## 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.