A researcher wishes to examine the relationship between years of schooling completed and the number of pregnancies in young women. Her research discovers a linear relationship, and the least squares line is: [tex]$\hat{y}=1-2 x$[/tex] where [tex]$x$[/tex] is the number of years of schooling completed and [tex]$y$[/tex] is the number of pregnancies. The slope of the regression line can be interpreted in the following way:
A. When amount of schooling increases by one year, the number of pregnancies decreases by 1.
B. When amount of schooling increases by one year, the number of pregnancies increases by 1.
C. When amount of schooling increases by one year, the number of pregnancies increases by 2.
D. When amount of schooling increases by one year, the number pregnancies decreases by 2.
Answer (1)
The slope of the regression line y ^ = 1 − 2 x is -2. This indicates that:
The equation represents a linear relationship between years of schooling and number of pregnancies.
The slope signifies the change in the number of pregnancies for each additional year of schooling.
A slope of -2 means the number of pregnancies decreases by 2 for each additional year of schooling.
Therefore, when the amount of schooling increases by one year, the number of pregnancies decreases by 2. $\boxed{When amount of schooling increases by one year, the number pregnancies decreases by 2 .}
Explanation
Understanding the Regression Equation The problem provides a linear regression equation y ^ = 1 − 2 x , where x represents years of schooling and y ^ represents the predicted number of pregnancies. We need to interpret the meaning of the slope, which is -2, in the context of this equation.
Interpreting the Slope The slope of a regression line indicates how much the dependent variable ( y ^ , number of pregnancies) is expected to change for each one-unit increase in the independent variable ( x , years of schooling). In this case, the slope is -2.
Meaning of the Negative Slope A slope of -2 means that for every additional year of schooling, the predicted number of pregnancies decreases by 2. This is because the coefficient of x in the equation is -2, indicating an inverse relationship between years of schooling and the number of pregnancies.
Final Answer Therefore, the correct interpretation is: When the amount of schooling increases by one year, the number of pregnancies decreases by 2.
Examples
In public health, understanding the relationship between education and family planning is crucial. For instance, if a study finds that for each additional year of schooling, the average number of children a woman has decreases by 0.2, this information can be used to design educational programs that promote smaller family sizes and better maternal health. This is based on the assumption that education empowers women to make informed choices about their reproductive health, leading to fewer pregnancies. The linear relationship can be expressed as y ^ = a − 0.2 x , where y ^ is the number of children, x is years of schooling, and a is a constant.
