Question

or
Find the equation for the least squares regression line of the data described below.
Jerry has trouble getting his math homework done on time, and his mother suspects it is due to lack of sleep.
For the next few nights, Jerry's mother notes the number of hours he sleeps, x, and the number of minutes it takes him to do his math homework the following day, y.
Hours slept Minutes needed to finish math homework
4.1 95
4.4 98
5.1 75
6.3 85
6.4 63
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation for the least squares regression line \(y = mx + b\), we need to calculate the slope \(m\) and the intercept \(b\) using the provided data. The data is as follows:

\[
\begin{array}{|c|c|}
\hline
\text{Hours slept (x)} & \text{Minutes needed to finish math homework (y)} \\
\hline
4.1 & 95 \\
4.4 & 98 \\
5.1 & 75 \\
6.3 & 85 \\
6.4 & 63 \\
\hline
\end{array}
\]

### Step 1: Calculate the Means
Calculate the means of \(x\) and \(y\):

\[
\bar{x} = \frac{4.1 + 4.4 + 5.1 + 6.3 + 6.4}{5} = \frac{26.3}{5} = 5.26
\]

\[
\bar{y} = \frac{95 + 98 + 75 + 85 + 63}{5} = \frac{416}{5} = 83.2
\]

### Step 2: Calculate the Slope \(m\)
Using the formula for the slope of the least squares regression line:

\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]

First, we calculate the necessary components.

\[
\sum (x_i - \bar{x})(y_i - \bar{y})
\]
\[
\sum (x_i - \bar{x})^2
\]

Calculating each component individually:

- For \(i = 1\): \((4.1 - 5.26)(95 - 83.2) = (-1.16)(11.8) = -13.688\)
- For \(i = 2\): \((4.4 - 5.26)(98 - 83.2) = (-0.86)(14.8) = -12.728\)
- For \(i = 3\): \((5.1 - 5.26)(75 - 83.2) = (-0.16)(-8.2) = 1.312\)
- For \(i = 4\): \((6.3 - 5.26)(85 - 83.2) = (1.04)(1.8) = 1.872\)
- For \(i = 5\): \((6.4 - 5.26)(63 - 83.2) = (1.14)(-20.2) = -23.028\)

Now, summing these values:

\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = -13.688 - 12.728 + 1.312 + 1.872 - 23.028 = -46.260
\]

Now for \(\sum (x_i - \bar{x})^2\):

\[
\sum (x_i - \bar{x})^2 = (-1.16)^2 + (-0.86)^2 + (-0.16)^2 + (1.04)^2 + (1.14)^2
\]
\[
= 1.3456 + 0.7396 + 0.0256 + 1.0816 + 1.2996 = 4.492
\]

Now we can calculate the slope \(m\):

\[
m = \frac{-46.260}{4.492} \approx -10.31
\]

### Step 3: Calculate the Intercept \(b\)
Using the formula:

\[
b = \bar{y} - m\bar{x}
\]

\[
b = 83.2 - (-10.31) \cdot 5.26 \approx 83.2 + 54.1646 \approx 137.3646
\]

### Step 4: Write the equation of the line
So, the least squares regression line is:

\[
y \approx -10.310x + 137.365
\]

### Final Answer (rounded to the nearest thousandth):
\[
y = -10.310x + 137.365
\]