Estimating r For each of the following, estimate the value of the linear correlation coefficient r for the given paired data obtained from 50 randomly selected adults.

a. Their heights are measured in inches and those same heights are recorded in centimeters .

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Understand the problem: The goal is to estimate the linear correlation coefficient (r) for the paired data. The data consists of heights measured in inches and the same heights recorded in centimeters. Since these two variables are directly proportional, we expect a strong positive linear relationship.

Recall the formula for the linear correlation coefficient (r): r measures the strength and direction of the linear relationship between two variables. It is calculated as:

r = \frac{\sum (x - x ¯) (y - y ¯)}{\sqrt{\sum (x - x ¯)^{2}} \sqrt{\sum (y - y ¯)^{2}}}

, where x and y are the paired data points, and x̄ and ȳ are their respective means.

Analyze the relationship: Since the heights in inches and centimeters are directly proportional (1 inch = 2.54 cm), the scatterplot of the data would form a perfect straight line with a positive slope. This indicates a very strong positive correlation.

Estimate the value of r: Based on the direct proportionality and the perfect linear relationship, the value of r is expected to be very close to 1. This is because r = 1 represents a perfect positive linear correlation.

Conclude: Without performing calculations, we can confidently estimate that the linear correlation coefficient r for this data is approximately 1, given the direct proportionality between the two variables.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Correlation Coefficient (r)

The linear correlation coefficient, denoted as r, quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. Understanding r is crucial for interpreting how closely two sets of data are related.

Units of Measurement

Units of measurement refer to the standard quantities used to express physical quantities, such as inches and centimeters for height. When analyzing data, it is important to recognize that while the numerical values may differ due to unit conversion, the underlying relationship between the variables remains unchanged. This concept is essential for correctly interpreting correlation in datasets measured in different units.

Paired Data

Paired data consists of two related sets of observations, where each pair corresponds to a single entity or subject. In this context, the heights of individuals measured in inches and centimeters represent paired data. Analyzing paired data allows for the assessment of relationships between two variables, making it fundamental for calculating correlation coefficients like r.

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