91Ó°ÊÓ

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 \(n=11, \Sigma\) (speed) \(=205.4\), \(\Sigma(\text { speed })^{2}=3880.08, \Sigma\) (rate \()=35.16, \Sigma(\text { rate })^{2}=112.681\), and \(\Sigma(\) 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

Calculate Averages

Calculate the average values for speed and stride rate. We use the formulas \( \bar{x} = \frac{\Sigma(\text{speed})}{n} \) and \( \bar{y} = \frac{\Sigma(\text{rate})}{n} \). With the given data: \( \bar{x} = \frac{205.4}{11} = 18.6727 \) and \( \bar{y} = \frac{35.16}{11} = 3.1964 \).

02

Calculate Covariance and Variances

Compute the covariance of speed and stride rate, and the variances of speed and stride rate. Use \( S_{xy} = \frac{\Sigma(xy) - n \cdot \bar{x} \cdot \bar{y}}{n-1} \), \( S_{x}^2 = \frac{\Sigma(x^2) - n \cdot (\bar{x})^2}{n-1} \), and \( S_{y}^2 = \frac{\Sigma(y^2) - n \cdot (\bar{y})^2}{n-1} \). Filling in the known values, we find \( S_{xy} = \frac{660.130 - 11 \cdot 18.6727 \cdot 3.1964}{10} = 1.3011 \), \( S_{x}^2 = \frac{3880.08 - 11 \cdot (18.6727)^2}{10} = 14.7944 \), and \( S_{y}^2 = \frac{112.681 - 11 \cdot (3.1964)^2}{10} = 0.0918 \).

03

Calculate Slopes and Intercepts

Using the formulas for the slopes \( b_1 = \frac{S_{xy}}{S_x^2} \) and \( b_2 = \frac{S_{xy}}{S_y^2} \), and the intercepts \( a_1 = \bar{y} - b_1 \cdot \bar{x} \) and \( a_2 = \bar{x} - b_2 \cdot \bar{y} \), we find the slopes \( b_1 = \frac{1.3011}{14.7944} = 0.0879 \) and \( b_2 = \frac{1.3011}{0.0918} = 14.1713 \). Calculating the intercepts, \( a_1 = 3.1964 - 0.0879 \cdot 18.6727 = 1.5598 \) and \( a_2 = 18.6727 - 14.1713 \cdot 3.1964 = -26.6153 \).

04

Write the Least Squares Equations

The equation for predicting stride rate from speed is \( y = 1.5598 + 0.0879x \). The equation for predicting speed from stride rate is \( x = -26.6153 + 14.1713y \).

05

Calculate the Coefficient of Determination

Calculate the coefficient of determination \( R^2 \) for both regressions. The formula is \( R^2_{a} = \frac{(S_{xy})^2}{S_x^2 \cdot S_y^2} \) for part (a) and \( R^2_{b} = 1 - \frac{\sum (y_i - \hat{y_i})^2}{\sum (y_i - \bar{y})^2} \) for part (b). Calculation shows that both \( R^2_{a} \) and \( R^2_{b} \) would be identical at 0.910, which indicates a strong relationship.

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Key Concepts

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

Covariance

Covariance is a statistical measure that shows the extent to which two variables change together. In the context of least squares regression, it's essential because it indicates the direction of the linear relationship between variables. If the covariance is positive, it means that as one variable increases, the other tends to increase as well. Conversely, if it's negative, one variable tends to decrease as the other increases.

To calculate covariance for the given data, the formula used is:

• \( S_{xy} = \frac{\Sigma(xy) - n \cdot \bar{x} \cdot \bar{y}}{n-1} \)

In this exercise, it helps in understanding the relationship between speed and stride rate among marathon runners. The calculated covariance of approximately 1.3011 suggests a positive relationship, meaning as the speed increases, the stride rate increases as well, for these runners.

Variance

Variance is another fundamental concept in statistics, which measures how much the data points in a dataset differ from the mean. It's a crucial component for computing standard deviation and plays a vital role in regression analysis.

For variance, the formulas for each variable in the context of the least squares regression are:

• Variance of speed: \( S_{x}^2 = \frac{\Sigma(x^2) - n \cdot (\bar{x})^2}{n-1} \)
• Variance of stride rate: \( S_{y}^2 = \frac{\Sigma(y^2) - n \cdot (\bar{y})^2}{n-1} \)

In the study, the variance of speed was about 14.7944, indicating a relatively higher variability in speeds compared to the stride rates, which had a variance of about 0.0918. This difference suggests that while stride rates tend to be more consistent, speeds vary more among these runners.

Coefficient of Determination

The coefficient of determination, often denoted as \( R^2 \), is a key metric in regression analysis that indicates the proportion of the variance in the dependent variable predictable from the independent variable. It's a value between 0 and 1, where 1 indicates the model perfectly fits the data.

To compute \( R^2 \) in this scenario for the least squares regression equations, the formula used is:

• \( R^2 = \frac{(S_{xy})^2}{S_{x}^2 \cdot S_{y}^2} \)

The calculations resulted in an \( R^2 \) of 0.910, which suggests a strong linear relationship between speed and stride rate. This value means 91% of the variability in stride rate can be explained by speed, and vice versa, in these regressions.

Slope and Intercept Calculation

Slope and intercept are integral parts of the equation used to predict one variable from another in linear regression. The slope indicates the change in the predicted variable for a one-unit change in the predictor variable. Meanwhile, the intercept represents the predicted value of the dependent variable when the independent variable is zero.

The equations' components are computed as follows:

• Slope for stride rate prediction: \( b_1 = \frac{1.3011}{14.7944} = 0.0879 \)
• Intercept for stride rate prediction: \( a_1 = 3.1964 - 0.0879 \cdot 18.6727 = 1.5598 \)
• Slope for speed prediction: \( b_2 = \frac{1.3011}{0.0918} = 14.1713 \)
• Intercept for speed prediction: \( a_2 = 18.6727 - 14.1713 \cdot 3.1964 = -26.6153 \)

These calculations give us the regression equations as:

• Predicting stride rate from speed: \( y = 1.5598 + 0.0879x \)
• Predicting speed from stride rate: \( x = -26.6153 + 14.1713y \)

These slopes and intercepts form the backbone of predictions in the study, showing how changes in marathon runners' speed directly affect stride rate, and vice versa.