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A survey is conducted on 700 Californians older than 30 years of age. The study wants to obtain inference on the relationship between years of education and yearly income in dollars. The response variable is income in dollars and the explanatory variable is years of education. A simple linear regression model is fit, and the output from R is below: Im(formula = Income ~ Education, data = CA) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 25200.25 1488.94 16.93 3.08e-10 *** Education 2905.35 112.61 25.80 1.49e-12 *** Residual standard error: 32400 on 698 degrees of freedom Multiple R-squared: 0.7602

A survey is conducted on 700 Californians older than 30 years of age. The study wants to obtain inference on the relationship between years of education and yearly income in dollars. The response variable is income in dollars and the explanatory variable is years of education. A simple linear regression model is fit, and the output from R is below: Im(formula = Income ~ Education, data = CA) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 25200.25 1488.94 16.93 3.08e-10 *** Education 2905.35 112.61 25.80 1.49e-12 *** Residual standard error: 32400 on 698 degrees of freedom Multiple R-squared: 0.7602

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 8E
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### Analysis of the Relationship Between Education and Income A survey was conducted on 700 Californians older than 30 years of age. The study aims to derive inferences about the relationship between years of education and yearly income in dollars. Here, the response variable is the income in dollars, and the explanatory variable is the number of years of education. A simple linear regression model was applied, and the results obtained from R are presented below: #### Linear Model ```R lm(formula = Income ~ Education, data = CA) ``` ### Coefficients: | Coefficients | Estimate | Std. Error | t value | Pr(>|t|) | |--------------|-----------|------------|---------|------------| | Intercept | 25200.25 | 1488.94 | 16.93 | 3.08e-10 *** | | Education | 2905.35 | 112.61 | 25.80 | 1.49e-12 *** | **Residual standard error**: 32400 on 698 degrees of freedom **Multiple R-squared**: 0.7602 ### Explanation: 1. **Intercept (Estimate = 25200.25)**: - This is the estimated average income (in dollars) when the number of years of education is zero. The high t-value (16.93) and the very small p-value (3.08e-10) indicate that this estimate is significantly different from zero. 2. **Education (Estimate = 2905.35)**: - This represents the estimated increase in income (in dollars) for each additional year of education. The high t-value (25.80) and the very small p-value (1.49e-12) indicate that this coefficient is also significantly different from zero. 3. **Residual Standard Error (32400)**: - This statistic measures the average amount that the observed values deviate from the predicted values. 4. **Multiple R-squared (0.7602)**: - This value indicates that approximately 76.02% of the variation in the income can be explained by the number of years of education. The presence of the three stars (***) next to the p-values in the coefficient table indicate a high level of statistical significance. This analysis suggests that there is a strong, positive relationship between years of education and yearly
Transcribed Image Text:### Analysis of the Relationship Between Education and Income A survey was conducted on 700 Californians older than 30 years of age. The study aims to derive inferences about the relationship between years of education and yearly income in dollars. Here, the response variable is the income in dollars, and the explanatory variable is the number of years of education. A simple linear regression model was applied, and the results obtained from R are presented below: #### Linear Model ```R lm(formula = Income ~ Education, data = CA) ``` ### Coefficients: | Coefficients | Estimate | Std. Error | t value | Pr(>|t|) | |--------------|-----------|------------|---------|------------| | Intercept | 25200.25 | 1488.94 | 16.93 | 3.08e-10 *** | | Education | 2905.35 | 112.61 | 25.80 | 1.49e-12 *** | **Residual standard error**: 32400 on 698 degrees of freedom **Multiple R-squared**: 0.7602 ### Explanation: 1. **Intercept (Estimate = 25200.25)**: - This is the estimated average income (in dollars) when the number of years of education is zero. The high t-value (16.93) and the very small p-value (3.08e-10) indicate that this estimate is significantly different from zero. 2. **Education (Estimate = 2905.35)**: - This represents the estimated increase in income (in dollars) for each additional year of education. The high t-value (25.80) and the very small p-value (1.49e-12) indicate that this coefficient is also significantly different from zero. 3. **Residual Standard Error (32400)**: - This statistic measures the average amount that the observed values deviate from the predicted values. 4. **Multiple R-squared (0.7602)**: - This value indicates that approximately 76.02% of the variation in the income can be explained by the number of years of education. The presence of the three stars (***) next to the p-values in the coefficient table indicate a high level of statistical significance. This analysis suggests that there is a strong, positive relationship between years of education and yearly
### Writing the Estimated Linear Equation #### Instructions: Write out the estimated linear equation. Round to two decimal places (x.xx). \[ \hat{Income_i} = \quad \_\_\_\_\_\_ \quad + \quad \_\_\_\_\_\_ \quad \text{Education}_i \] *Note:* The equation is in the form of a linear regression equation where \(\hat{Income_i}\) is the estimated income based on the level of education (\(\text{Education}_i\)). The coefficients to be determined are the intercept and the slope for the variable Education.
Transcribed Image Text:### Writing the Estimated Linear Equation #### Instructions: Write out the estimated linear equation. Round to two decimal places (x.xx). \[ \hat{Income_i} = \quad \_\_\_\_\_\_ \quad + \quad \_\_\_\_\_\_ \quad \text{Education}_i \] *Note:* The equation is in the form of a linear regression equation where \(\hat{Income_i}\) is the estimated income based on the level of education (\(\text{Education}_i\)). The coefficients to be determined are the intercept and the slope for the variable Education.
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