Question
or
Find the equation for the least squares regression line of the data described below.
Pedro is preparing for the national spelling bee and is following a strict study plan. To create the plan, he timed how long it had taken him to memorize several words.
Pedro recorded the number of letters in each word, x, and how many minutes, y, it had taken him to memorize it.
Letters Minutes
9 3.4
10 4.3
11 3.7
13 5.4
20 6.9
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the data provided, we need to follow these steps:
1. **Calculate the averages** of \( x \) (number of letters) and \( y \) (minutes).
2. **Calculate the slope (m)** of the regression line using the formula:
\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
3. **Calculate the y-intercept (b)** using the formula:
\[
b = \bar{y} - m\bar{x}
\]
4. **Write the equation** of the regression line in the form \( y = mx + b \).
### Step 1: Calculate averages
Given data:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
9 & 3.4 \\
10 & 4.3 \\
11 & 3.7 \\
13 & 5.4 \\
20 & 6.9 \\
\hline
\end{array}
\]
- \( n = 5 \) (number of data points)
- \( \Sigma x = 9 + 10 + 11 + 13 + 20 = 73 \)
- \( \Sigma y = 3.4 + 4.3 + 3.7 + 5.4 + 6.9 = 23.7 \)
- \( \Sigma xy = (9 \cdot 3.4) + (10 \cdot 4.3) + (11 \cdot 3.7) + (13 \cdot 5.4) + (20 \cdot 6.9) = 30.6 + 43 + 40.7 + 70.2 + 138 = 322.5 \)
- \( \Sigma x^2 = (9^2) + (10^2) + (11^2) + (13^2) + (20^2) = 81 + 100 + 121 + 169 + 400 = 871 \)
Now calculate averages:
\[
\bar{x} = \frac{\Sigma x}{n} = \frac{73}{5} = 14.6
\]
\[
\bar{y} = \frac{\Sigma y}{n} = \frac{23.7}{5} = 4.74
\]
### Step 2: Calculate the slope (m)
Using the slope formula:
\[
m = \frac{5(322.5) - (73)(23.7)}{5(871) - (73)^2}
\]
Calculating the components:
- \( 5(322.5) = 1612.5 \)
- \( (73)(23.7) = 1729.1 \)
- \( 5(871) = 4355 \)
- \( (73)^2 = 5329 \)
Now plug in:
\[
m = \frac{1612.5 - 1729.1}{4355 - 5329} = \frac{-116.6}{-974} \approx 0.119
\]
### Step 3: Calculate the y-intercept (b)
Using the intercept formula:
\[
b = \bar{y} - m\bar{x} = 4.74 - (0.119)(14.6)
\]
Calculating:
\[
0.119 \cdot 14.6 \approx 1.7354
\]
\[
b \approx 4.74 - 1.7354 \approx 3.0046
\]
### Step 4: Write the equation
Now round the values to the nearest thousandth:
- Slope \( m \approx 0.119 \)
- Intercept \( b \approx 3.005 \)
### Final Equation
The least squares regression line is:
\[
y = 0.119x + 3.005
\]
1. **Calculate the averages** of \( x \) (number of letters) and \( y \) (minutes).
2. **Calculate the slope (m)** of the regression line using the formula:
\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
3. **Calculate the y-intercept (b)** using the formula:
\[
b = \bar{y} - m\bar{x}
\]
4. **Write the equation** of the regression line in the form \( y = mx + b \).
### Step 1: Calculate averages
Given data:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
9 & 3.4 \\
10 & 4.3 \\
11 & 3.7 \\
13 & 5.4 \\
20 & 6.9 \\
\hline
\end{array}
\]
- \( n = 5 \) (number of data points)
- \( \Sigma x = 9 + 10 + 11 + 13 + 20 = 73 \)
- \( \Sigma y = 3.4 + 4.3 + 3.7 + 5.4 + 6.9 = 23.7 \)
- \( \Sigma xy = (9 \cdot 3.4) + (10 \cdot 4.3) + (11 \cdot 3.7) + (13 \cdot 5.4) + (20 \cdot 6.9) = 30.6 + 43 + 40.7 + 70.2 + 138 = 322.5 \)
- \( \Sigma x^2 = (9^2) + (10^2) + (11^2) + (13^2) + (20^2) = 81 + 100 + 121 + 169 + 400 = 871 \)
Now calculate averages:
\[
\bar{x} = \frac{\Sigma x}{n} = \frac{73}{5} = 14.6
\]
\[
\bar{y} = \frac{\Sigma y}{n} = \frac{23.7}{5} = 4.74
\]
### Step 2: Calculate the slope (m)
Using the slope formula:
\[
m = \frac{5(322.5) - (73)(23.7)}{5(871) - (73)^2}
\]
Calculating the components:
- \( 5(322.5) = 1612.5 \)
- \( (73)(23.7) = 1729.1 \)
- \( 5(871) = 4355 \)
- \( (73)^2 = 5329 \)
Now plug in:
\[
m = \frac{1612.5 - 1729.1}{4355 - 5329} = \frac{-116.6}{-974} \approx 0.119
\]
### Step 3: Calculate the y-intercept (b)
Using the intercept formula:
\[
b = \bar{y} - m\bar{x} = 4.74 - (0.119)(14.6)
\]
Calculating:
\[
0.119 \cdot 14.6 \approx 1.7354
\]
\[
b \approx 4.74 - 1.7354 \approx 3.0046
\]
### Step 4: Write the equation
Now round the values to the nearest thousandth:
- Slope \( m \approx 0.119 \)
- Intercept \( b \approx 3.005 \)
### Final Equation
The least squares regression line is:
\[
y = 0.119x + 3.005
\]