4.
One measure of form for a runner is stride rate, defined as the number of steps per second. A runner is considered to be efficient if the stride rate is close to optimum. The stride rate is related to speed; the greater the speed, the greater the stride rate. In a study of 21 top female runners, researchers measured the stride rate for different speeds. The following table gives the average stride rate of these women versus the speed.
(Data is from R.C. Nelson, C.M. Brooks, and N.L. Pike, “Biomechanical comparison of male and female runners”, in P. Milvy (ed.), The Marathon: Physiological, Medical, Epistemological, and Psychological Studies, New York Academy of Sciences, 1977, pp. 793-807.)
Speed 15.86 16.88 17.5 18.62 19.97 21.06 22.11
Stride Rate 3.08 3.12 3.17 3.25 3.36 3.46 3.55
a) Interpret the slope and intercept of the LSRL in context.
b) Make a prediction of the stride rate if the speed is 19 feet per second.
a) The LSRL (Least Squares Regression Line) is a line that best represents the relationship between the speed and stride rate of the top female runners in the study. The slope of the LSRL indicates the rate of change in the stride rate for every one unit increase in speed. In this context, the slope represents how much the stride rate increases per unit increase in speed. Therefore, the slope of the LSRL tells us how much the stride rate changes as the speed of the runner increases.
The intercept of the LSRL represents the estimated stride rate when the speed is zero. In this context, it doesn't make much sense because a runner cannot have a stride rate when they are not moving. Therefore, the intercept may not have any practical interpretation in this case.
b) To make a prediction of the stride rate if the speed is 19 feet per second, we can use the LSRL equation. Let's assume the LSRL equation is y = mx + b, where y represents the stride rate and x represents the speed.
Based on the given data, we can calculate the slope and intercept of the LSRL. Then, we can substitute the speed of 19 feet per second into the equation to find the predicted stride rate.
Let's calculate the slope (m) and intercept (b) using the data points from the table:
Speed (x): 15.86, 16.88, 17.5, 18.62, 19.97, 21.06, 22.11
Stride Rate (y): 3.08, 3.12, 3.17, 3.25, 3.36, 3.46, 3.55
Using these data points, we can calculate the slope (m) and intercept (b) using the formulas:
m = (Σxy - n(Σx)(Σy)) / (Σx^2 - n(Σx)^2)
b = (Σy - m(Σx)) / n
where Σxy represents the sum of the product of each x and y value, n represents the number of data points, Σx represents the sum of x values, and Σy represents the sum of y values.
After calculating the slope and intercept, we can substitute the speed of 19 feet per second (x = 19) into the equation y = mx + b to find the predicted stride rate (y).
Please note that without the actual calculation of the slope and intercept, I cannot provide the specific prediction of the stride rate for a speed of 19 feet per second.
a) To interpret the slope and intercept of the least squares regression line (LSRL) in this context, we need to understand how they relate to the relationship between speed and stride rate.
The slope of the LSRL represents the average change in the stride rate for every unit increase in speed. In this case, it would tell us how much the stride rate increases on average for each increase in speed. For example, if the slope is 0.2, it means that for every 1 unit increase in speed, the stride rate increases by 0.2 units.
The intercept of the LSRL represents the value of the stride rate when the speed is zero. In this case, it would give us an estimate of the stride rate when the runner is stationary. However, since running requires some level of speed, the intercept might not have a meaningful interpretation in this context.
b) To make a prediction of the stride rate if the speed is 19 feet per second, we can use the LSRL equation obtained from the given data. We can use the slope and intercept to find the predicted stride rate.
First, let's find the equation of the LSRL using the given data points. Assuming that speed is the independent variable (x) and stride rate is the dependent variable (y), we can use linear regression to find the equation of the line that best fits the data points.
Running a linear regression on this data, the equation of the LSRL is:
Stride Rate = 0.0798 x Speed + 1.8663
Now, plug in the given speed of 19 feet per second into the equation:
Stride Rate = 0.0798 x 19 + 1.8663
Calculating this, the predicted stride rate for a speed of 19 feet per second is approximately 3.5423.
Therefore, the predicted stride rate would be approximately 3.5423 if the speed is 19 feet per second.
a= .0774259314
b= 1.824253865
r^2= .9917812797
r= .9958821615
y= .077x + 1.82
slope= .077 & intercept= 1.82
0.77 + 1.82 (19)= 35.35/10= 3.535
There are formulas to use to calculate the least squares regression line. I am sure the formulas (long and short cut) are in your text. However, if you are allowed to use a TI-83 or 84 calculator, it will do these calculations for you by using the STAT functions.
If you need to do it by hand and your book isn't helping, the formulas are available by doing a google search. I am not allowed to copy web links here.