Assignment 1: Understanding Correlation Coefficients
Problem Description: Evaluate and explain the nature of correlation coefficients and their implications for causality.
Answer: Correlation coefficients measure the strength and direction of a relationship between two variables but do not imply causation. The Spearman correlation specifically assesses non-linear relationships.
Assignment 2: Correlation Analysis in JMP
Problem Description: Examine the correlations among various variables, highlighting significant relationships based on p-values.
Answer: The correlation matrix indicates relationships, and p-values identify significant correlations. In this case, Waist & Weight and Waist & Situps are significant.
Assignment 3: Parameter Estimates Interpretation
Problem Description: Explore the significance of parameters, focusing on the estimate, standard error, t-ratio, and p-value for each variable.
Answer: Only the variable "waist" is statistically significant based on its p-value (< 0.05).
Assignment 4: Model Comparison and Selection
Problem Description: Compare multiple models, considering adjusted R-squares and root mean square errors to identify the most suitable one.
Answer: Model M1, including Waist, Weight, and Pulse, appears to be the most appropriate based on the adjusted R-square and RMSE.
Assignment 5: Adjusted R-squares and Model Selection
Problem Description: Assess adjusted R-squares to determine the model with the best fit.
Answer: Model M3 has the largest adjusted R-square (35.46%), indicating its superior fit.
Assignment 6: Model Evaluation and Selection
Problem Description: Evaluate and select a model based on adjusted R-squares, R-squares consistency, and root mean square error.
Answer: Model M3 is preferred due to the largest adjusted R-square, consistent R-squares, and lower RMSE.
Assignment 7: Normality of Residuals
Problem Description: Validate the normality assumption for residuals in the preferred model.
Answer: The null hypothesis, assuming normality, is accepted for the residuals of the preferred model.
Assignment 8: Parameter Estimation and Prediction
Problem Description: Estimate parameters and make predictions using the preferred model, interpreting the coefficients.
Answer: Hence estimated model would be
Situps = 843.83477 + 0.823045*Weight -24.03094*Waist
If we put the value of weight and waist in the above expression, we can get the expected number of situps
Hence Situps = 843.83477 + 0.823045*191 -24.03094*36 = 135.92
Hence the expected number of stiups would be 136 approximately
Assignment 9: Confidence Intervals for Predicted Values
Problem Description: Establish 95% confidence intervals for predicted values based on the selected model.
| weight ( lbs ) |
waist ( in ) |
pulse (BPM) |
chins | situps | jumps | Predicted situps | Lower 95 % Indiv situps | Upper 95 % Indiv situps |
|---|---|---|---|---|---|---|---|---|
| 191 | 679 36 | 50 | 5 | 162 | 60 | 135.9224 | 24.35773 | 234.6987 |
| 189 | 37 | 52 | 2 | 110 | 60 | 110.2453 | 4.069573 | 218.6264 |
| 193 | 38 20 | 58 | 12 | 101 | 101 | 89.50656 | -17.4062 | 203.7419 |
| 162 | in 35 | 62 | 12 | 105 | 37 | 136.085 | 43.38623 | 252.0306 |
| 189 | 35 | 46 | 13 | 155 | 58 | 158.3072 | 43.38623 | 252.0306 |
| 182 | 36 | 56 | 4 | 101 | 42 | 128.515 | 24.35773 | 234.6987 |
| 211 | 38 | 56 | 00 8 | 101 | 38 | 104.3214 | -17.4062 | 203.7419 |
| 167 | 34 | 60 | 05 6 | 125 | 40 | 164.2312 | 61.12445 | 270.6527 |
| 176 | 31 | 74 | 15 | 200 | 40 | 243.7314 | 106.9677 | 333.8906 |
| 154 | 33 | 56 | 17 | 251 | 250 | 177.5625 | 77.58846 | 290.5491 |
Table 1: Intervals for predicted values
Answer: The required 95% CI for predicted values is [24.35, 234.69].
Assignment 10: Interpretation of Coefficients
Problem Description: Interpret negative coefficients and their impact on the dependent variable.
Answer: The negative coefficient for the Waist implies that a one-unit increase in waist size is associated with a decrease of approximately 18 situps.
Assignment 11: Factor Interaction Analysis
Problem Description: Investigate the interaction effect between two factors, focusing on their significance.
Answer: The correct option is Sepal Width * Species.
Assignment 12: Interaction Effect Significance
Problem Description: Assess the significance of the interaction term through effect tests.
Answer: The p-value for the interaction term Sepal Width * Species is 0.001, indicating significance.
Assignment 13: Identifying Interaction Terms
Problem Description: Identify and explain the relevant interaction term in a model.
Answer: Correct option - Interaction term of Sepal width and species (Sepal width * species).
Assignment 14: Group Comparison Significance
Problem Description: Determine significant differences between specific groups within a dataset.
Answer: Significant differences exist between Versicolor, Virginica, and Setosa, Virginica.
Assignment 15: Confidence Interval for Mean Difference
Problem Description: Compute a 95% confidence interval for the difference in means.
Answer: The difference in sepal length at a 95% CI is 0.0897.
Assignment 16: Percentage Difference Calculation
Problem Description: Calculate the percentage difference between two values.
Answer: The correct option is 9.63%.
Assignment 17: Logistic Regression Analysis - Job Satisfaction
Problem Description: Assess the logistic regression model for job satisfaction and interpret the odds ratio and relative risk.
Answer: The odds ratio is 1.278, indicating a concerning increase in the odds of being unsatisfied. The relative risk for individuals aged over 40 is 3.743, emphasizing a significantly higher risk of dissatisfaction.
Assignment 18: Logistic Regression - Odds Ratios
Problem Description: Explore the alarming nature of the odds ratio in the logistic regression model.
Answer: The odds ratio of 1.278 is alarming, signifying an increased likelihood of dissatisfaction.
Assignment 19: Logistic Regression - Relative Risk
Problem Description: Investigate the relative risk in the logistic regression model.
Answer: The relative risk (RR) for individuals aged over 40 is 3.743, indicating a substantially higher risk of job dissatisfaction.
Assignment 20: Logistic Regression - Odds Ratios and Relative Risks
Problem Description: Explore the relationship between odds ratios and relative risks in logistic regression.
Answer: Odds ratios and relative risks exhibit similarities when the probability of job dissatisfaction is high (> 90%) in each age group.
Assignment 21: Hypothesis Testing Conclusion
Problem Description: Conclude the results of hypothesis testing based on p-values.
Answer: The relationship is statistically significant at alpha = 0.05.
Assignment 22: Significance Assessment
Problem Description: Determine the statistical significance of a relationship at a given significance level.
Answer: The correct option is a p-value < 0.05.
Assignment 23: Logistic Regression - Odds Ratios Interpretation
Problem Description: Interpret the odds ratios in a logistic regression model.
Answer: The odds ratio for the waist variable is interpreted as a unit increase in waist size being associated with a decrease of approximately 18 situps.
Assignment 24: Logistic Regression - Probability Calculation
Problem Description: Calculate the probability for a specific condition in a logistic regression model.
Answer: The probability of someone in the family surviving for passenger class = 1 is 0.4578.
Assignment 25: Logistic Regression - Probability Calculation (Another Scenario)
Problem Description: Calculate the probability for a different condition in a logistic regression model.
Answer: The probability of someone in the family surviving for passenger class = 2 is -0.6785.
Assignment 26: Logistic Regression - Odds Ratios for Different Scenarios
Problem Description: Examine odds ratios for various conditions in a logistic regression model.
Answer: The odds ratio for the event "Anybody in family survived = 1" is 2.3754, while for "Anybody in family survived = 0" is 1.8674.
Assignment 27: Hypothesis Testing Criteria
Problem Description: Define the criteria for accepting or rejecting the null hypothesis.
Answer: The null hypothesis is accepted if p-value < 0.001 and p-value < 0.05.
Assignment 28: Survival Analysis - Median Time Calculation
Problem Description: Calculate the median time of survival in a survival analysis.
Answer: The median time of survival is determined to be 10 units of time.
Assignment 29: Survival Analysis - Probability Estimation (New Treatment Group)
Problem Description: Estimate the probability of survival for a specific group in a survival analysis.
Answer: The probability that an animal assigned to the New Treatment group will survive at least 10 units of time is 0.7983.
Assignment 30: Survival Analysis - Probability Estimation (Placebo Group)
Problem Description: Estimate the probability of survival for another group in a survival analysis.
Answer: The probability that an animal assigned to the Placebo group will survive at least 10 units of time is 0.3567.
Assignment 31: Nonparametric Test - Categorical Data
Problem Description: Perform a nonparametric test for categorical data.
Answer: The correct option for the nonparametric test is Option 7.
Assignment 32: Survival Analysis - Time Calculation
Problem Description: Calculate the time of survival in a survival analysis.
Answer: The calculated time of survival is 9.76 hours.
Assignment 33: Survival Analysis - U Statistic Calculation
Problem Description: Calculate the U statistic in a survival analysis.
Answer: The U statistic for the survival analysis is calculated to be 7.896.
Assignment 34: Meta-analysis Results
Problem Description: Conclude the meta-analysis results based on the Q test statistic.
Answer: The null hypothesis is rejected, indicating a difference between treatments.
Assignment 35: Q Test Statistic Calculation
Problem Description: Calculate the Q test statistic for meta-analysis.
Answer: The calculated Q test statistic is 8.5674.