Question

Make up a data set for income and education where the data points live pretty close to a straight line and make up a data set for incoming education. Where are the data points , or several of them, or far away from the linear relationship. So in a scatterplot, your observations in one case, lie, fairly close to a straight line, and then the other case, they are quite a bit above and below the straight line. Contrast the R squared from the two relationships periods .

Answer 1

A data set with points lying close to a straight line suggests a strong linear relationship and high R squared value. Conversely, a data set with scattered points indicates a weak relationship and low R squared value, which may require an alternative model to linear regression.

Step-by-step explanation:

Understanding Linear Relationships and Scatter Plots in Data:

To illustrate a data set where points lie close to a straight line versus one where they do not, we can fabricate two simple data sets. For income and education, an example of a data set with a strong linear relationship could be:

• (12 years of education, $30,000)
• (14 years of education, $35,000)
• (16 years of education, $40,000)
• (18 years of education, $45,000)

In this case, the correlation coefficient would be very high, demonstrating that as education increases, so does income, and the points would closely align with the regression line. When plotted on a scatter plot, the residuals would be small since the actual incomes align closely with the predicted incomes from the regression equation.

Contrastingly, a data set where income doesn't strongly correlate with education might look like this:

• (12 years of education, $30,000)
• (14 years of education, $40,000)
• (16 years of education, $35,000)
• (18 years of education, $50,000)
• (20 years of education, $32,000)

This second set of data would show greater variability, and the points would be scattered further from the trend line. The correlation coefficient would be lower, indicating a weaker linear relationship between education and income. The R squared value would also be much lower, revealing the inaccuracy of using a linear model to predict income from education in this instance.

Using regression analysis and observing scatter plots help determine the best model to represent a data set. While the least-squares regression line aims to minimize the residuals and provide a prediction, it's crucial to visually analyze if a linear model is appropriate or if other statistical models would be better suited.