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Chapter 13: Problem 48

If the sample correlation coefficient is equal to 1 , is it necessarily true that \(\rho=1\) ? If \(\rho=1\), is it necessarily true that \(r=1\) ?

Short Answer

Expert verified
No, if \(r=1\), it doesn't necessarily mean that \(\rho=1\). Likewise, if \(\rho=1\), it doesn't automatically mean that \(r=1\). This is due to potential sample variance.

Step by step solution

01

Understanding the Concepts

The sample correlation coefficient (\(r\)) measures the strength and direction of the linear relationship between two variables for a sample from the population. The population correlation (\(\rho\)), measures the same for the whole population. While both are correspondent, they are not always equal due to sample variance.
02

Relationship when \(r=1\)

If \(r=1\), it indicates a perfect linear relationship among the variables in the sample. However, it doesn't necessarily mean that \(\rho=1\) for the population. This is due to the sample variance which may not fully represent the population's properties. Therefore, \(\rho\) could be less than 1.
03

Relationship when \(\rho=1\)

Conversely, if \(\rho=1\), it indicates a perfect linear relationship among variables in the entire population. In this case, \(r\) could indeed be 1 if the sample perfectly aligns with the population model and there is no sample variance. However, due to the presence of sample variance, \(r\) might be less than 1, consistent with the idea of a slightly weaker relationship in the sample.

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Key Concepts

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

Sample Correlation
A sample correlation coefficient, often denoted as \(r\), plays a crucial role in statistics for understanding relationships between two variables in a sample. The term "sample" refers to a subset of data taken from a larger population. The sample correlation coefficient measures the strength and direction of a linear relationship for these specific data points.

Key points about sample correlation:
  • Value Range: \(r\) always varies between -1 and +1.
  • Interpretation: An \(r\) of +1 indicates a perfect positive linear relationship, while an \(r\) of -1 indicates a perfect negative linear relationship.
  • Direction: If \(r\) is positive, as one variable increases, so does the other. If \(r\) is negative, one variable increases while the other decreases.
It is important to understand that the sample correlation coefficient is specific to the sample and may not always reflect the overall correlation in the population due to sample variability. This means when \(r=1\), it implies a perfect linear relationship in the sample but might not necessarily indicate the same for the entire population.
Population Correlation
The population correlation coefficient, symbolized by \(\rho\), provides insight into how two variables are related across an entire population. Unlike the sample correlation, which only involves a sample, the population correlation aims to capture the true correlation for every individual in the overarching group.

Significant aspects of population correlation include:
  • True Measure: \(\rho\) is considered the true measure of linear relationship, devoid of the variability inherent in samples.
  • Fixed Value: Unlike \(r\), the value of \(\rho\) does not vary when selecting different samples from the same population, assuming no sampling error.
  • Relationship to \(r\): The sample correlation \(r\) estimates \(\rho\) but might not be equal due to sample-specific fluctuations.
Thus, if \(\rho=1\), it unambiguously signifies a perfect linear relationship across the entire population. This doesn't automatically mean any given sample will exhibit the same correlation due to sample variance, which can cause \(r\) to deviate from \(\rho\).
Linear Relationship
Understanding a linear relationship is pivotal to grasping how correlation coefficients function. A linear relationship between two variables suggests that when plotted on a graph, they form a straight line. This implies that the change in one variable consistently corresponds to a proportionate change in the other.

Characteristics of a linear relationship include:
  • Predictability: A constant rate of change makes predictions about one variable possible when the other is known.
  • Correlation Coefficient: Positive values of correlation suggest a positive linear relationship, whereas negative values indicate a negative linear relationship.
  • Strength: The closer the correlation coefficient (either \(r\) or \(\rho\)) is to 1 or -1, the stronger the linear relationship.
These relationships are essential for interpreting both \(r\) and \(\rho\), as a stronger linear relationship provides more accurate predictions and insights into the data structure. When both \(r\) and \(\rho\) approach 1, they signify nearly perfect predictability within the dataset as a whole.

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Most popular questions from this chapter

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A random sample of \(n=347\) students was selected, and each one was asked to complete several questionnaires, from which a Coping Humor Scale value \(x\) and a Depression Scale value \(y\) were determined ("Depression and Sense of Humor." (Psychological Reports \([1994]: 1473-1474)\). The resulting value of the sample correlation coefficient was -.18 . a. The investigators reported that \(P\) -value \(<.05 .\) Do you agree? b. Is the sign of \(r\) consistent with your intuition? Explain. (Higher scale values correspond to more developed sense of humor and greater extent of depression.) c. Would the simple linear regression model give accurate predictions? Why or why not?

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