Problem Description:
In this statistical assignment, the focus is on understanding the distribution of SchoolSES categories (High, Low, Medium) and investigating the potential relationship between Gender and SchoolSES. The objective is to employ rigorous statistical analyses, including chi-squared tests, to assess whether the proportions of SchoolSES categories are equal and if there exists an association between Gender and SchoolSES. By delving into these aspects, the assignment aims to provide valuable insights into the distribution of socioeconomic status among students and whether this distribution is influenced by gender.
Testing Proportions of SchoolSES Categories
State the Hypotheses
- Null Hypothesis (H₀): The proportions of each of the three categories of SchoolSES (High, Low, Medium) are equal.
- Alternative Hypothesis (Hₐ): The proportions of at least one of the SchoolSES categories are different from the others.
Set the Significance Level
- α = 0.05
Perform the Test and Analyze Results
Chi-squared test for given probabilities
data: SchoolSES
X-squared = 19, df = 2, p-value = 7.485e-05
Expected frequencies
High Low Medium
66.66667 66.66667 66.66667
The chi-squared test reveals a significant relationship (χ²(2) = 19, p < 0.001). As the p-value is less than the significance level, we reject the null hypothesis, suggesting that the proportions of SchoolSES categories are not equal.
Testing Independence between Gender and SchoolSES
State the Hypotheses
- Null Hypothesis (H₀): There is no association between Gender and SchoolSES.
- Alternative Hypothesis (Hₐ): There is an association between Gender and SchoolSES.
Set the Significance Level
- α = 0.05
Perform the Test and Analyze Results
Pearson's Chi-squared test
data: contingency_table
X-squared = 0.25819, df = 2, p-value = 0.8789
Expected frequencies
High Low Medium
Female 20.8 46.8 36.4
Male 19.2 43.2 33.6
The test does not find a significant association (χ²(2) = 0.25819, p = 0.8789). As the p-value is greater than the significance level, we fail to reject the null hypothesis, indicating no significant relationship between Gender and SchoolSES.
APA Write-up
A chi-squared test was conducted to examine whether the proportions of the three categories of SchoolSES (High, Low, Medium) are equal. The test revealed a significant relationship between SchoolSES categories (χ²(2) = 19, p < 0.001). Thus, we reject the null hypothesis, indicating that the proportions of SchoolSES categories are not equal.
A Pearson's chi-squared test was performed to assess the relationship between Gender and SchoolSES. The test did not find a significant association between Gender and SchoolSES (χ²(2) = 0.25819, p = 0.8789). Therefore, we fail to reject the null hypothesis, suggesting that there is no significant relationship between Gender and SchoolSES.
In both analyses, the assumptions related to expected cell frequencies were met, indicating that the sample sizes were adequate for conducting the chi-squared tests. As a result, we can have confidence in the validity of the chi-squared test results.
Conclusion
In conclusion, the analyses revealed that the proportions of SchoolSES categories (High, Low, Medium) are not equal. Additionally, there is no significant association between Gender and SchoolSES. These findings suggest that the SchoolSES categories are unevenly distributed, and Gender does not appear to be associated with different SchoolSES categories.