Question

what is the residual of the data point (6, 7.82)

Answer 1

The residual is the difference between the observed value and the predicted value. To calculate the residual, we need to know the predicted value. If we don't have a specific prediction model, we can use a simple linear regression line as an estimate of the predicted value.

Let's assume the linear regression line equation is y = mx + b, where m is the slope and b is the y-intercept. We can calculate the slope and y-intercept using two given data points and then use the equation to estimate the predicted value.

If we use the points (6, 7.82) and another point (x1, y1) to calculate the slope and y-intercept, we have:

m = (y1 - 7.82) / (x1 - 6)

Now suppose we have another point (3, 5.2), we can use this point and (6, 7.82) to find the slope and y-intercept:

m = (5.2 - 7.82) / (3 - 6) = -2.62 / -3 = 0.8733

Now we can plug in the slope into the equation:

y = 0.8733x + b

Using point (6, 7.82) to calculate the y-intercept:

7.82 = 0.8733(6) + b
7.82 = 5.2398 + b
b = 2.5802

So, the equation of the line is y = 0.8733x + 2.5802

Now, we plug in x = 6 to find the predicted value:

y = 0.8733(6) + 2.5802 = 7.794

The residual is the observed value minus the predicted value:

Residual = 7.82 - 7.794 = 0.026

Therefore, the residual of the data point (6, 7.82) is 0.026.