Case Study: Latinx Household Income Analysis

Introduction:

This case study revisits the concerns regarding income levels of Latinx households within a specific geographic area, with an updated statistical analysis to assess any significant changes in the mean income and the proportion of households earning below a defined income threshold.

Hypotheses:

For the population mean ():

• Null Hypothesis (H0): The population mean income of Latinx households is greater than or equal to $50,000. ( $50,000)
• Alternative Hypothesis (HA): The population mean income of Latinx households is less than $50,000. ( < $50,000)

For the population proportion (p):

• Null Hypothesis (H0): The proportion of Latinx households making less than $30,000 is 20% or less. (p 0.20)
• Alternative Hypothesis (HA): The proportion of Latinx households making less than $30,000 is greater than 20%. (p > 0.20)

Data Collection and Calculation:

Data was collected from a sample of 36 Latinx households. The recalculated statistics are:

• Sample mean household income: $47,278
• Sample standard deviation: $19,524
• Sample proportion making less than $30,000: 22.22%

Hypothesis Testing:

For the population mean ():

• Calculated t-statistic: approximately -0.837
• Corresponding one-tailed p-value for the t-test: approximately 0.204

The t-test results suggest that there is insufficient evidence to conclude that the sample mean income is significantly less than the hypothesized population mean of $50,000 since the one-tailed p-value is greater than the common alpha level of 0.05. Therefore, we do not reject the null hypothesis.

For the population proportion (p):

• Calculated z-statistic: approximately 0.333
• Corresponding one-tailed p-value for the z-test: approximately 0.369

The z-test results also indicate that the sample proportion does not significantly differ from the hypothesized population proportion of 20%. The p-value is greater than 0.05, leading us to fail to reject the null hypothesis.

Interpretation of Results:

• For population mean (): The one-tailed p-value for the t-test (0.204) is above the alpha level of 0.05, indicating that there is not enough evidence to support the claim that the mean income of Latinx households has fallen below $50,000.
• For population proportion (p): The one-tailed p-value for the z-test (0.369) also exceeds 0.05, suggesting that the proportion of Latinx households making less than $30,000 does not significantly exceed 20%.

Summary Report:

The re-evaluated data from the sample of 36 Latinx households shows a mean income of approximately $47,278 with a standard deviation of $19,524. The proportion of households earning less than $30,000 was about 22.22%. The updated findings reaffirm the original conclusion, indicating no significant decrease in the mean income or increase in the proportion of households earning below $30,000.

Conclusion:

The refined statistical analysis upholds the previous findings, providing no substantial evidence to the claims of a decrease in mean income level below $50,000 or an increase in the proportion of households earning less than $30,000 among the Latinx households in the sample.

Implications:

This methodical approach to hypothesis testing reinforces the use of statistical evidence in making informed decisions. The clear and precise presentation of this analysis confirms the original implications and supports the integrity of the conclusions drawn.