In Exercise 26, add data for an international soccer player who can perform the half squat with a maximum of 210 kilograms and can sprint 10 meters in 2.00 seconds. Describe how this affects the correlation coefficient r.

Verified step by step guidance

1. Understand the problem: The task involves adding a new data point to an existing dataset and analyzing how this new data point affects the correlation coefficient (r). The correlation coefficient measures the strength and direction of the linear relationship between two variables—in this case, maximum squat weight and sprint time.
2. Identify the variables: The two variables are (1) maximum squat weight (in kilograms) and (2) sprint time (in seconds). The new data point is (210, 2.00).
3. Determine the impact of the new data point: Consider whether the new data point aligns with the existing trend in the dataset. If the new data point strengthens the linear relationship, the magnitude of r will increase (closer to 1 or -1). If it weakens the relationship, the magnitude of r will decrease (closer to 0).
4. Recalculate the correlation coefficient: Use the formula for the Pearson correlation coefficient: $r=\frac{\sum (x-x¯)(y-y¯)}{\sqrt{\sum (x-x¯{)}^2}\sqrt{\sum (y-y¯{)}^2}}$. Here, x represents squat weight, y represents sprint time, and x̄ and ȳ are their respective means. Add the new data point to the dataset and compute the updated r.
5. Interpret the result: After recalculating r, compare it to the original value. If the new data point is consistent with the existing trend, r will likely increase in magnitude. If it deviates significantly from the trend, r will decrease in magnitude. This interpretation helps understand the influence of the new data point on the relationship between squat weight and sprint time.

Key Concepts

Correlation Coefficient (r)

The correlation coefficient, denoted as r, quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation. Understanding r is crucial for interpreting how changes in one variable may relate to changes in another, such as the relationship between an athlete's strength and speed.

Half Squat Performance

The half squat is a strength training exercise that targets the lower body, particularly the quadriceps, hamstrings, and glutes. An athlete's ability to perform a half squat with a maximum weight, such as 210 kilograms, can indicate their overall strength and power. This performance metric can be correlated with other athletic abilities, such as sprinting speed, to assess how strength influences speed in sports.

Sprinting Speed

Sprinting speed refers to the time it takes an athlete to cover a specific distance, in this case, 10 meters in 2.00 seconds. This measure is critical in sports performance, as it reflects an athlete's explosive power and acceleration. Analyzing the relationship between sprinting speed and strength metrics, like half squat performance, can provide insights into how these physical attributes interact and contribute to overall athletic performance.