Interpreting Categorical & Quantitative Data - Common Core: High School - Statistics and Probability

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We start to solve this problem by picking out the graph with obvious positive or negative trends and excluding them. The following two graphs possess a positive and moderately positive trendline, respectively:

Two other graphs possess a negative and slightly negative trendline, respectively:

Last, one graph has a near horizontal line, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

Correct Answer: INSERT plot9.5.png

Wrong Answer 1: INSERT plot9.4.png

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Wrong Answer 4: INSERT plot9.1.png

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

In order to solve this problem, we need to understand several key concepts associated with correlations. First, let's discuss what is meant by the term "correlation." A correlation exists when two variables possess a statistical relationship with one another. It is important to note that correlation in no way relates to causation. Causation implies that one variable causes change in the other, while correlation simply denotes the observation of a trend between two variables.

Second, let's observe the differences between slope and the correlation coefficient. The correlation coefficient is denoted by the following variable.

It is mathematically defined as a goodness of fit measure that is calculated by dividing the covariance of the samples by the product of the sample's standard deviations. This is also known as Pearson's r and it describes the strength and direction of a linear relationship between two variables. On the other hand, the slope is described as the gradient of a line and is key component of the slope intercept formula:

This formula provides information about two key parts of a line: the slope and y-intercept.

The slope is commonly defined as rise over run. In other words it is the change in y-values across points divided by the change in x-values. It is calculated using the following formula:

In this formula, the x and y-values come from two points from the line written in the following format:

It is important to note that slopes can be positive or negative. A positive slope moves upward from left to right while a negative slope moves downward. Even though the correlation coefficient will share the same sign as the slope, they mean entirely different things.

We have discussed the following distinctions: the differences between what is meant correlation and causation as well as the differences between the correlation coefficient and the slope. Now, we can start to solve the problem.

First, lets learn how calculate the correlation coefficient from coefficient of determination. The coefficient of determination is denoted by the following:

We can calculate the correlation coefficient by taking the square root of the coefficient of determination:

After we calculate the correlation coefficient, we need to know how to evaluate what the number means. We can pick the sign based on the position of the trendline or slope. If the slope is negative then the trendline travels downward from the left to the right of the graph. On the other hand, if the slope is positive then the trendline travels upwards from the left to the right side of the graph. Below is a table of values that explains the relationships between points based upon the correlation coefficient. A correlation coefficient close to zero indicates a random distribution.

We will start solving this problem by picking out the graph with obvious positive or negative trends and excluding them.

The last, two graph's have a near horizontal trendline, which is indicative of a random distribution:

This graph possesses a trendline with the following coefficient of determination:

We must take the square root of this measure calculated using statistical technology (this standard does not require you to calculate the correlation coefficient, only to interpret it).

This trendline's correlation coefficient is most indicative of a random distribution.

No explanation available

In order to properly solve this question, we need to understand the differences between what is meant by correlation and causation. A correlation refers to the strength of the linear association between two quantitative variables. On the other hand, causation indicates that the change in one variable is the cause of change in another.

Correlation can be used as an indicator of causal relationships; however, experimentation is needed to properly identify which variable is actually causing the observed change. Scientific experimentation identifies causality through he implementation of laboratory procedures in a controlled setting. When variables are controlled, causation can be determined through observation and repeated tests.

Several logical fallacies explain why correlation does not directly imply causation. First, cause-and-effect is not determined by two events occurring simultaneously. In other words, events that occur together do not necessarily cause one another. Second, causality is not determined by an event preceding another temporally. In other words, this means that event B is not always a consequence of event A simply because event A occurs before event B.

Lurking or linking variables can cause events that are highly correlated to one another appear to have a casual relationship. This is because a third separate factor may be inducing change in the two variables.

Now, let's solve this problem. It asks us to describe the relationship in the scatterplot. We know that there is a positive relationship between the two variables; however, if we think critically we know that shark attacks and ice cream sales are independent of one another. The answers that suggest causality are incorrect. A linking or lurking variable—in this case warm temperatures—is causing change in both of the variables. In other words, warmer temperatures cause individuals to purchase ice cream and frequent the beach. Greater populations of beach goers increase the probability of shark attacks.

In order to properly solve this question, we need to understand the differences between what is meant by correlation and causation. A correlation refers to the strength of the linear association between two quantitative variables. On the other hand, causation indicates that the change in one variable is the cause of change in another.

Correlation can be used as an indicator of causal relationships; however, experimentation is needed to properly identify which variable is actually causing the observed change. Scientific experimentation identifies causality through he implementation of laboratory procedures in a controlled setting. When variables are controlled, causation can be determined through observation and repeated tests.

Several logical fallacies explain why correlation does not directly imply causation. First, cause-and-effect is not determined by two events occurring simultaneously. In other words, events that occur together do not necessarily cause one another. Second, causality is not determined by an event preceding another temporally. In other words, this means that event B is not always a consequence of event A simply because event A occurs before event B.

Lurking or linking variables can cause events that are highly correlated to one another appear to have a casual relationship. This is because a third separate factor may be inducing change in the two variables.

Now, let's solve this problem. It asks us to describe the relationship in the scatterplot. We know that there is a positive relationship between the two variables; however, if we think critically we know that beach attendance and shark attacks do not cause one another. The answers that suggest causality are incorrect. There are many factors that influence shark attacks on beaches—beach attendance is one of them. For example, if no one goes to the beach, then a shark located at the beach can attack no one. Increased beach attendance is positively correlated with shark attacks but further investigation is needed to determine if this causes the attacks. A mating cycle, global warming, or changes in food sources could all induce a shark attack. Beach attendance is only one factor correlated with this phenomenon.

When data are presented using a scatter plot that is fitted with a trend line, we can calculate estimations based on the association between variables. Several conditions must be met before one can use a scatter plot to make estimates off of correlations.

First, the points must possess some type of relationship between one another. This relationship can be positive or negative. Positive relationships occur when data move upwards from the left side to the right side of the graph; however, when the data slopes downward the relationship is negative. Second, we must identify if the relationship is strong or weak. A strong correlation exists when the data is clustered closer together and the trend line. On the other hand, a weak correlation occurs when data is spread apart from each other and the trend line. After we have evaluated these characteristics of the graph, we can use the scatter plot to make predictions.

How do we make predictions? Predictions are made using several methods: qualitative and quantitative observations. One can qualitatively use the associations present in the graph to make estimates based on spread, clustering, and the trend line’s position. Spread between points in a given area can provide a range of values for a given coordinate, while clustering of plots in a given area can give an average value for a given coordinate. On the other hand, the trend line can be used to estimate points by drawing lines that intersect from each axis. Last, we can quantitatively estimate a point using the equation of the trend line and solving for either the x or y variable.

Let’s work with these methods and solve the question.

First let’s observe if we can make a prediction using the following data:

First, let’s observe if we can make a prediction using the following data:

We can see that the data possesses a positive correlation: that is, as the runner’s weight increases then so does the time it takes them to run a mile. Also, we can see that the points are weakly to moderately clustered with one another and the trend line; therefore, we can use qualitative and quantitative means to estimate how fast a 185 pound male can run a mile.

Next, let’s make a qualitative estimate by drawing a line from 185 pounds on the x-axis to the trend line. Next, we will draw a line from that point to the y-axis. This will qualitatively estimate our value. Observe this method below:

From this information, we can make the prediction that a male of this weight should be able to run a mile in just under ten minutes.

Now, lets predict this value quantitatively using the equation of the line:

Plug in 185 for the x-coordinate.

Solve.

Round to two decimal places.

According to the data, we can predict that it would take a male 9.71 minutes for a 185 pound male to run a single mile.

Remember, we can also use this method to estimate the y-value of the data.

When data are presented using a scatter plot that is fitted with a trend line, we can calculate estimations based on the association between variables. Several conditions must be met before one can use a scatter plot to make estimates off of correlations.

First, the points must possess some type of relationship between one another. This relationship can be positive or negative. Positive relationships occur when data move upwards from the left side to the right side of the graph; however, when the data slopes downward the relationship is negative. Second, we must identify if the relationship is strong or weak. A strong correlation exists when the data is clustered closer together and the trend line. On the other hand, a weak correlation occurs when data is spread apart from each other and the trend line. After we have evaluated these characteristics of the graph, we can use the scatter plot to make predictions.

How do we make predictions? Predictions are made using several methods: qualitative and quantitative observations. One can qualitatively use the associations present in the graph to make estimates based on spread, clustering, and the trend line’s position. Spread between points in a given area can provide a range of values for a given coordinate, while clustering of plots in a given area can give an average value for a given coordinate. On the other hand, the trend line can be used to estimate points by drawing lines that intersect from each axis. Last, we can quantitatively estimate a point using the equation of the trend line and solving for either the x- or y-variable.

Let’s work with these methods and solve the question.

First let’s observe if we can make a prediction using the following data:

We can see that the data possesses a positive correlation: that is, as the runner’s weight increases then so does the time it takes them to run a mile. Also, we can see that the points are weakly to moderately clustered with one another and the trend line; therefore, we can use qualitative and quantitative means to estimate how fast a 185 pound male can run a mile.

Next, let’s make a qualitative estimate by drawing a line from 185 pounds on the x-axis to the trend line. Next, we will draw a line from that point to the y-axis. This will qualitatively estimate our value. Observe this method below:

From this information, we can make the prediction that a male of this weight should be able to run a mile in just under ten minutes.

Now, lets predict this value quantitatively using the equation of the line:

Plug in 185 for the x-coordinate.

Solve.

Round to two decimal places.

According to the data, we can predict that it would take a male 9.71 minutes for a 185 pound male to run a single mile.

Remember, we can also use this method to estimate the y-value of the data.

When data are presented using a scatter plot that is fitted with a trend line, we can calculate estimations based on the association between variables. Several conditions must be met before one can use a scatter plot to make estimates off of correlations.

First, the points must possess some type of relationship between one another. This relationship can be positive or negative. Positive relationships occur when data move upwards from the left side to the right side of the graph; however, when the data slopes downward the relationship is negative. Second, we must identify if the relationship is strong or weak. A strong correlation exists when the data is clustered closer together and the trend line. On the other hand, a weak correlation occurs when data is spread apart from each other and the trend line. After we have evaluated these characteristics of the graph, we can use the scatter plot to make predictions.

How do we make predictions? Predictions are made using several methods: qualitative and quantitative observations. One can qualitatively use the associations present in the graph to make estimates based on spread, clustering, and the trend line’s position. Spread between points in a given area can provide a range of values for a given coordinate, while clustering of plots in a given area can give an average value for a given coordinate. On the other hand, the trend line can be used to estimate points by drawing lines that intersect from each axis. Last, we can quantitatively estimate a point using the equation of the trend line and solving for either the x or y variable.

Let’s work with these methods and solve the question.

First let’s observe if we can make a prediction using the following data:

First, let’s observe if we can make a prediction using the following data:

We can see that the data possesses a positive correlation: that is, as the runner’s weight increases then so does the time it takes them to run a mile. Also, we can see that the points are weakly to moderately clustered with one another and the trend line; therefore, we can use qualitative and quantitative means to estimate how fast a 185 pound male can run a mile.

Next, let’s make a qualitative estimate by drawing a line from 185 pounds on the x-axis to the trend line. Next, we will draw a line from that point to the y-axis. This will qualitatively estimate our value. Observe this method below:

From this information, we can make the prediction that a male of this weight should be able to run a mile in just under ten minutes.

Now, lets predict this value quantitatively using the equation of the line:

Plug in 185 for the x-coordinate.

Solve.

Round to two decimal places.

According to the data, we can predict that it would take a male 9.71 minutes for a 185 pound male to run a single mile.

Remember, we can also use this method to estimate the y-value of the data.

When data are presented using a scatter plot that is fitted with a trend line, we can calculate estimations based on the association between variables. Several conditions must be met before one can use a scatter plot to make estimates off of correlations.

First, the points must possess some type of relationship between one another. This relationship can be positive or negative. Positive relationships occur when data move upwards from the left side to the right side of the graph; however, when the data slopes downward the relationship is negative. Second, we must identify if the relationship is strong or weak. A strong correlation exists when the data is clustered closer together and the trend line. On the other hand, a weak correlation occurs when data is spread apart from each other and the trend line. After we have evaluated these characteristics of the graph, we can use the scatter plot to make predictions.

How do we make predictions? Predictions are made using several methods: qualitative and quantitative observations. One can qualitatively use the associations present in the graph to make estimates based on spread, clustering, and the trend line’s position. Spread between points in a given area can provide a range of values for a given coordinate, while clustering of plots in a given area can give an average value for a given coordinate. On the other hand, the trend line can be used to estimate points by drawing lines that intersect from each axis. Last, we can quantitatively estimate a point using the equation of the trend line and solving for either the x- or y-variable.

Let’s work with these methods and solve the question.

First let’s observe if we can make a prediction using the following data:

We can see that the data possesses a positive correlation: that is, as the runner’s weight increases then so does the time it takes them to run a mile. Also, we can see that the points are weakly to moderately clustered with one another and the trend line; therefore, we can use qualitative and quantitative means to estimate how fast a 188 pound male can run a mile.

Next, let’s make a qualitative estimate by drawing a line from 188 pounds on the x-axis to the trend line. Next, we will draw a line from that point to the y-axis. This will qualitatively estimate our value. Observe this method below:

Now, lets predict this value quantitatively using the equation of the line:

Plug in 188 for the x-coordinate.

Solve

Round to two decimal places.

According to the data, we can predict that it would take a male 8.79 minutes for a 188 pound male to run a single mile.

Remember, we can also use this method to estimate the y-value of the data.