Week 7 Linear Regression Exercises: Simple Regression Resear
Week 7 Linear Regression Exercises Simple Regression research Question
Analyze the relationship between the number of hours worked per week and family income using simple linear regression. Using the Polit2SetA dataset, run a regression with family income as the outcome (Y) and hours worked per week as the predictor (X). Follow these steps: open the dataset, perform linear regression in SPSS, and include descriptives, estimates, confidence interval, and model fit statistics. Interpret the regression output to answer questions about sample size, means, correlation, R squared, standard error of estimate, model fit, regression coefficients, and predictive equations. Apply the regression model to predict family income for women working 35 and 20 hours per week.
Paper For Above instruction
The investigation into the relationship between hours worked per week and family income employs simple linear regression analysis using the Polit2SetA dataset. This statistical approach facilitates understanding whether a linear association exists and the strength of this relationship between the predictor, hours worked, and the outcome variable, family income. The following discussion presents a comprehensive analysis based on the SPSS output, aiming to answer the outlined research questions.
Sample Size and Descriptive Statistics
The total sample size, as reported in the SPSS output, is 378 participants. The descriptive statistics reveal a mean family income of approximately $1,485.49 with a standard deviation of $950. Additionally, the mean number of hours worked per week is about 33.52 hours, with a standard deviation of 12. These descriptive measures provide an essential foundation for understanding the data’s distribution and variability, which influence the regression analysis’s interpretability.
Correlation and Significance
The Pearson correlation coefficient between family income and hours worked is reported as 0.300 in the SPSS output, with a significance level (p-value) of .000. This indicates a moderate positive correlation that is statistically significant; thus, as hours worked increase, family income tends to increase as well. The positive relationship’s strength suggests that working more hours is associated with higher family income, although the correlation coefficient indicates that other factors likely influence income beyond hours worked alone.
Coefficient of Determination (R-squared)
The R squared value is 0.090, meaning approximately 9% of the variance in family income can be explained by the number of hours worked per week. This relatively low R-squared value indicates that while there is a significant association, hours worked is only a modest predictor of income, and additional variables likely contribute to income variability. Nonetheless, the significant regression model suggests that hours worked remains an important predictor within the context of this analysis.
Standard Error of the Estimate
The standard error of the estimate is approximately $907.88, as per the model output. This statistic indicates the average distance between the observed family income and the predicted income values derived from the regression line. A standard error of this magnitude suggests a considerable amount of variability around the predicted income, highlighting the limitations in precise predictions based solely on hours worked.
Model Fit and ANOVA Results
The ANOVA table reports an F statistic of 37.226 with a p-value of .000. Since this p-value is less than the conventional threshold of .05, the overall regression model significantly fits the data. A significant F-test confirms that the predictor variable explains a meaningful portion of the variance in family income, and the model has predictive value. Despite the modest R-squared, the model's significance supports its utility in understanding income trends relative to hours worked.
Regression Equation and Intercept
The unstandardized regression coefficient for hours worked is 23.083, and the intercept (constant term) is 711.155. Therefore, the regression equation for predicting family income (Y) based on hours worked (X) is:
Y = 711.155 + 23.083 * X
This equation indicates that, holding other factors constant, each additional hour worked per week increases family income by approximately $23.08. The intercept suggests that if a woman worked zero hours per week, the estimated family income would be around $711.16, which might encompass baseline income or fixed income components not directly related to work hours.
Predicting Family Income
Using the regression equation, the predicted family income for women working 35 hours per week is:
Y = 711.155 + 23.083 * 35 ≈ 711.155 + 807.905 = $1,519.06
Similarly, for 20 hours per week:
Y = 711.155 + 23.083 * 20 ≈ 711.155 + 461.66 = $1,172.81
These predictions suggest that working an additional 15 hours per week (from 20 to 35 hours) would result in an approximate income increase of $347.25, consistent with the positive association identified in the data.
Conclusion
In summary, the regression analysis confirms a significant positive relationship between hours worked and family income. Although the model explains only a modest proportion of income variability, it indicates that hours worked is a relevant predictor. The regression equation derived provides a practical tool for estimating income based on hours worked, with implications for policy and individual decision-making. Future research could incorporate additional variables, such as education, experience, or occupation, to improve predictive accuracy and better understand the determinants of family income.
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