Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.
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a. Find the value of the linear correlation coefficient r.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to calculate the linear correlation coefficient (r), which measures the strength and direction of the linear relationship between two variables: time (x) and height (y).
2. Step 2: Use the formula for the linear correlation coefficient: r = (nΣ(xy) - ΣxΣy) / sqrt([(nΣ(x^2) - (Σx)^2)][(nΣ(y^2) - (Σy)^2)]). Here, n is the number of data points, Σ(xy) is the sum of the product of x and y, Σx and Σy are the sums of x and y respectively, Σ(x^2) is the sum of the squares of x, and Σ(y^2) is the sum of the squares of y.
3. Step 3: Organize the data into a table with columns for x (time), y (height), x^2, y^2, and xy. Compute the values for each column and sum them up to find Σx, Σy, Σ(x^2), Σ(y^2), and Σ(xy).
4. Step 4: Substitute the computed sums and the number of data points (n) into the formula for r. Simplify the numerator and denominator separately before dividing.
5. Step 5: Interpret the value of r. If r is close to 1 or -1, it indicates a strong linear relationship. If r is close to 0, it indicates a weak or no linear relationship.

Key Concepts

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 signifies no correlation. Understanding how to calculate and interpret r is essential for analyzing the relationship between the time elapsed and the height of the soccer ball in the given experiment.

Data Representation

Data representation involves organizing and displaying data in a way that makes it easier to analyze and interpret. In this context, the height of the soccer ball over time can be represented in a table or graph, which helps visualize the relationship between the two variables. Proper data representation is crucial for identifying trends and patterns that inform the calculation of the correlation coefficient.

Statistical Significance

Statistical significance refers to the likelihood that a relationship observed in data is not due to random chance. In the context of calculating the correlation coefficient, it is important to assess whether the correlation found is statistically significant, which can be determined through hypothesis testing. This concept helps in making informed conclusions about the relationship between the time and height of the soccer ball in the experiment.