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Association Suppose you were to collect data for each pair of variables. You want to make a
- a) Apples: weight in grams, weight in ounces
- b) Apples: circumference (inches), weight (ounces)
- c) College freshmen: shoe size, grade point average
- d) Gasoline: number of miles you drove since filling up, gallons remaining in your tank
a.
Find the explanatory variable and response variable to plot a scatterplot.
Find the direction, form and strength of the scatterplot.
Answer to Problem 1E
Either weight in grams or weight in ounces could be the explanatory or response variable.
The association between the variables is straight, positive and strong.
Explanation of Solution
Given info:
The variables of the apples are given one is weight in grams and the other is weight in ounces.
Justification:
Associated variables:
Two variables are associated or related if the value of one variable gives you information about the value of the other variable.
The two variables weight in grams and weight in ounces are associated variables.
Response variable:
The variable to be measured or observed in regression analysis is called as response variable. In other words it can also be defined as, the variable that is changed due to the impact of the explanatory variable is defined as response variable.
Therefore, the dependent variables which is measured by the independent variables is called the response variable.
Here, given two variables are weight in grams of apple and weight in ounces of apple.
That is, each apple’s weight is measured in two different scales.
Therefore, there will be chances for weight in grams to depend on weight in ounces and vice versa.
Thus, either weight in grams or weight in ounces could be the explanatory or response variable.
Explanatory variable:
The variable used to predict or explain the response variable is called as predictor variable or explanatory variable. In other words it can also be defined as, the variable that explains the changes in the response variable is defined as explanatory variable.
Therefore, the independent variables to predict the response variable is called the predictor variable.
Here, given two variables are weight in grams of apple and weight in ounces of apple.
That is, each apple’s weight is measured in two different scales.
Therefore, there will be chances for weight in grams to depend on weight in ounces and vice versa.
Thus, either weight in grams or weight in ounces could be the explanatory or response variable.
Form of the association between variable:
The form of the association describes whether the data points follow a linear pattern or some other complicated curves. For data if it appears that a line would do a reasonable job of summarizing the overall pattern in the data. Then, the association between two variables is linear.
Here, weight in ounces increases or decreases with the increase or decrease in the weight in grams.
The pattern of the relationship between weight in ounces and weight in grams represents a straight line.
Hence, the association between the weight in ounces and weight in grams is linear.
Direction of association:
If the increase in the values of one variable increases the values of another variable, then the direction is positive. If the increase in the values of one variable decreases the values of another variable, then the direction is negative.
Here, weight in ounces increases or decreases with the increase or decrease in the weight in grams.
Hence, the direction of the association is positive.
Strength of the association:
The association is said to be strong if all the points are close to the straight line. It is said to be weak if all points are far away from the straight line and it is said to be moderate if the data points are moderately close to straight line.
Here, the variables will have perfect correlation between them.
Hence, the association between the variables is strong.
b.
Find the explanatory variable and response variable to plot a scatterplot.
Find the direction, form and strength of the scatterplot.
Answer to Problem 1E
Circumference of apple is explanatory variable and weight is the response variable.
The association between the variables is straight, positive and strong.
Explanation of Solution
Given info:
The variables of the apples are given one is circumference in inches and the other is weight in ounces.
Justification:
Associated variables:
Two variables are associated or related if the value of one variable gives you information about the value of the other variable.
The two variables circumference in inches and weight in ounces are associated variables.
Response variable:
The variable to be measured or observed in regression analysis is called as response variable. In other words it can also be defined as, the variable that is changed due to the impact of the explanatory variable is defined as response variable.
Therefore, the dependent variables which is measured by the independent variables is called the response variable.
Here, given two variables are circumference in inches of apple and weight in ounces of apple.
Three dimensional volume is nothing but the weight and one dimensional circumference explains the three dimensional volume.
Therefore, weight of the apple is predicted with the circumference of the apple.
That is, weight of the apple is depend on the circumference of the apple.
Thus, weight in ounces is dependent or response variable.
Explanatory variable:
The variable used to predict or explain the response variable is called as predictor variable or explanatory variable. In other words it can also be defined as, the variable that explains the changes in the response variable is defined as explanatory variable.
Therefore, the independent variables to predict the response variable is called the predictor variable.
Here, given two variables are circumference in inches of apple and weight in ounces of apple.
Weight of the apple is predicted with the circumference of the apple.
Thus, circumference in inches is independent or explanatory variable.
Form of the association between variable:
The form of the association describes whether the data points follow a linear pattern or some other complicated curves. For data if it appears that a line would do a reasonable job of summarizing the overall pattern in the data. Then, the association between two variables is linear.
Here, weight in ounces increases or decreases with the increase or decrease in the circumference in inches of apple.
The pattern of the relationship between weight in ounces and circumference in inches of apple represents a straight line for same size apples.
Hence, the association between the weight in ounces and circumference in inches of apple is linear for same size apples.
The association curve will be apparent if the sample contains very large and very small apples.
Direction of association:
If the increase in the values of one variable increases the values of another variable, then the direction is positive. If the increase in the values of one variable decreases the values of another variable, then the direction is negative.
Here, weight in ounces increases or decreases with the increase or decrease in the circumference in inches of apple.
Hence, the direction of the association is positive.
Strength of the association:
The association is said to be strong if all the points are close to the straight line. It is said to be weak if all points are far away from the straight line and it is said to be moderate if the data points are moderately close to straight line.
Here, the variables will have perfect correlation between them.
Hence, the association between the variables is strong.
c.
Find the explanatory variable and response variable to plot a scatterplot.
Find the direction, form and strength of the scatterplot.
Answer to Problem 1E
The variables shoe size and grade point average are not associated with each other.
Explanation of Solution
Given info:
The variables of the college freshmen are given one is shoe size and the other is grade point average.
Justification:
Associated variables:
Two variables are associated or related if the value of one variable gives you information about the value of the other variable.
There is no relationship between the variables shoe size and grade point average.
Therefore, there is no association between the variables.
Hence, the discussion will not go further.
d.
Find the explanatory variable and response variable to plot a scatterplot.
Find the direction, form and strength of the scatterplot.
Answer to Problem 1E
Circumference of apple is explanatory variable and weight is the response variable.
The association between the variables is straight, negative and strong.
Explanation of Solution
Given info:
The variables of the gasoline are given one is number of miles drove since filling up and the other is gallons remaining in the tank.
Justification:
Associated variables:
Two variables are associated or related if the value of one variable gives you information about the value of the other variable.
The two variables number of miles drove since filling up and gallons remaining in the tank are associated variables.
Response variable:
The variable to be measured or observed in regression analysis is called as response variable. In other words it can also be defined as, the variable that is changed due to the impact of the explanatory variable is defined as response variable.
Therefore, the dependent variables which is measured by the independent variables is called the response variable.
Here, given two variables are number of miles drove since filling up and gallons remaining in the tank.
The fuel that is remained in the tank is dependent on the fuel that is used for driving.
Therefore, gallons remaining in the tank is predicted with the number of miles drove since filling up.
That is, gallons remaining in the tank is depend on the number of miles drove since filling up.
Thus, gallons remaining in the tank is dependent or response variable.
Explanatory variable:
The variable used to predict or explain the response variable is called as predictor variable or explanatory variable. In other words it can also be defined as, the variable that explains the changes in the response variable is defined as explanatory variable.
Therefore, the independent variables to predict the response variable is called the predictor variable.
Here, given two variables are number of miles drove since filling up and gallons remaining in the tank.
Gallons remaining in the tank is predicted with the number of miles drove since filling up.
Thus, the number of miles drove since filling up is independent or explanatory variable.
Form of the association between variable:
The form of the association describes whether the data points follow a linear pattern or some other complicated curves. For data if it appears that a line would do a reasonable job of summarizing the overall pattern in the data. Then, the association between two variables is linear.
Here, gallons remaining in the tank decreases with the increase in the number of miles drove since filling up.
The pattern of the relationship between gallons remaining in the tank and the number of miles drove since filling up represents a straight line.
Hence, the association between the gallons remaining in the tank and the number of miles drove since filling up is linear.
Direction of association:
If the increase in the values of one variable increases the values of another variable, then the direction is positive. If the increase in the values of one variable decreases the values of another variable, then the direction is negative.
Here, gallons remaining in the tank decreases with the increase in the number of miles drove since filling up and gallons remaining in the tank increases with the decrease in the number of miles drove since filling up.
Hence, the direction of the association is negative.
Strength of the association:
The association is said to be strong if all the points are close to the straight line. It is said to be weak if all points are far away from the straight line and it is said to be moderate if the data points are moderately close to straight line.
Here, the variables will have moderate correlation between them.
Hence, the association between the variables is moderate.
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