Question

or Watch a video Find the equation for the least squares regression line of the data described below. To plan its upcoming programming, a local television station wanted to study the television watching habits of people in the area. Station executives called a random sample of people in their broadcast area and asked them a variety of questions. In particular, they asked people their ages, x, and how many hours of television they had watched last week, y. Age Hours 17 0 26 7 68 19 68 15 76 20 Round your answers to the nearest thousandth. y=x+

Answer 1

To find the equation for the least squares regression line of the given data, we will need to calculate the slope (m) and y-intercept (b) of the regression line. The general form of the least squares regression line is:

\[ y = mx + b \]

Where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept

### Step 1: Preparation

We have the following data points:

\[
\begin{array}{cc}
\text{Age (x)} & \text{Hours (y)} \\
17 & 0 \\
26 & 7 \\
68 & 19 \\
68 & 15 \\
76 & 20 \\
\end{array}
\]

### Step 2: Calculate the necessary sums

Let's find the required sums:

- \( n = 5 \) (number of data points)
- Sum of X: \( \sum x = 17 + 26 + 68 + 68 + 76 = 255 \)
- Sum of Y: \( \sum y = 0 + 7 + 19 + 15 + 20 = 61 \)
- Sum of XY: \( \sum xy = (17 \cdot 0) + (26 \cdot 7) + (68 \cdot 19) + (68 \cdot 15) + (76 \cdot 20) \)

Calculating the XY sums:
- For (17, 0): \( 17 \cdot 0 = 0 \)
- For (26, 7): \( 26 \cdot 7 = 182 \)
- For (68, 19): \( 68 \cdot 19 = 1292 \)
- For (68, 15): \( 68 \cdot 15 = 1020 \)
- For (76, 20): \( 76 \cdot 20 = 1520 \)

Thus:
\[
\sum xy = 0 + 182 + 1292 + 1020 + 1520 = 3014
\]
- Sum of \( x^2 \): \( \sum x^2 = 17^2 + 26^2 + 68^2 + 68^2 + 76^2 = 289 + 676 + 4624 + 4624 + 5776 = 11589 \)

### Step 3: Calculate slope (m) and y-intercept (b)

Using the formulas:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Calculating \( m \):
\[
m = \frac{5(3014) - (255)(61)}{5(11589) - (255)^2}
\]
Calculating:
\[
m = \frac{15070 - 15555}{57945 - 65025} = \frac{-485}{-7070} \approx 0.0686
\]

Calculating \( b \):
\[
b = \frac{61 - 0.0686 \cdot 255}{5} \approx \frac{61 - 17.491}{5} \approx \frac{43.509}{5} \approx 8.702
\]

### Step 4: Final Equation

Substituting \( m \) and \( b \):
\[
y \approx 0.069x + 8.702
\]

Rounded to the nearest thousandth:
\[
y = 0.069x + 8.702
\]

So, the least squares regression line of the data is:
\[
\boxed{y = 0.069x + 8.702}
\]