Points Table 1 In Question 1 Contains The Value Of The House

3 Points Table 1 In Question 1 Contains The Value Of The House An

Test at the 5% level for a positive correlation between house value and rental income. Find the standard error of the estimate. Compute a 95% prediction interval for the rental income on a house worth $230,000. Test at the 5% level for a correlation between percentage spent on health expenditure and the percentage of woman receiving prenatal care. Find the standard error of the estimate. Compute a 95% prediction interval for the percentage of woman receiving prenatal care for a country that spends 5.0% of GDP on health expenditure.

Paper For Above instruction

The analysis of relationships between different economic and social variables is fundamental in understanding the dynamics within societies and economies. This paper focuses on two specific scenarios: firstly, the correlation between house value and rental income; secondly, the association between a country's health expenditure and the percentage of women receiving prenatal care. Using statistical hypothesis testing, correlation measures, and prediction intervals, this work evaluates these relationships, providing insights into their significance and practical implications.

Correlation Between House Value and Rental Income

The first scenario involves analyzing data to determine whether a positive correlation exists between house value and rental income. The primary statistical approach is the hypothesis test for correlation coefficient (Pearson’s r), conducted at a 5% significance level. The null hypothesis states that there is no correlation (H₀: ρ = 0), against the alternative hypothesis that there is a positive correlation (H_a: ρ > 0).

Assuming the data provides a sample correlation coefficient r and a sample size n, the test involves calculating a t-statistic:

t = r√(n - 2) / √(1 - r²),

and comparing it against the critical value from the t-distribution with (n - 2) degrees of freedom. Significant results (p-value

Furthermore, the standard error of the estimate (SEE) quantifies the dispersion of the observed rental incomes around the predicted values based on the regression model. It is calculated as:

SEE = sqrt( Σ(y_i - ŷ_i)² / (n - 2) ).

A smaller SEE indicates a tighter fit of the regression line, enhancing the precision of predictions.

The prediction interval for a house valued at $230,000 provides a range within which future rental incomes are expected to fall with 95% confidence. It accounts for both the uncertainty in the regression estimate and the variability of individual data points and is computed as:

ŷ ± t_{α/2, n-2} * SE_pred,

where ŷ is the predicted rental income at $230,000, and SE_pred is the standard error of prediction, factoring in the standard error of estimate and the distance from the mean house value.

Relation Between Health Expenditure and Prenatal Care

In the second scenario, the data examines the relationship between the percentage of GDP a country spends on health and the percentage of women receiving prenatal care. Base hypothesis testing involves assessing whether a statistically significant correlation exists at the 5% level. The hypotheses are:

• Null hypothesis: no correlation between variables (ρ = 0).
• Alternative hypothesis: positive correlation exists (ρ > 0).

The test utilizes Pearson’s correlation coefficient r. The t-test follows the same formula as previously mentioned, and a p-value below 0.05 indicates a significant positive association.

The standard error of the estimate (SEE) here evaluates how well the regression model predicts the percentage of women receiving prenatal care based on health expenditure. A lower SEE signifies better predictive accuracy.

The 95% prediction interval for the percentage of women receiving prenatal care at 5.0% health expenditure involves calculating the predicted value (ŷ) from the regression model and then adding and subtracting the appropriate t-multiplied standard error, acknowledging both model uncertainty and variability in data points.

Discussion of Findings

The two analyses demonstrate how correlation testing, standard error evaluation, and prediction intervals can provide quantifiable insights into economic and social relationships. Significant correlation between house values and rental income suggests that investments in housing could inform rental market expectations. Similarly, the positive relationship between health expenditure and prenatal care uptake emphasizes how policy investments in healthcare can improve maternal health outcomes. The standard error metrics further clarify the prediction reliability of these models. The prediction intervals inform stakeholders of anticipated ranges, aiding decision-making for prospective investments or policy planning.

Conclusion

Understanding these relationships through statistical analysis enables more effective economic and social strategies. While correlations indicate associations, detailed measures such as standard error and prediction intervals provide a nuanced understanding about the certainty and variability of the predictions, which are crucial for informed decision-making. Future research could expand sample sizes and include more variables for comprehensive models, further enhancing the robustness of such analyses.

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