strength and direction; in this context, a negative r indicates the drug reduces cholesterol levels as dose increases. The coefficient of determination (r²) reveals the proportion of variance in cholesterol explained by dose, illustrating the drug's effectiveness.

Statistical testing shows whether the correlation is significant; a p-value less than 0.05 indicates significance. Predictions for cholesterol levels at specific doses (e.g., 4 mg and 0 mg) are obtained by plugging these values into the regression equation. The predicted cholesterol level at 4 mg reflects the dose's impact, while at 0 mg, it indicates the baseline cholesterol without the drug.

Part 8: Job Stress Survey Analysis

The survey on employees' job stress involved descriptive statistics, frequency distributions, and ANOVA analysis to compare stress across age groups. Variables with the highest mean scores are most favorable, and the variable with the most variation has the highest standard deviation. The percentage of employees perceiving stress as reasonable and not is calculated from frequency data.

The null hypothesis for the ANOVA tests whether stress levels are equal across age groups, while the alternative suggests differences exist. The ANOVA results with significance p-values below 0.05 indicate significant differences in stress levels among age groups. Group means identify which age cohort experiences the highest and lowest stress, and the group with the greatest variance demonstrates more inconsistent perceptions of stress.

The multiple regression analysis further explores how factors like resources, work-life balance, involvement, learning opportunities, and rewards influence stress. R-squared indicates the proportion of variance in stress explained by these predictors. Significance testing of regression coefficients pinpoints which factors significantly contribute to stress, guiding targeted organizational interventions.

Summary

This comprehensive analysis demonstrates how correlation, regression, and ANOVA are applied to understand the relationships among variables within organizational and health research contexts. Stronger correlations and higher explained variance suggest more substantial relationships, while significance tests inform the reliability of these associations. Effective interpretation of statistical outputs enables informed decision-making to improve employee well-being and health outcomes.

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