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.

d. The 50 adults all drove cars from Jacksonville, Florida, to Richmond, Virginia. Their average (mean) speeds are recorded along with the times it took to complete that trip.

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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 the average speeds of the cars and the times it took to complete the trip.

Recall the properties of the correlation coefficient (r): It ranges from -1 to 1. A value close to -1 indicates a strong negative linear relationship, a value close to 1 indicates a strong positive linear relationship, and a value near 0 indicates no linear relationship.

Analyze the relationship between the variables: As the average speed increases, the time taken to complete the trip is expected to decrease. This suggests a negative linear relationship between speed and time.

Visualize the data: If possible, create a scatterplot of the data points with average speed on the x-axis and time on the y-axis. Look for a downward trend in the points, which would confirm a negative correlation.

Estimate the value of r: Based on the observed trend in the scatterplot or the relationship described, estimate r as a negative value closer to -1 if the points form a strong linear pattern, or closer to 0 if the pattern is weak or scattered.

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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 changes in one variable may relate to changes in another.

Mean Speed and Time Relationship

In the context of the given data, the mean speed of the cars and the time taken for the trip are two variables that can be analyzed for correlation. Typically, as speed increases, the time taken to complete a trip decreases, suggesting a negative correlation. Analyzing this relationship helps in understanding how these two factors interact in real-world scenarios.

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 is essential for obtaining unbiased data, which enhances the validity of statistical analyses. In this case, the 50 randomly selected adults provide a representative sample for estimating the correlation between speed and time.

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