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Chapter 4: Problem 52

Write a paragraph that explains the concept of correlation. Include a discussion of the role that \(x_{i}-\bar{x}\) and \(y_{i}-\bar{y}\) play in the computation.

Short Answer

Expert verified
Correlation measures the linear relationship between two variables. \(x_i - \bar{x}\) and \(y_i - \bar{y}\) show how each value deviates from their means, helping compute the correlation coefficient.

Step by step solution

01

Define Correlation

Correlation is a statistical measure that describes the extent to which two variables are linearly related. It indicates how a change in one variable is associated with a change in another variable.
02

Explain the Role of \(x_{i}-\bar{x}\)

In correlation computation, \(x_i - \bar{x}\) represents the deviation of each value of the variable x from its mean \(\bar{x}\). This measurement helps in understanding how each individual data point x differs from the average value of x.
03

Explain the Role of \(y_{i}-\bar{y}\)

Similarly, \(y_i - \bar{y}\) represents the deviation of each value of the variable y from its mean \(\bar{y}\). This helps in understanding how each individual data point y differs from the average value of y.
04

Relate Deviations to Correlation

The product of these deviations for corresponding pairs of x and y values, \( (x_i - \bar{x})(y_i - \bar{y}) \), is computed to examine how x and y move in relation to each other. Summing these products over all data points and normalizing it gives the correlation coefficient.
05

Summarize Correlation Coefficient

The correlation coefficient, often denoted as r, ranges from -1 to 1. A value close to 1 indicates a strong positive linear relationship, a value close to -1 indicates a strong negative linear relationship, and a value around 0 indicates little to no linear relationship.

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Key Concepts

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

Linear Relationship
A linear relationship describes a situation in which two variables are connected in such a way that their values change in a proportional manner. If you were to plot these variables on a graph, they would form a straight line. This means that any change in one variable will consistently result in a change in the other variable. For instance, if studying harder leads to higher grades, a linear relationship exists between study time and grades.
Linear relationships are crucial in understanding how two variables interact and are the foundation of correlation. In a linear relationship, the degree of change from one variable directly affects the degree of change in the other.
Deviation from Mean
Deviation from the mean measures how far a particular value is from the average of all values. For a dataset of variable x, the mean is denoted by \(\bar{x}\). Each value in the dataset is represented by \(x_i\). The deviation of a value from the mean is calculated as \(x_i - \bar{x}\). Similarly, for variable y, the deviation is \(y_i - \bar{y}\).
These deviations are fundamental in computing correlation because they show us how individual points differ from the average and help us understand the data's distribution and variability. In a way, these deviations are the building blocks that make up the correlation coefficient.
Correlation Coefficient
The correlation coefficient, often represented as \(r\), quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. Here鈥檚 what the values indicate:
  • \(r\) close to 1: Strong positive linear relationship
  • \(r\) close to -1: Strong negative linear relationship
  • \(r\) around 0: Little to no linear relationship
To compute \(r\), we use the formula:
\[ r = \frac{\frac{1}{n-1} \times \text{Sum}((x_i - \bar{x})(y_i - \bar{y}))}{\text{Standard Deviation of } x \times \text{Standard Deviation of } y} \]
This formula combines all deviations of x and y, multiplies them for each data point, sums them up, and divides by the standard deviations of x and y, giving a normalized measure of correlation.
Positive and Negative Correlation
Correlation can either be positive or negative. A positive correlation means that as one variable increases, the other variable also increases. For example, the more hours you study, the higher your test scores might be. In this case, the correlation coefficient \(r\) will be positive.
A negative correlation means that as one variable increases, the other variable decreases. For instance, if you spend more time on social media, your productivity might decrease. Here, the correlation coefficient \(r\) will be negative.
Understanding these correlations helps identify the type of relationship between variables, making it easier to predict future behavior based on existing data.

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Most popular questions from this chapter

Explain what is wrong with the following statement: "We have concluded that a high correlation exists between the gender of drivers and rates of automobile accidents" Suggest a better way to write the sentence.

The General Social Survey asks questions about one's happiness and health. One would think that health plays a role in one's happiness. Use the data in the table to determine whether healthier people tend to also be happier. Treat level of health as the explanatory variable. $$\begin{array}{lrcccc} & \text { Poor } & \text { Fair } & \text { Good } & \text { Excellent } & \text { Total } \\\\\hline \text { Not too happy } & 696 & 1,386 & 1,629 & 732 & 4,443 \\\\\hline \text { Pretty happy } & 950 & 3,817 & 9,642 & 5,195 & 19,604 \\\\\hline \text { Very happy } & 350 & 1,382 & 4,520 & 5,095 & 11,347 \\ \hline \text { Total } & 1.996 & 6.585 & 15.791 & 11.022 & 35.394\end{array}$$

32\. Putting It Together: Exam Scores The data in the next column represent scores earned by students in Sullivan's Elementary Algebra course for Chapter 2 (Linear Equations and Inequalities in One Variable) and Chapter 3 (Linear Equations and Inequalities in Two Variables). Completely summarize the relation between Chapter 2 and Chapter 3 exam scores, treating Chapter 2 exam scores as the explanatory variable. Write a report detailing the results of the analysis including the presence of any outliers or influential points. What does the relationship say about the role Chapter 2 plays in a student's understanding of Chapter \(3 ?\) $$ \begin{array}{cc} \text { Chapter 2 Score } & \text { Chapter 3 Score } \\ \hline 71.4 & 76.1 \\ \hline 76.2 & 82.6 \\ \hline 60.3 & 60.9 \\ \hline 88.1 & 91.3 \\ \hline 95.2 & 78.3 \\ \hline 82.5 & 100 \\ \hline 100 & 95.7 \\ \hline 87.3 & 81.5 \\ \hline 71.4 & 50.7 \\ \hline 95.2 & 81.5 \\ \hline 95.2 & 87.0 \\ \hline 85.7 & 73.9 \\ \hline 71.4 & 74.6 \\ \hline 33.3 & 45.7 \\ \hline 78.0 & 37.3 \\ \hline 83.3 & 88.0 \\ \hline 100 & 100 \\ \hline 81.0 & 76.1 \\ \hline 76.2 & 63.0 \end{array} $$

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

Suppose that two variables, \(X\) and \(Y\), are negatively associated. Does this mean that above-average values of \(X\) will always be associated with below- average values of \(Y ?\) Explain.
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