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.

c. Their pulse rates are measured and their IQ scores are measured .

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Understand the problem: The goal is to estimate the linear correlation coefficient (r), which measures the strength and direction of the linear relationship between two variables. Here, the variables are pulse rates and IQ scores for 50 adults.

Visualize the data: Create a scatterplot of the paired data (pulse rates on one axis and IQ scores on the other). This will help you observe the pattern of the relationship (e.g., positive, negative, or no correlation).

Assess the direction of the relationship: Look at the scatterplot to determine if the points trend upward (positive correlation), downward (negative correlation), or show no clear trend (no correlation).

Evaluate the strength of the relationship: Observe how closely the points cluster around a straight line. If the points are tightly clustered, the correlation is strong. If they are widely scattered, the correlation is weak.

Estimate the value of r: Based on the scatterplot, assign an approximate value to r. Recall that r ranges from -1 to 1, where values close to -1 or 1 indicate strong negative or positive correlation, respectively, and values near 0 indicate no correlation.

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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 the relationship between pulse rates and IQ scores in the given data.

Paired Data

Paired data refers to two sets of related observations, where each pair consists of one measurement from each variable. In this context, each adult's pulse rate is paired with their corresponding IQ score. Analyzing paired data allows for the examination of potential relationships or correlations between the two variables, which is essential for estimating r.

Random Sampling

Random sampling is a technique used to select a subset of individuals from a larger population, ensuring that every individual has an equal chance of being chosen. This method helps to eliminate bias and allows for generalization of results. In this question, the pulse rates and IQ scores are obtained from 50 randomly selected adults, which enhances the validity of the correlation analysis.

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