Question

bivariate data for the quantitative variables x and y are given in the table below. these data are plotted in the scatter plot shown next to the table.

in the scatter plot, sketch an approximation of the least-squares regression line for the data.

Explanation:

Calculate summary statistics

We first calculate the sample means of the variables \(x\) and \(y\) to find the centroid \((\bar{x}, \bar{y})\) of the data.
Using the Sample Mean knowledge point:
The number of data points is \(n = 18\).
Sum of \(x\):

$$ \sum x = 2.1 + 6.5 + 9.6 + 10.2 + 8.7 + 1.2 + 2.2 + 3.3 + 3.4 + 7.2 + 6.1 + 4.0 + 7.3 + 7.8 + 8.3 + 4.3 + 5.2 + 5.4 = 96.8 $$

$$ \bar{x} = \frac{96.8}{18} \approx 5.378 $$

Sum of \(y\):

$$ \sum y = 4.2 + 5.8 + 7.8 + 9.5 + 8.3 + 2.5 + 3.1 + 3.8 + 4.4 + 5.5 + 7.4 + 4.9 + 7.1 + 7.7 + 6.9 + 5.1 + 5.2 + 6.4 = 105.6 $$

$$ \bar{y} = \frac{105.6}{18} \approx 5.867 $$

Calculate sums of squares

To find the slope \(b_1\) of the least-squares regression line, we compute the sum of squares \(SS_{xx}\) and sum of products \(SS_{xy}\).
The formulas are:

$$ SS_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} $$

$$ SS_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} $$

Calculating the individual terms:

$$ \sum x^2 = 2.1^2 + 6.5^2 + \dots + 5.4^2 = 641.16 $$

$$ SS_{xx} = 641.16 - \frac{96.8^2}{18} = 641.16 - 520.569 = 120.591 $$

$$ \sum xy = (2.1)(4.2) + (6.5)(5.8) + \dots + (5.4)(6.4) = 644.02 $$

$$ SS_{xy} = 644.02 - \frac{(96.8)(105.6)}{18} = 644.02 - 567.893 = 76.127 $$

Determine the regression line equation

Now we calculate the slope \(b_1\) and the y-intercept \(b_0\) of the regression line:

$$ b_1 = \frac{SS_{xy}}{SS_{xx}} = \frac{76.127}{120.591} \approx 0.631 $$

$$ b_0 = \bar{y} - b_1 \bar{x} = 5.867 - (0.631)(5.378) \approx 5.867 - 3.394 = 2.473 $$

Thus, the equation of the least-squares regression line is:

$$ \hat{y} = 0.631x + 2.473 $$

Identify key plotting points

To sketch the line accurately on the scatter plot, we find two points on the line:

• For \(x = 0\): \(\hat{y} \approx 2.47\)
• For \(x = 10\): \(\hat{y} = 0.631(10) + 2.473 \approx 8.78\)

Answer:

The least-squares regression line is approximately given by the equation:

$$ \hat{y} = 0.63x + 2.47 $$

To sketch this line on the scatter plot, draw a straight line passing through the following approximate coordinates:

• \(y\)-intercept at \((0, 2.5)\)
• Point at \((10, 8.8)\)