Question

using the least - squares line.

1. will i bomb the final? we expect that students who do well on the midterm exam in a course will usually also do well on the final exam. gary smith of pomona college looked at the exam scores of all 346 students who took his statistics class over a 10 - year period. assume that both the midterm and final scores were scored out of 100 points.

(a) state the equation of the least - squares regression line if each student scored the same on the midterm and the final.

Explanation:

Step1: Recall least - squares regression line concept

The least - squares regression line for a simple linear regression $y = a+bx$ is used to predict the relationship between two variables. In the case where $y$ (final exam score) and $x$ (midterm exam score) are the same for all students, the relationship is $y=x$. The general form of the least - squares regression line is $\hat{y}=b_0 + b_1x$, where $b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$ and $b_0=\bar{y}-b_1\bar{x}$. When $y_i=x_i$ for all $i = 1,\cdots,n$, we have:
First, $\bar{y}=\bar{x}$. And $\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=\sum_{i = 1}^{n}(x_i-\bar{x})(x_i - \bar{x})=\sum_{i=1}^{n}(x_i-\bar{x})^2$. So, $b_1 = 1$ and $b_0=\bar{y}-1\times\bar{x}=0$.

Step2: Write the regression line equation

The least - squares regression line equation is $\hat{y}=x$.

Answer:

The equation of the least - squares regression line is $\hat{y}=x$.