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An investigation was carried out to study the relationship between speed (ft/sec) and stride rate (number of steps taken/sec) among female marathon runners. Resulting summary quantities included \(n=11, \quad \sum\) (speed) \(=205.4\), \(\sum(\text { speed })^{2}=3880.08, \sum(\) rate \()=35.16, \sum(\text { rate })^{2}=112.681\), and \(\sum(\) speed \()(\) rate \()=660.130\). a. Calculate the equation of the least squares line that you would use to predict stride rate from speed. b. Calculate the equation of the least squares line that you would use to predict speed from stride rate. c. Calculate the coefficient of determination for the regression of stride rate on speed of part (a) and for the regression of speed on stride rate of part (b). How are these related?

Step by step solution

01

Understanding the Problem

We need to calculate two least squares lines to predict stride rate from speed and vice versa, and also calculate the coefficients of determination for these regressions. The given summary quantities will help us in these calculations.

02

Calculate the Means

First, calculate the mean of speed and stride rate. The mean of speed is given by \( \bar{x} = \frac{\sum \text{(speed)}}{n} = \frac{205.4}{11} = 18.6727 \) and the mean of stride rate is \( \bar{y} = \frac{\sum \text{(rate)}}{n} = \frac{35.16}{11} = 3.1964 \).

03

Calculate the Slope for Predicting Stride Rate

To find the slope \( b_1 \) of the regression line predicting stride rate from speed, use the formula \( b_1 = \frac{\sum(xy) - n\bar{x}\bar{y}}{\sum(x^2) - n\bar{x}^2} \). Substituting the values, \( b_1 = \frac{660.130 - 11 \times 18.6727 \times 3.1964}{3880.08 - 11 \times (18.6727)^2} = 0.0867 \).

04

Calculate the Intercept for Predicting Stride Rate

The intercept \( b_0 \) is calculated using \( b_0 = \bar{y} - b_1\bar{x} \). Substituting the values we have \( b_0 = 3.1964 - 0.0867 \times 18.6727 = 1.5743 \). Thus, the regression equation is \( \hat{y} = 1.5743 + 0.0867x \).

05

Calculate the Slope for Predicting Speed

For the inverse regression predicting speed from stride rate, \( b_1 = \frac{\sum(xy) - n\bar{x}\bar{y}}{\sum(y^2) - n\bar{y}^2} \). Substituting the values, \( b_1 = \frac{660.130 - 11 \times 18.6727 \times 3.1964}{112.681 - 11 \times (3.1964)^2} = 10.8314 \).

06

Calculate the Intercept for Predicting Speed

The intercept \( b_0 \) is \( \bar{x} - b_1\bar{y} \). Thus, \( b_0 = 18.6727 - 10.8314 \times 3.1964 = -15.8975 \). Hence, the regression equation is \( \hat{x} = -15.8975 + 10.8314y \).

07

Calculate Coefficient of Determination for Stride Rate on Speed

The coefficient of determination \( R^2 \) for the regression of stride rate on speed is calculated using \( R^2 = \left(\frac{\sum(xy) - n\bar{x}\bar{y}}{\sqrt{(\sum(x^2) - n\bar{x}^2)(\sum(y^2) - n\bar{y}^2)}}\right)^2 \). Substituting values gives \( R^2 = (0.2832^2) = 0.0802 \).

08

Calculate Coefficient of Determination for Speed on Stride Rate

Using the same formula for the inverse relationship, the \( R^2 \) value remains the same because \( R^2 \) is symmetric in x and y. Thus, \( R^2 = 0.0802 \).

09

Relationship between Both R-squared Values

The coefficients of determination for both regressions are identical, as expected in linear regression and correlation analysis. They both explain the same proportion of variance when swapping dependent and independent variables.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Squares Method

In statistics, the Least Squares Method is widely used for finding the best-fitting linear relationship between two variables. This method minimizes the sum of the squared differences, also known as the residuals, between the observed values and those predicted by the model. This approach ensures that the resulting linear equation provides the smallest possible error.When calculating this for our data, you begin by determining the mean values of both variables, which helps in centering the data.

• For strides and speed in our example, calculate individual means to understand averages.
• Then, you proceed with finding the slope (\( b_1 \)) using the formula: \( b_1 = \frac{\sum(xy) - n\bar{x}\bar{y}}{\sum(x^2) - n\bar{x}^2} \).
• The intercept (\( b_0 \)) is found as: \( b_0 = \bar{y} - b_1\bar{x} \).

These two components, slope and intercept, form the linear equation. This calculation can be done in either direction, predicting one variable from the other, and is fundamental in regression analysis.

Coefficient of Determination

The Coefficient of Determination, often denoted as \( R^2 \), is crucial in linear regression as it quantifies how well data fits a statistical model. Specifically, it explains the proportion of the variance in the dependent variable that's predictable from the independent variable.In simpler terms, \( R^2 \) tells us how much of the change in one variable can be explained by its relationship with another.

• In our exercise, the \( R^2 \) value for both regressions was found to be 0.0802, meaning roughly 8% of the variability in one variable is explained by the other.
• This value is consistent for both directional regressions, highlighting its symmetric property.

An \( R^2 \) close to 1 indicates a strong correlation, while one near 0 suggests a weak relationship, thereby drawing significant insights about the data's behavior.

Linear Regression Equations

Linear Regression Equations are foundational modeling tools in statistics, allowing predictions of one variable based on the relationship with another. These equations take the form of \( y = b_0 + b_1x \), capturing a straight-line relationship between variables.In our case of speed and stride rate:

• The equation predicting stride rate from speed is: \( \hat{y} = 1.5743 + 0.0867x \). This tells us for every unit increase in speed, the stride rate increases by roughly 0.0867 steps/sec.
• Conversely, the equation for predicting speed from stride rate is: \( \hat{x} = -15.8975 + 10.8314y \), indicating a stronger increase in speed with stride rate change.

Understanding these equations involves not just calculating but interpreting the slope and intercept values in a meaningful way, thereby bridging real-world data with statistical interpretations efficiently.