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Now say using the same dataset an analysis is done but using Euros as the unit of measurement of income, not US dollars (1 dollar=0.85 euros). Thus this model is all in units of measurement being euros for income. What happens to the slope estimate on Education? Will be larger. Will be smaller. Stays the same. Will be 0.

Now say using the same dataset an analysis is done but using Euros as the unit of measurement of income, not US dollars (1 dollar=0.85 euros). Thus this model is all in units of measurement being euros for income. What happens to the slope estimate on Education? Will be larger. Will be smaller. Stays the same. Will be 0.

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter1: Equations And Graphs
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## Relationship Between Education and Income in Californians Over 30 ### Study Overview A survey was conducted on 700 Californians older than 30 years of age. The study aims to understand the relationship between years of education and yearly income in dollars. In this context: - The **response variable** is the income in dollars. - The **explanatory variable** is the years of education. ### Data Analysis using Linear Regression A simple linear regression model was used to analyze the data. The following output was generated using R: ``` lm(formula = Income ~ Education, data = CA) ``` The coefficients obtained from the model are: #### 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 *** | The notation `***` indicates that the p-values are highly significant. - **Intercept (25200.25)**: This is the estimated income for individuals with 0 years of education. - **Education (2905.35)**: This coefficient indicates the increase in yearly income for each additional year of education. #### Model Summary: - **Residual standard error**: 32400 on 698 degrees of freedom. This value indicates the standard deviation of the residuals, implying how much the observed values deviate from the fitted values. - **Multiple R-squared**: 0.7602. This value represents the proportion of variance in the income that can be explained by the years of education. A value of 0.7602 suggests a strong positive relationship between education and income. ### Interpretation From the results, it is evident that there is a significant and positive relationship between years of education and yearly income. For each additional year of education, the income increases by approximately $2905.35. The high R-squared value indicates that education is a good predictor of income within this dataset.
Transcribed Image Text:## Relationship Between Education and Income in Californians Over 30 ### Study Overview A survey was conducted on 700 Californians older than 30 years of age. The study aims to understand the relationship between years of education and yearly income in dollars. In this context: - The **response variable** is the income in dollars. - The **explanatory variable** is the years of education. ### Data Analysis using Linear Regression A simple linear regression model was used to analyze the data. The following output was generated using R: ``` lm(formula = Income ~ Education, data = CA) ``` The coefficients obtained from the model are: #### 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 *** | The notation `***` indicates that the p-values are highly significant. - **Intercept (25200.25)**: This is the estimated income for individuals with 0 years of education. - **Education (2905.35)**: This coefficient indicates the increase in yearly income for each additional year of education. #### Model Summary: - **Residual standard error**: 32400 on 698 degrees of freedom. This value indicates the standard deviation of the residuals, implying how much the observed values deviate from the fitted values. - **Multiple R-squared**: 0.7602. This value represents the proportion of variance in the income that can be explained by the years of education. A value of 0.7602 suggests a strong positive relationship between education and income. ### Interpretation From the results, it is evident that there is a significant and positive relationship between years of education and yearly income. For each additional year of education, the income increases by approximately $2905.35. The high R-squared value indicates that education is a good predictor of income within this dataset.
### Analyzing Slope Estimate Change with Currency Conversion **Context**: When conducting statistical analyses, changing the unit of measurement can impact the results. In this scenario, we consider the effect of switching from US dollars to euros in an analysis of income. **Situation**: Suppose an analysis is conducted on a dataset using US dollars as the unit of measurement for income. The conversion rate is given as 1 dollar = 0.85 euros. If the analysis is redone using euros instead of dollars, the model will now use euros for the income variable. **Question**: What happens to the slope estimate on Education when the unit of measurement for income is changed from US dollars to euros? **Options**: - Will be larger. - Will be smaller. - Stays the same. - Will be 0. #### Explanation: To answer the above question, let's consider the implications of changing the currency unit on the slope estimate: When converting the income from dollars to euros, the value of the income variable is multiplied by 0.85 (since 1 dollar = 0.85 euros). This directly impacts the scale of the slope in a linear regression model. **Graph/Diagram Explanation**: Although no specific graph or diagram is provided in this context, consider a general linear regression line where the slope represents the change in income per unit change in education. If we convert income from dollars to euros, the numeric values of income will reduce (as euros have higher value per unit compared to dollars), which scales down the slope correspondingly. ### Answer: - **Will be smaller**: The correct option suggests that the slope estimate will be smaller since the conversion decreases the numeric values of the income variable, thus decreasing the slope accordingly. Understanding these changes is crucial for interpreting statistical models correctly when switching between different units of measurement.
Transcribed Image Text:### Analyzing Slope Estimate Change with Currency Conversion **Context**: When conducting statistical analyses, changing the unit of measurement can impact the results. In this scenario, we consider the effect of switching from US dollars to euros in an analysis of income. **Situation**: Suppose an analysis is conducted on a dataset using US dollars as the unit of measurement for income. The conversion rate is given as 1 dollar = 0.85 euros. If the analysis is redone using euros instead of dollars, the model will now use euros for the income variable. **Question**: What happens to the slope estimate on Education when the unit of measurement for income is changed from US dollars to euros? **Options**: - Will be larger. - Will be smaller. - Stays the same. - Will be 0. #### Explanation: To answer the above question, let's consider the implications of changing the currency unit on the slope estimate: When converting the income from dollars to euros, the value of the income variable is multiplied by 0.85 (since 1 dollar = 0.85 euros). This directly impacts the scale of the slope in a linear regression model. **Graph/Diagram Explanation**: Although no specific graph or diagram is provided in this context, consider a general linear regression line where the slope represents the change in income per unit change in education. If we convert income from dollars to euros, the numeric values of income will reduce (as euros have higher value per unit compared to dollars), which scales down the slope correspondingly. ### Answer: - **Will be smaller**: The correct option suggests that the slope estimate will be smaller since the conversion decreases the numeric values of the income variable, thus decreasing the slope accordingly. Understanding these changes is crucial for interpreting statistical models correctly when switching between different units of measurement.
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