Consider the following data.
[tex]$x =\#$[/tex] of packs of cigarettes smoked;
[tex]$y=$[/tex] longevity
According to your analysis, how many years of life are lost or gained per two additional packs of cigarettes a day?
A. loss of 2.175 years of life
B. loss of 8.7 years of life
C. gain of 4.35 years of life
D. loss of 4.35 years of life
Answer (1)
Calculate the sums: ∑ x , ∑ y , ∑ x y , and ∑ x 2 .
Calculate the slope a using the formula: a = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y ) .
Multiply the slope by 2 to find the change in longevity per two packs of cigarettes: 2 a .
The result is a loss of 8.7 years of life.
Explanation
Understanding the Problem We are given a dataset of cigarette packs smoked ( x ) and longevity ( y ). Our goal is to determine how many years of life are lost or gained per two additional packs of cigarettes smoked a day, based on a linear regression analysis.
Setting up Linear Regression To find the relationship between cigarette smoking and longevity, we will perform a linear regression. The equation for the regression line is y = a x + b , where y is longevity, x is the number of packs of cigarettes smoked, a is the slope (change in longevity per pack of cigarettes), and b is the y-intercept.
Calculating the Slope The slope a can be calculated using the formula: a = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y ) where n is the number of data points.
Calculating Sums First, let's calculate the necessary sums: ∑ x = 0 + 0 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 = 20 ∑ y = 80 + 70 + 72 + 70 + 68 + 65 + 69 + 60 + 58 + 55 = 667 ∑ x y = ( 0 ∗ 80 ) + ( 0 ∗ 70 ) + ( 1 ∗ 72 ) + ( 1 ∗ 70 ) + ( 2 ∗ 68 ) + ( 2 ∗ 65 ) + ( 3 ∗ 69 ) + ( 3 ∗ 60 ) + ( 4 ∗ 58 ) + ( 4 ∗ 55 ) = 0 + 0 + 72 + 70 + 136 + 130 + 207 + 180 + 232 + 220 = 1247 ∑ x 2 = 0 2 + 0 2 + 1 2 + 1 2 + 2 2 + 2 2 + 3 2 + 3 2 + 4 2 + 4 2 = 0 + 0 + 1 + 1 + 4 + 4 + 9 + 9 + 16 + 16 = 60 n = 10
Calculating the Slope Value Now, substitute these values into the formula for a :
a = 10 ( 60 ) − ( 20 ) 2 10 ( 1247 ) − ( 20 ) ( 667 ) = 600 − 400 12470 − 13340 = 200 − 870 = − 4.35 So, the slope a = − 4.35 .
Finding the Change in Longevity Since we want to find the change in longevity per two additional packs of cigarettes, we multiply the slope by 2: 2 a = 2 ∗ ( − 4.35 ) = − 8.7 This means that for every two additional packs of cigarettes smoked per day, a person loses 8.7 years of life.
Final Answer Therefore, according to our analysis, a person loses 8.7 years of life per two additional packs of cigarettes smoked a day.
Examples
Understanding the impact of smoking on longevity can be crucial in public health campaigns. For instance, if a campaign aims to reduce smoking by discouraging people from smoking two packs a day, this analysis shows that, on average, they could potentially gain 8.7 years of life. This kind of data helps illustrate the tangible benefits of reducing smoking, making the message more impactful and easier to understand. By quantifying the years of life lost, it provides a clear and compelling reason for individuals to quit or reduce their cigarette consumption.
