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Chapter 4: Problem 2

If the linear correlation between two variables is negative, what can be said about the slope of the regression line?

Short Answer

Expert verified
The slope of the regression line is negative.

Step by step solution

01

Understanding Correlation

The correlation between two variables quantifies the strength and direction of their linear relationship. A negative linear correlation means that as one variable increases, the other variable tends to decrease.
02

Interpreting Negative Correlation

If the correlation is negative, the regression line, which best fits the data points, will have a negative slope. This implies that the line will go downwards from left to right.
03

Relationship Between Correlation and Slope

The slope of the regression line represents the rate of change of the dependent variable with respect to the independent variable. A negative correlation indicates that this rate of change is negative, meaning the slope is less than zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear relationship
Understanding a linear relationship is crucial in statistics. It means that there is a straight-line association between two variables. In other words, if you plot the data points of these variables, you could draw a straight line close to them. This line can either go upwards or downwards. If the values of one variable increase as the values of the other variable also increase, you have a positive linear relationship. If one value increases while the other decreases, you have a negative linear relationship. Knowing this helps in predicting and analyzing how changes in one variable may impact the other.
regression line
The regression line is a crucial concept in understanding statistical models. This line, also known as the 'line of best fit,' shows the direction and strength of the relationship between two variables. You get this line by using a statistical method that minimizes the distance between the actual data points and the predicted values on the line. When plotted on a graph, this line helps in making predictions. For example, if you have data on study hours and exam scores, the regression line can help predict scores based on hours studied. It's a visual and mathematical way to represent the connection between two variables.
slope of the regression line
The slope of the regression line is a single number that measures how steep the line is. It indicates how many units the dependent variable (y) changes for a one-unit increase in the independent variable (x). In a simple formula, if the regression line is y = mx + b, the slope (m) is the value that multiplies x. A positive slope means that as x increases, y also increases. A negative slope means as x increases, y decreases. For example, if the slope is -2, for every one unit increase in x, y decreases by 2 units. Understanding the slope helps in understanding the rate and direction of the change.

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Most popular questions from this chapter

True or False: The least-squares regression line always travels through the point \((\bar{x}, \bar{y})\)

What does it mean to say two variables are positively associated? Negatively associated?

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{llllll} \hline x & 0 & 5 & 7 & 8 & 9 \\ \hline y & 3 & 8 & 6 & 9 & 4 \end{array} $$

Explain the difference between correlation and causation. When is it appropriate to state that the correlation implies causation?

In Problems \(17-20,\) (a) draw a scatter diagram of the data,(b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{llllll} x & 2 & 6 & 6 & 7 & 9 \\ \hline y & 8 & 7 & 6 & 9 & 5 \end{array} $$
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