Statistical Analysis and Interpretation of Various Research Questions

In the comprehensive analysis of various research questions, we explore statistical methods to uncover meaningful insights. These questions range from examining disparities in pretrial release days among different attorney types to investigating correlations between crime rates and the number of police agencies. Additionally, we delve into the intriguing relationship between prison spending and violent crime rates. Our exploration extends to the application of ANOVA and correlation tests using real-world data. This study offers a valuable demonstration of how statistical tools can aid in addressing diverse research inquiries.

Assignment 1: Evaluating Attorney Type in Relation to Pretrial Release Days

Problem Description: An ongoing issue in the criminal court system revolves around the potential advantages wealthier defendants may have over poorer ones, especially when it comes to hiring their own attorneys. It is commonly believed that privately retained attorneys are more skilled and dedicated compared to their publicly appointed counterparts. In this data analysis assignment, we will examine this issue using a sample of rape defendants from the 2018 State Court Processing Statistics data set. We will focus on the relationship between attorney type (independent variable) and the number of days to pretrial release (dependent variable) for rape defendants who were released pending trial.

Solution: Step 1: Definition of Hypotheses

• H0: The mean days to pretrial release for individuals with a public defender, assigned counsel, and private attorney are equal.
• H1: At least one group's mean is different.

Step 2: Degrees of Freedom for F-table

• Df numerator = K - 1 = 3 - 1 = 2
• Df denominator = N - K = 23 - 3 = 20

Step 3: Reading the F-value from the F-table

• F (denominator=20, numerator=2) at alpha 0.05 = 3.493

Step 4: Comparing Obtained Value and F-table Value

• Obtained value = 13.55
• F-table value = 3.493
• F-obtained > F-table value

Step 5: Conclusion

• Since the obtained value is greater than the critical value, we reject the null hypothesis and conclude that H1 is true. At least one group's mean is different.

Assignment 2: Examining Police Agencies and Crime Rates

Problem Description: We want to determine if there is a correlation between the number of police agencies in areas and their crime rates. The independent variable is "crimes per square mile," and the dependent variable is "law enforcement agencies per 1,000 square miles." We aim to test for a positive correlation between these variables.

Table 1: Correlation between state police agencies in areas and their crime rates

Solution: Step 1: Definition of Hypotheses

• H0:There is no correlation between crimes per square miles and law enforcement agencies per 1,000 square miles.
• H1:There is a positive correlation between crimes per square miles and law enforcement agencies per 1,000 square miles.

Step 2: Degrees of Freedom

Df = N-2

Df = 9-2

Df = 7

Step 3: Finding the Critical Value

• For a one-tailed test at 0.01 alpha level with Df=7, the critical value is 2.998.

Step 4: Comparing Critical Value and Obtained Value

• Obtained value = 0.93
• Critical value = 2.998
• Critical value > Obtained value

Step 5: Conclusion

• Since the obtained value 0.93 is less than the critical value 2.998, we fail to reject the null hypothesis, concluding that there is no correlation between crimes per square miles and law enforcement agencies per 1,000 square miles.

Assignment 3: Analyzing Prison Spending and Violent Crime Rates

Problem Description: The objective is to investigate the correlation between the amount of money states spend on prisons and their violent crime rates. The data includes information on "Prison Dollars per Capita" and "Violent Crime Rate" for a sample of 5 states.

Table 2: Correlation between the amount of money spent on state prisons and their violent crime rates

Solution: Step 1: Definition of Hypotheses

• H0:There is no correlation between the amount of money states spend on prisons and their violent crime rates.
• H1: There exists a correlation between prison spending and violent crime rates.

Step 2: Degrees of Freedom

• Df = N - 2 = 5 - 2 = 3

Step 3: Finding the Critical Value

• For a two-tailed test at 0.01 alpha level with Df=3, the critical values range from -5.841 to +5.841.

Step 4: Comparing Critical Value and Obtained Value

• Obtained value = 17.22
• Critical value range = -5.841 to +5.841

Step 5: Conclusion

• The obtained value falls outside the critical value range, so we reject the null hypothesis and conclude there is a correlation between prison spending and violent crime rates.
• Further analysis shows a weak positive correlation (r = 0.45) between these variables. Approximately 20% of the change in violent crime rates can be explained by prison spending.

Assignment 4: ANOVA and Correlation Tests

Problem Description: In this assignment, we will conduct ANOVA and correlation tests using the GSS18 dataset for specific sets of variables.

Solution:

A) ANOVA: #630 PADEG & #129 – CONINC

• From the computed F-statistic (4.799) with significance = 0.000, we reject the null hypothesis. We conclude that at least one group's mean is different.
• Multiple comparisons reveal significant differences between individuals with the highest family income of $158,202 and various family incomes.

B) ANOVA: #983 - WEEKSWRK & #31 – ANCESTRS

The F-statistic is 1.245 with a significance of 0.144. We fail to reject the null hypothesis, indicating that the means for respondents' weeks worked last year are equal for people who believe in the supernatural power of deceased ancestors and those who don't.

C) Correlation: #722 – REALRINC & #305 – HEIGHT

• A significant correlation is found (P-value = 0.000), leading to the rejection of the null hypothesis.
• A weak positive correlation (r = 0.192) suggests that an increase in height corresponds to an increase in income.
• R-squared indicates that 3.69% of the change in income can be explained by height.

D) Correlation: #108– COLSCINM & #199 – EDUC

• A significant correlation (P-value = 0.000) is observed, leading to the rejection of the null hypothesis.
• A weak positive correlation (r = 0.367) indicates that taking more college-level science courses is associated with a higher level of education.
• R-squared shows that 13.47% of the change in courses taken can be explained by the highest year of school completed.