Which of the following statements about correlation is true? Choose the correct answer below.
A. We say that there is a positive correlation between x and y if there is no distinct pattern in the scatterplot.
B. We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values increase.
C. We say that there is a negative correlation between x and y if the x-values increase as the corresponding y-values increase.
D. We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values decreas
The Correct Answer and Explanation is:
The correct answer is B. We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values increase.
Explanation:
Correlation refers to the relationship between two variables, typically represented as x and y, and how changes in one variable relate to changes in the other. The correlation can be positive, negative, or there can be no correlation at all. Understanding the different types of correlation helps in interpreting the relationship between the variables. Let’s break down each of the choices:
- A. “We say that there is a positive correlation between x and y if there is no distinct pattern in the scatterplot.”
- This statement is incorrect. If there is no distinct pattern in the scatterplot, it indicates no correlation or a very weak correlation between the variables. A positive correlation implies that as x increases, y also increases, and this would be visible in the scatterplot as a clear upward trend.
- B. “We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values increase.”
- This statement is correct. A positive correlation means that as the x-values increase, the corresponding y-values also increase. In a scatterplot, this would be represented by points that tend to form an upward trend. A real-world example could be the relationship between hours studied and test scores: generally, as study time (x) increases, test scores (y) also tend to increase, indicating a positive correlation.
- C. “We say that there is a negative correlation between x and y if the x-values increase as the corresponding y-values increase.”
- This statement is incorrect. A negative correlation means that as x increases, y decreases. If the x-values increase and the corresponding y-values increase, it is a positive correlation, not negative. For a negative correlation, you would expect the scatterplot to show a downward trend.
- D. “We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values decrease.”
- This statement is incorrect. If x-values increase and y-values decrease, this indicates a negative correlation, not a positive one. The scatterplot for this would show a downward trend, meaning as x increases, y decreases.
Summary:
A positive correlation is observed when both variables move in the same direction, i.e., as x increases, y also increases. This is visually represented in a scatterplot where the data points generally rise from left to right. Conversely, a negative correlation occurs when one variable increases while the other decreases. Thus, option B is the correct answer, as it accurately describes a positive correlation between x and y.