Problem Description
This Statistical Analysis assignment focuses on investigating the association between a child's insurance status and their highest level of education. We aim to determine if the two variables are related. To do this, we formulate a null hypothesis (H₀) and an alternative hypothesis (HA) to guide our analysis:
H0: The insurance status of the child is not associated with the level of education.
HA: The insurance status of the child is associated with the level of education.
Statistical Analysis:
Hypothesis Testing: We employ the Chi-square test for contingency tables to evaluate the relationship between the child's insurance status and their educational level. Both variables are categorical, and our significance level (alpha) is set to 0.05. With degrees of freedom (υ) calculated as 6, the critical value of the Chi-square statistic at α = 0.05 is 12.60.
χ_(α,υ)^2= χ_(.05,6 )^²= 12.6
Expected Cell Counts: To proceed, we calculate the expected cell counts for each combination of insurance status and education level. The expected cell counts help us understand what we would anticipate if there were no association between the variables. Here are the results:
- No HS: Uninsured (1.90), Medicaid (6.70), Private Insurance (3.40)
- Some HS: Uninsured (15.03), Medicaid (53.07), Private Insurance (26.90)
- HS Grad: Uninsured (24.83), Medicaid (87.71), Private Insurance (44.46)
- Some College or Higher: Uninsured (20.24), Medicaid (71.51), Private Insurance (36.24)
| Highest Education Level | Child Insurance Status | |||||
|---|---|---|---|---|---|---|
| Uninsured | Medicaid | Private Insurance | Total | |||
| No HS | 392 ×62392 × 12392=1.90 | 392 ×219392 × 12392= 6.70 | 392 ×111392 × 12392= 3.40 | 12 | ||
| Some HS | 392 ×62392 × 95392=15.03 | 392 ×219392 × 95392=53.07 | 392 ×111392 × 95392=26.90 | 95 | ||
| HS Grad | 392 ×62392 × 157392=24.83 | 392 ×219392 × 157392=87.71 | 392 ×111392 × 157392=44.46 | 157 | ||
| Some College or Higher | 392 ×62392 × 128392=20.24 | 392 ×219392 × 128392=71.51 | 392 ×111392 × 128392=36.24 | 128 | ||
| Total | 62 | 219 | 111 | 392 | ||
Table 1: Expected Cell Counts
Variable Measurement: The child's insurance status is a nominal variable as it includes categories (Uninsured, Medicaid, Private Insurance) with no inherent order. In contrast, the educational level is an ordinal variable, as it presents four categories (No HS, Some HS, HS Grad, Some College or Higher) that can be ranked in a specific order.
Conclusion: Our calculated Chi-square test statistic (χ²) is 24.60, which exceeds the critical value of 12.60. We conclude that a child's insurance status is indeed associated with their educational level.
χ^2> χ_υ^2
24.60> 12.60
Percentage Calculation: To gain further insights, we compute the percentages for each education level and insurance status as a fraction of the total for each row. These percentages highlight the relationship between the two variables.
| Highest Education Level | Child Insurance Status | |||
|---|---|---|---|---|
| Uninsured | Medicaid | Private Insurance | Total | |
| No HS | 012×100=0% | 1112×100=91.67% | 112×100=8.33% | 100% |
| Some HS | 1495×100=14.74% | 6695×100=69.47% | 1595×100=15.79% | 100% |
| HS Grad | 30157×100=19.11% | 83157×100=52.87% | 44157×100=28.03% | 100% |
| Some College or Higher | 18128×100=14.06% | 59128×100=46.09% | 51128×100=39.84% | 100% |
Table 2: Computing Percentages as Fraction of ROW TOTAL on row-by-row basis
Expected Cell Counts Violating Assumption: It's essential to note that Chi-square tests assume expected cell counts should be higher than 5 for a large contingency table. In our analysis, two cell counts are less than 5 (No HS by Uninsured and No HS by Private Insurance). While this may be due to the majority of individuals attending high school, we should exercise caution when interpreting these results as they violate the expected cell count assumption.
Summary: This analysis demonstrates the association between a child's insurance status and their highest level of education. The Chi-square test reveals that these two variables are not independent, and there is a significant relationship between them. However, it's important to acknowledge that certain cell counts violate the Chi-square test's assumption, which can impact the validity of the results.
Analyzing the Relationship Between High School Graduation Rate and Disciplinary Issues
Problem Description: In this assignment, we investigate whether the rate of high school graduation differs based on the presence of disciplinary issues during the first grade. We formulate a null hypothesis (H₀) and an alternative hypothesis (HA) to guide our analysis:
- H0: The rate of high school graduation is not associated with the classification of disciplinary issues in the first grade.
- HA: The rate of high school graduation is associated with the classification of disciplinary issues in the first grade.
Statistical Analysis:
Hypothesis Testing: We employ the Chi-square test for contingency tables to examine the association between the rate of high school graduation and the classification of disciplinary issues. Both variables are categorical, and our significance level (alpha) is set to 0.05. With degrees of freedom (υ) calculated as 1, the critical value of the Chi-square statistic at α = 0.05 is 3.84.
χ_(α,υ)^2= χ_(.05,1 )^²= 3.84
Expected Cell Counts: To conduct the Chi-square test, we calculate the expected cell counts for each combination of graduation rate and disciplinary issue classification:
- Had First Grade Consistent Discipline Issues: Graduated HS (20.67), Dropped out of HS (6.33)
- Did NOT Have Consistent Discipline Issues in 1st Grade: Graduated HS (175.33), Dropped out of HS (53.67)
Conclusion: The calculated Chi-square test statistic (χ²) is 26.28, which surpasses the critical value of 3.84. Therefore, there is substantial evidence to reject the null hypothesis. We conclude that the rate of high school graduation is indeed associated with the classification of disciplinary issues during the first grade.
Summary: This analysis reveals a significant relationship between the rate of high school graduation and the presence or absence of disciplinary issues during the first grade. The Chi-square test supports our conclusion that these two variables are associated. However, it's essential to acknowledge that Chi-square test assumptions may be violated due to low expected cell counts, which should be considered when interpreting the results.