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Chapter 4: Problem 19

What are all the values that a correlation \(r\) can possibly take? (a) \(r \geq 0\) (b) \(0 \leq r \leq 1\) (c) \(-1 \leq r \leq 1\)

Short Answer

Expert verified
Option (c) \(-1 \leq r \leq 1\) is correct.

Step by step solution

01

Understanding Correlation

The correlation coefficient, denoted by \(r\), is a statistical measure that describes the strength and direction of a linear relationship between two variables. It is scaled to be between -1 and 1.
02

Exploring Possible Values

The correlation coefficient can take any value from -1 to 1, inclusive. This range allows for various relationships: -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.
03

Analyzing Given Options

We analyze each option to see which describes the possible values of \(r\). Option (a) states \(r \geq 0\), suggesting \(r\) can only be positive, which is incorrect as \(r\) can be negative. Option (b) proposes \(0 \leq r \leq 1\), implying only non-negative values, which is partially correct but incomplete because \(r\) can also be negative. Option (c) \(-1 \leq r \leq 1\) accurately reflects the full range of values \(r\) can take.

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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 between two variables means that as one variable changes, the other variable tends to change in a proportional manner. This kind of relationship can be represented graphically as a straight line on a scatter plot. The line can slope upwards or downwards, indicating whether there's a positive or negative correlation, respectively.

In statistics, the correlation coefficient is used to measure the direction and strength of this linear relationship. The closer the correlation coefficient ( ")) is to 1 or -1, the stronger the linear relationship; the closer it is to 0, the weaker the relationship. When you look at two variables and they seem to increase together, or one increases while the other decreases, you might describe the relationship as linear if the pattern closely resembles a straight line.
Negative Correlation
Negative correlation exists when two variables move in opposite directions. As one variable increases, the other decreases, and vice versa. In terms of the correlation coefficient, a negative correlation will have an r-value between -1 and 0.

- An r-value of -1 indicates a perfect negative linear relationship. This means that every increase in one variable is met with a proportional decrease in the other. - An r-value closer to 0, but still negative, suggests a weaker negative relationship.

You can visualize a negative correlation as a downward-sloping line when plotting two variables on a graph. Negative correlation is found in various real-world data scenarios, such as the relationship between sea level pressure and altitude.
Positive Correlation
A positive correlation indicates that two variables tend to move in the same direction. This means that as one variable increases, the other variable also increases, and as one decreases, the other does too.

- The value of the correlation coefficient (r) for a positive correlation ranges from 0 to 1. - An r-value of 1 reflects a perfect positive linear relationship, meaning that the data points lie exactly on a line with an upward slope. - If the r-value is closer to 0, it suggests a weaker positive linear relationship.

You can identify positive correlation on a scatter plot by an upward trend in the data points. One common example of positive correlation is the relationship between height and weight in humans; generally, as height increases, so does weight.

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Most popular questions from this chapter

Toucan's beak . The toco toucan, the largest member of the toucan family, possesses the largest beak relative to body size of all birds. This exaggerated feature has received various interpretations, such as being a refined adaptation for feeding. However, the large surface area may also be an important mechanism for radiating heat (and hence cooling the bird) as outdoor temperature increases. Here are data for beak heat loss, as a percent of total body heat loss, at various temperatures in degrees Celsius: \({ }^{25}\) III ToucAN $$ \begin{array}{l|cccccccc} \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 \\ \text { Percent heat loss from beak } & 32 & 34 & 35 & 33 & 37 & 46 & 55 & 51 \\\ \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \text { Percent heat loss from beak } & 43 & 52 & 45 & 53 & 58 & 60 & 62 & 62 \\\ \hline \end{array} $$ Investigate the relationship between outdoor temperature and beak heat loss, as a percentage of total body heat loss.

Explanatory and Response Variables? You have data on a large group of college students. Here are four pairs of variables measured on these students. For each pair, is it more reasonable to simply explore the relationship between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? In the latter case, which is the explanatory variable, and which is the response variable? (a) Number of lectures attended in your statistics course and grade on the final exam for the course (b) Number of hours per week spent exercising and calories burned per week (c) Hours per week spent online using Facebook and grade point average (d) Hours per week spent online using Facebook and IQ

Thinking about correlation. Exercise \(4.27\) presents data on wine intake and the relative risk of breast cancer in women. (a) If wine intake is measured in ounces of alcohol per day rather than grams per day, how would the correlation change? (There is \(0.035\) ounce in a gram.) (b) How would \(r\) change if all the relative risks were \(0.25\) less than the values given in the table? Does the correlation tell us that among women who drink, those who drink more wine tend to have a greater relative risk of cancer than women who don't drink at all? (c) If drinking an additional gram of alcohol each day raised the relative risk of breast cancer by exactly \(0.01\), what would be the correlation between alcohol in wine intake and relative risk of breast cancer? (Hint: Draw a

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Sloppy writing about correlation. Each of the following statements contains a blunder. Explain in each case what is wrong. (a) "There is a high correlation between the sex of American workers and their income." (b) "We found a high correlation \((r=1.09)\) between students' ratings of faculty teaching and ratings made by other faculty members." (c) "The correlation between height and weight of the subjects was \(r=0.63\) centimeter per kilogram."
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