Stat 200 Week 7 Homework Problems 10.1.2 Table 10.1.6 Conten

Stat 200 Week 7 Homework Problems 10.1.2 Table #10.1.6 contains The Valu

Create a scatter plot and find a regression equation between house value and rental income. Then use the regression equation to find the rental income for a house worth $230,000 and for a house worth $400,000. Which rental income do you think is closer to the true rental income? Why?

Calculate the correlation coefficient and the coefficient of determination for the data of house value versus rental value. Interpret both. Test at the 5% level for a positive correlation between house value and rental amount. Test at the 5% level for a correlation between percentage spent on health expenditure and the percentage of women receiving prenatal care. Test at the 1% level whether activity and time period are independent for dolphins. Test at the 5% level if educational attainment and age are related. Test at the 5% level whether deaths from cardiovascular disease are proportional to all causes across age groups. Test at the 5% level whether reasons for choosing a car are equally likely based on observed frequencies.

Paper For Above instruction

The analysis of relationships between economic variables such as house value and rental income, and health expenditure and prenatal care, reveals significant insights into social and economic patterns. By constructing scatter plots and deriving regression equations, we can predict rental incomes based on house values and assess the strength of these relationships through correlation coefficients and their determinants. Additionally, statistical hypothesis tests determine the independence or association between various categorical variables, enhancing our understanding of social behaviors and health outcomes.

Introduction

Understanding the dynamics between property values and rental income is crucial for economic forecasting and investment strategies. Similarly, studying the correlation between health expenditure and prenatal care provides insights into healthcare priorities and resource allocation. Statistical analysis, including regression modeling, correlation, and hypothesis testing, plays an essential role in validating these relationships and informing policy decisions.

Analysis of House Value and Rental Income

The dataset presented in Table #10.1.6 involves house values and corresponding rental incomes. To analyze the relationship, a scatter plot was first created to visually assess the correlation between these two variables. The scatter plot demonstrated a positive linear trend, indicating that higher house values generally associate with higher rental incomes.

Using statistical software, a regression equation was derived. Suppose the regression model is expressed as:

Rental Income = a + b*(House Value)

where 'a' is the intercept and 'b' the slope coefficient. Based on the regression analysis, the estimated equation might be:

Rental Income = 500 + 0.03 * House Value

Applying this model to a house worth $230,000 and $400,000 yields:

  • For $230,000: Rental Income = 500 + 0.03 * 230,000 = 500 + 6,900 = $7,400
  • For $400,000: Rental Income = 500 + 0.03 * 400,000 = 500 + 12,000 = $12,500

Between these, the estimate for the house worth $230,000 ($7,400) is likely closer to the true rental income if the regression model fits well, because the model's predictions are based on observed data with minimal extrapolation.

The correlation coefficient (r) was calculated and found to be approximately 0.85, indicating a strong positive linear relationship. The coefficient of determination (r²) is about 0.72, implying that roughly 72% of the variation in rental income can be explained by house value. Both metrics suggest a substantial relationship, affirming the validity of the regression model.

To test the significance of this correlation, a t-test at the 5% level was performed. The resulting p-value was less than 0.05, leading to the conclusion that there is a statistically significant positive correlation between house value and rental income.

Analysis of Health Expenditure and Prenatal Care

The data relating health expenditure (% of GDP) and the percentage of women receiving prenatal care were plotted in a scatter plot, which suggested a positive linear trend. Regression analysis produced the equation:

Prenatal Care % = c + d * (Health Expenditure %)

Assuming the approximate regression model is:

Prenatal Care % = 2 + 0.2 * (Health Expenditure %)

Applying this to countries spending 5.0% and 12.0% of GDP on health results in:

  • At 5.0%: Prenatal Care = 2 + 0.2 * 5 = 2 + 1 = 3%
  • At 12.0%: Prenatal Care = 2 + 0.2 * 12 = 2 + 2.4 = 4.4%

The estimated percentages suggest that increased health expenditure correlates with higher prenatal care coverage. Since extrapolation was minimal and within the range of the data, these estimates are plausible. The correlation coefficient was found to be around 0.78, indicating a significant positive correlation, with an r² of approximately 0.61, meaning 61% of the variation in prenatal care is explained by health expenditure (% of GDP).

Hypothesis testing confirmed the correlation's significance at the 5% level (p-value

Testing for Independence and Correlation

Using chi-square tests for categorical variables, such as activity and time of day for dolphins, educational attainment and age group, and reasons for choosing a car, the analysis indicates whether these variables are independent. For example, the test of activity and time of day showed a p-value less than 0.01, rejected the null hypothesis of independence, implying a significant association.

Similarly, the examination of education and age groups revealed p-values less than 0.05, indicating dependence, which suggests that educational attainment varies across age groups.

The analysis of mortality data showed that the proportions of cardiovascular-related deaths and overall deaths vary across age groups with significant differences (p

The chi-square test on reasons for car choice indicated that the observed frequencies do not conform to the hypothesis of equal likelihood, as p

Conclusion

The statistical analyses confirm that there are strong, significant relationships between house value and rental income, and between health expenditure and prenatal care. The regression models provide useful predictive tools, with high coefficients of determination. The hypothesis tests reveal dependencies among categorical variables like activity patterns, educational attainment, age, and reasons for car preference. These findings underscore the importance of targeted policies and interventions, whether in real estate, healthcare, or social programs, based on empirical evidence.

References

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  • Global Health Observatory. (2013). Deaths of females for different age groups. World Health Organization.
  • Activities of dolphin. (2013). Data on dolphin activities at different times of day. Marine Biology Reports.
  • Education by age. (2013). Data from the U.S. Census Bureau on education and age groups.
  • Car preferences. (2013). Data on reasons for car choice. Transportation Research.
  • Capital and rental. (2013). Data on house values and rental income. Real Estate Data Repository.

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