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Chapter 13: Problem 65

For a sample data set on two variables, the value of the linear correlation coefficient is (close to) zero. Does this mean that these variables are not related? Explain.

Short Answer

Expert verified
No, a linear correlation coefficient close to zero does not mean these variables are not related at all. It merely indicates there is no linear relationship between the two. There could be a non-linear relationship between the variables.

Step by step solution

01

Understand the Concept of Correlation Coefficients

Correlation Coefficients provide a measure of the strength and direction of a linear relationship between two variables. Correlation coefficients range between -1 and 1. A correlation of 1 means a perfect positive linear relation, -1 means a perfect negative linear relation, and 0 means no linear relation.
02

Interpret the Zero Correlation

If the correlation coefficient is close to zero, it suggests that there is no linear relationship between the two variables. In other words, changes in one variable do not coincide with changes in the other variable in a systematic way that can be described with a straight line.
03

Differentiate between No Correlation and Non-linear Correlation

A correlation close to zero does not completely exclude the possibility of a relationship between the two variables. The relationship, if it exists, could be of a non-linear nature, meaning it cannot be described with a straight line. Examples of non-linear relationships include quadratic or exponential relationships.

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Most popular questions from this chapter

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. $$ \begin{array}{l|rrrrrrrrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the amount of sugar and the number of calories per serving? b. Find the predictive regression equation of the number of calories on the amount of sugar. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the predictive regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Calculate the predicted calorie count for a cereal with 16 grams of sugar per serving. f. Estimate the calorie count for a cereal with 52 grams of sugar per serving. Comment on this finding.

Explain the least squares method and least squares regression line. Why are they called by these names?

A researcher took a sample of 25 electronics companies and found the following relationship between \(x\) and \(y\), where \(x\) is the amount of money (in millions of dollars) spent on advertising by a company in 2011 and \(y\) represents the total gross sales (in millions of dollars) of that company for 2011 . $$ \hat{y}=3.6+11.75 x $$ a. An electronics company spent $$\$ 2$$ million on advertising in 2011 . What are its expected gross sales for 2011 ? b. Suppose four electronics companies spent $$\$ 2$$ million each on advertising in \(2011 .\) Do you expect these four companies to have the same actual gross sales for 2011 ? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

The following table, reproduced from Exercise \(13.28\), lists the percentages of space for eight magazines that contain advertisements and the prices of these magazines. $$ \begin{array}{l|rrrrrrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 & 70 & 28 & 65 & 32 \\ \hline \text { Price }(\$) & 5.50 & 6.95 & 4.95 & 5.75 & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x $$ a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?
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