the trend line for the following data set is y = 5x + 3.4 with a correlation coefficient of 0.87. complete the table to find the residual of each point.

Question

actual point | point on trend line | residual
(1,9) | (__) | (__)
(2, 12.3) | (__) | (__)
(3, 18.625) | (__) | (__)

Answer 1

To complete the table with the actual points, points on the trend line, and the residuals, we follow these steps:

1. Calculate the point on the trend line using the equation \( y = 5x + 3.4 \) where \( x \) is the x-value of the actual points.
2. Calculate the residual for each point, which is the difference between the actual y-value and the y-value on the trend line. The formula for the residual is: \[ \text{Residual} = \text{Actual y} - \text{Point on trend line y} \]

Now, let's fill in the table:

For point (1, 9):

1. Point on Trend Line:
   \( y = 5(1) + 3.4 = 5 + 3.4 = 8.4 \)
   So, the point on the trend line is \( (1, 8.4) \).

2. Residual:
   \( \text{Residual} = 9 - 8.4 = 0.6 \)

For point (2, 12.3):

1. Point on Trend Line:
   \( y = 5(2) + 3.4 = 10 + 3.4 = 13.4 \)
   So, the point on the trend line is \( (2, 13.4) \).

2. Residual:
   \( \text{Residual} = 12.3 - 13.4 = -1.1 \)

For point (3, 18.625):

1. Point on Trend Line:
   \( y = 5(3) + 3.4 = 15 + 3.4 = 18.4 \)
   So, the point on the trend line is \( (3, 18.4) \).

2. Residual:
   \( \text{Residual} = 18.625 - 18.4 = 0.225 \)

Now we can summarize this in the completed table:

| Actual Point | Point on Trend Line | Residual | |----------------|---------------------|-----------| | (1, 9) | (1, 8.4) | 0.6 | | (2, 12.3) | (2, 13.4) | -1.1 | | (3, 18.625) | (3, 18.4) | 0.225 |