the data in the chart shows the maximum and corresponding resting heart rate of a healthy 20-year old man. resting 50,60,70,80,90. maximum 162,163,166,167,170. Find the equation of the trend line. Choose the correct answer below
To find the equation of the trend line, we need to determine the relationship between the resting heart rate and the maximum heart rate. We will use linear regression to find the equation of the line that best fits the data points.
First, let's label the resting heart rate as x and the maximum heart rate as y. We can establish the following data points:
(x, y) = (50, 162), (60, 163), (70, 166), (80, 167), (90, 170)
We will use these data points to calculate the equation of the trend line. The formula for the equation of a linear regression line is:
y = mx + b
Where m is the slope of the line and b is the y-intercept.
Using the formula:
Mean of x (x̄) = (50 + 60 + 70 + 80 + 90) / 5 = 70
Mean of y (ȳ) = (162 + 163 + 166 + 167 + 170) / 5 = 165.6
Let's calculate the values needed to find the slope:
Σ(x - x̄) = (50 - 70) + (60 - 70) + (70 - 70) + (80 - 70) + (90 - 70) = -20 + -10 + 0 + 10 + 20 = 0
Σ(y - ȳ) = (162 - 165.6) + (163 - 165.6) + (166 - 165.6) + (167 - 165.6) + (170 - 165.6) = -3.6 + -2.6 + 0.4 + 1.4 + 4.4 = 0
Σ(x - x̄)(y - ȳ) = 0 * 0 + -20 * -3.6 + -10 * -2.6 + 0 * 0.4 + 10 * 1.4 + 20 * 4.4 = 72 + 26 + 0 + 14 + 88 = 200
Σ(x - x̄)^2 = (-20)^2 + (-10)^2 + 0^2 + 10^2 + 20^2 = 400 + 100 + 0 + 100 + 400 = 1000
Now, we can calculate the slope (m):
m = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)^2 = 200 / 1000 = 0.2
Finally, we can calculate the y-intercept (b):
b = ȳ - m * x̄ = 165.6 - 0.2 * 70 = 152.6
Therefore, the equation of the trend line is:
y = 0.2x + 152.6