Question

Find the equation for the least squares regression line of the data described below.
Playing video games makes Debbie so happy that she thinks it even helps her complete more schoolwork. Debbie's roommate Lacey is skeptical, so over the next few days, Lacey asks Debbie about her progress on her daily reading.
For each day, Lacey notes the number of minutes Debbie spends playing video games, x. She also takes the number of pages Debbie reads and divides it by the total number of pages assigned, y.
Minutes playing video games Percentage of reading assignment
34 58
48 43
71 31
79 43
95 31
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line for the given data, we first need to compute the slope (m) and y-intercept (b) of the regression line using the formula for the least squares method.

The data points provided are:
- (34, 58)
- (48, 43)
- (71, 31)
- (79, 43)
- (95, 31)

We can calculate the necessary sums for the formulas:
- \( n = \) number of points
- \( \sum x = \) sum of the x-values
- \( \sum y = \) sum of the y-values
- \( \sum xy = \) sum of the product of x and y
- \( \sum x^2 = \) sum of the square of the x-values

Calculating these sums step by step:

1. **Count (n)**:
There are 5 data points, so \( n = 5 \).

2. **Sum of x values**:
\[
\sum x = 34 + 48 + 71 + 79 + 95 = 327
\]

3. **Sum of y values**:
\[
\sum y = 58 + 43 + 31 + 43 + 31 = 206
\]

4. **Sum of xy** (x multiplied by y for each pair):
\[
\sum xy = (34 \cdot 58) + (48 \cdot 43) + (71 \cdot 31) + (79 \cdot 43) + (95 \cdot 31)
\]
\[
= 1972 + 2064 + 2201 + 3397 + 2945 = 12579
\]

5. **Sum of x²**:
\[
\sum x^2 = 34^2 + 48^2 + 71^2 + 79^2 + 95^2
\]
\[
= 1156 + 2304 + 5041 + 6241 + 9025 = 23767
\]

Now we can use the formulas for \( m \) and \( b \):

### Calculate the slope (m):
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
Plugging in the values:
\[
m = \frac{5(12579) - (327)(206)}{5(23767) - (327)^2}
\]
\[
= \frac{62895 - 67362}{118835 - 106929}
\]
\[
= \frac{-4467}{11806} \approx -0.379
\]

### Calculate the y-intercept (b):
\[
b = \frac{\sum y - m \sum x}{n}
\]
Plugging in the values:
\[
b = \frac{206 - (-0.379)(327)}{5}
\]
\[
= \frac{206 + 123.153}{5}
\]
\[
= \frac{329.153}{5} \approx 65.831
\]

### Final equation for the least squares regression line:
The least squares regression line will be in the form \( y = mx + b \):

\[
y = -0.379x + 65.831
\]

Rounding to the nearest thousandth, the final equation is:

\[
y = -0.379x + 65.831
\]