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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b. Based on the result from part (a), what do you conclude about a linear correlation between time and height?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine whether there is a linear correlation between time and height based on the data provided. Linear correlation measures the strength and direction of a linear relationship between two variables.
2. Step 2: Review the data. Examine the table of time (independent variable) and height (dependent variable). Ensure the data is complete and ready for analysis.
3. Step 3: Calculate the correlation coefficient (r). Use the formula for Pearson's correlation coefficient: $r=\frac{\sum (x-x¯)\bullet (y-y¯)}{\sqrt{\sum (x-x¯{)}^2}\bullet \sqrt{\sum (y-y¯{)}^2}}$, where x and y are the variables, and x̄ and ȳ are their respective means.
4. Step 4: Interpret the correlation coefficient. If r is close to 1 or -1, there is a strong linear correlation. If r is close to 0, there is little to no linear correlation. Positive r indicates a positive relationship, while negative r indicates a negative relationship.
5. Step 5: Draw a conclusion. Based on the calculated r value, determine whether the data supports a linear correlation between time and height. Consider the context of the experiment and whether the relationship aligns with expectations from physics (e.g., parabolic motion).

Key Concepts

Linear Correlation

Linear correlation refers to the relationship between two variables where a change in one variable is associated with a proportional change in another. This relationship can be quantified using the correlation coefficient, which ranges from -1 to 1. A value close to 1 indicates a strong positive correlation, while a value close to -1 indicates a strong negative correlation. A value around 0 suggests no linear correlation.

Scatter Plot

A scatter plot is a graphical representation of two variables, where each point represents an observation in the dataset. It helps visualize the relationship between the variables, making it easier to identify patterns, trends, or correlations. In the context of the soccer ball experiment, plotting time against height can reveal whether a linear relationship exists between these two variables.

Regression Analysis

Regression analysis is a statistical method used to determine the relationship between a dependent variable and one or more independent variables. In this case, it can help quantify how height (dependent variable) changes with time (independent variable). By fitting a regression line to the data, one can assess the strength and nature of the correlation, providing insights into the dynamics of the soccer ball's motion.