Chapter ISG, Problem 4.6.2.1P

a.

To determine

The interpretation of the given scatter plot.

Answer to Problem 4.6.2.1P

The given scatter is agreeable since the weight of a person increases with age as is depicted in the scatter plot.

Explanation of Solution

Given: The scatter plot showing the relation between the weight and age is as follows-

  

Concept Used: A scatter plot shows the relationship between a set of data with two variables graphed as ordered pairs on a coordinate plane. A scatter plot determines the correlation between the two quantities being plotted.

According to the given scatter plot, the weight increases as the age of the person increases. Hence it depicts a positive correlation which is agreeable since the weight of any person increases with age, up to a certain extent.

b.

To determine

The weight at the age of 40.

Answer to Problem 4.6.2.1P

The weight cannot be correctly determined from the line of regression as it will give an unreasonable value.

Explanation of Solution

Given: The scatter plot showing the relation between the weight and age is as follows-

  

Concept Used: A scatter plot shows the relationship between a set of data with two variables graphed as ordered pairs on a coordinate plane. A scatter plot determines the correlation between the two quantities being plotted. A line of best fit is a line through the scatter plot which is as close to all the points in the plot as possible. The slope of the line of best fit can be found using the formula-

  $m=\frac{y_2-y_1}{x_2-x_1}$ , where $(x_1,y_1)$ and $(x_2,y_2)$ are any two points on the line.

The equation of the line of best fit can help finding the values of one quantity if the value of the other quantity in the scatter plot is known. The equation can be determined using the formula-

  $y-y_1=m(x-x_1)$.

Calculation: For the given scatter plot, the line of best fit is as follows-

  

Since the line passes through the points $(100,12)$ and $(150,16)$ , hence the slope of the line is-

  $m=\frac{16-12}{150-100}$

  $=\frac{2}{25}$

Therefore the equation of the line passing through $(100,12)$ and having slope $\frac{2}{25}$ is given by-

  $y-12=\frac{2}{25}(x-100)$

  $\Rightarrow 25(y-12)=2(x-100)$

  $\Rightarrow 25y-300=2x-200$

  $\Rightarrow 25y=2x+100$

  $\Rightarrow y=\frac{2x+100}{25}$

  $\Rightarrow y=\frac{2x}{25}+4$

When the age is 40 years, the weight can be determined by putting $y=40$ in the above equation. Thus,

  $40=\frac{2x}{25}+4$

  $\Rightarrow \frac{2x}{25}=36$

  $\Rightarrow x=\frac{36\times 25}{2}$

  $\Rightarrow x=18\times 25$

  $\Rightarrow x=450$

This is an unreasonable value for the weight of a 40year old person. The reason for getting this absurd result is because unlike real world conditions where the weight after a certain age ceases to increase unreasonably, the line of best fit of the given scatter plot has a positive slope meaning that the weight keeps on increasing as age increases. Thus, the line of regression doesn’t provide a reasonable value of the weight at the age of 40.