Stat 350 Spring 2017 Homework 11 (20 Points + 1 Bonus Point) ✓ Solved
Stat 350 Spring 2017 Homework 11 (20 points + 1 point BONUS) Practice Problems
Stat 350 Spring 2017 Homework 11 involves analyzing various datasets and interpreting statistical results, including regression analysis, ANOVA tables, and understanding relationships between variables in health-related studies. The tasks include identifying the form, direction, and strength of associations in graphical data, computing regression lines, interpreting regression coefficients, performing ANOVA analyses, calculating variances and proportions of explained variability, and evaluating the significance of associations based on statistical measures. Additionally, it involves understanding the use of transformations such as logarithms in modeling and making critical assessments about the nature of relationships between variables, especially in health sciences contexts.
Sample Paper For Above instruction
Introduction
This paper addresses the statistical analysis of relationships between variables as presented in a series of practice problems from the Stat 350 Spring 2017 coursework. The key focus areas include regression modeling, analysis of variance (ANOVA), interpretation of regression coefficients, and understanding the implications of transformations like logarithms in data modeling. The context spans environmental science, health sciences, and real-world application scenarios, highlighting how statistical tools elucidate relationships that inform scientific and health-related decisions.
Regression Analysis of Wind Speed and Wave Height
The second problem presents a dataset on wave forecasts, where wind speed (in knots) is hypothesized to linearly relate to wave height (in feet). The provided summary statistics and scatter plot enable estimation of the regression model. The least squares regression line is derived as \(\hat{Y} = \beta_0 + \beta_1 X\). Using the formulas for regression coefficients:
\[
\hat{\beta}_1 = \frac{S_{XY}}{S_{XX}} = \frac{36.4}{91.75} \approx 0.3968,
\]
and
\[
\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} = 1.68 - 0.3968 \times 9.25 \approx 1.68 - 3.669 \approx -1.989.
\]
Hence, the estimated regression line is \(\hat{Y} = -1.989 + 0.397 X\).
Interpreting the y-intercept’s physical meaning reveals it as the estimated wave height when wind speed is zero. Since negative wave heights are non-physical, the y-intercept lacks real-world applicability but is mathematically necessary for the regression equation.
The regression slope \(\hat{\beta}_1\) indicates that for each additional knot increase in wind speed, the wave height is expected to increase by approximately 0.397 feet. This linear estimate presumes other variables remain constant, highlighting the direct influence of wind speed on wave height.
Calculating the expected wave height at a wind speed of 8.6 knots:
\[
\hat{Y} = -1.989 + 0.397 \times 8.6 \approx -1.989 + 3.412 \approx 1.423\, \text{feet}.
\]
This prediction provides a practical estimate for maritime operations, illustrating how wind speed forecasts can inform safety protocols.
The ANOVA table is constructed based on the sums of squares and degrees of freedom:
- Total degrees of freedom: \(n - 1 = 19\).
- Regression degrees of freedom: 1.
- Error degrees of freedom: 18.
- Sum of squares for regression (\(\text{SSR}\)) and error (\(\text{SSE}\)) are derived from the sums of squares totals, with a provided regression sum of squares of 36.4 (which likely sums to the regression sum itself), but precise values depend on the complete data.
The mean square error (MSE) is calculated as:
\[
\text{MSE} = \frac{\text{SSE}}{\text{df}_\text{Error}}.
\]
Given the information, an estimated variance of residuals can be approximated, aiding in assessing model fit.
The proportion of variance explained by wind speed is given by:
\[
R^2 = \frac{\text{SSR}}{\text{SST}},
\]
where SST is the total sum of squares. Calculating \(R^2\) indicates how well wind speed predicts wave height, with higher values signifying stronger associations.
The statistical significance of the model can be judged by the F-test comparing regression mean square to error mean square. A significant F-value indicates a likely real association, consistent with the positive correlation indicated by the slope.
Cholesterol Ratio and Triglyceride Levels in Health Studies
The third problem examines the relationship between cholesterol ratio and triglyceride concentration. The regression analysis uses the natural logarithm of triglyceride level minus a constant, \(x_2 = \ln(\text{TG} - 129)\), against cholesterol ratio \(y\).
The coefficient of determination, \(R^2\), is derived from the regression sum of squares relative to total sum of squares:
\[
R^2 = \frac{\text{SSR}}{\text{SST}} = \frac{103.16}{106} \approx 0.973,
\]
indicating that approximately 97.3% of the variability in cholesterol ratio is explained by the model.
Interpreting causality from this model suggests that an increase in triglyceride levels is associated with an increase in cholesterol ratio, but causation cannot be conclusively established from correlation alone. Other confounding factors may influence both variables, and a longitudinal or experimental study would be necessary to infer causality.
The logarithmic transformation is employed to stabilize variance and normalize data, especially when triglyceride levels exhibit skewness. Log transformations are common practices in biological data to meet the assumptions of linear regression, such as constant variance and normality of residuals.
Conclusion
The statistical analyses presented underscore the power of regression and ANOVA tools in elucidating relationships in environmental and health sciences contexts. Estimating regression equations facilitates predictions and understanding of variable influence, while ANOVA tests validate the significance of these models. Recognizing when transformations are necessary enhances the robustness of conclusions, especially in biological data prone to skewness and heteroscedasticity. Overall, these methods provide vital insights for research and practical decision-making in fields like oceanography and medicine.
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