Consider the following data.
[tex]$x =\#$[/tex] of packs of cigarettes smoked;
[tex]$y=$[/tex] longevity
Find the (least-squares) regression equation.
A. [tex]$y=-4.35+75.4 x$[/tex]
B. [tex]$y=75.4-4.35 x$[/tex]
C. [tex]$y=7.5-4.35 x$[/tex]
D. [tex]$y=75.4+4.35 x$[/tex]
Answer (1)
Calculate the means of x and y : x ˉ = 2 and y ˉ = 66.7 .
Calculate the slope b and y-intercept a using the formulas: b = − 4.35 and a = 75.4 .
Form the least-squares regression equation: y = a + b x = 75.4 − 4.35 x .
The correct regression equation is y = 75.4 − 4.35 x .
Explanation
Understanding the Problem We are given a set of data points ( x , y ) where x represents the number of packs of cigarettes smoked and y represents longevity. Our goal is to find the least-squares regression equation that best fits this data. The general form of a linear regression equation is y = a + b x , where a is the y-intercept and b is the slope.
Formulas for Slope and Intercept To find the least-squares regression equation, we need to calculate the slope ( b ) and the y-intercept ( a ). The formulas for these are:
b = ∑ i = 1 n ( x i − x ˉ ) 2 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ )
a = y ˉ − b x ˉ
where x ˉ and y ˉ are the means of the x and y values, respectively.
Calculating the Means First, let's calculate the means of x and y :
x ˉ = 10 0 + 0 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 = 10 20 = 2
y ˉ = 10 80 + 70 + 72 + 70 + 68 + 65 + 69 + 60 + 58 + 55 = 10 667 = 66.7
Calculating Slope and Intercept Now, we calculate the slope b :
Using python calculation tool, we find that: a = 75.4 b = − 4.35
The Regression Equation Therefore, the least-squares regression equation is: y = 75.4 − 4.35 x
Final Answer Comparing this equation with the given options, we find that it matches the second option: y = 75.4 − 4.35 x .
Examples
Linear regression is a powerful tool used in various fields. For instance, in healthcare, it can model the relationship between a patient's dosage of a drug and their blood pressure. By analyzing data from multiple patients, doctors can create a regression equation that predicts how blood pressure changes with different dosages. This helps in determining the optimal dosage for individual patients, ensuring effective treatment while minimizing potential side effects. Understanding and applying linear regression enables informed decision-making in healthcare and beyond.
