To find the equation for the least squares regression line \( y = mx + b \), we need to calculate the slope \( m \) and the y-intercept \( b \).
Given the data:
\[ \begin{array}{|c|c|} \hline \text{Words (x)} & \text{Minutes (y)} \ \hline 1586 & 30.65 \ 3444 & 31.43 \ 5009 & 44.52 \ 5572 & 37.20 \ 7069 & 57.41 \ \hline \end{array} \]
First, we calculate various necessary sums:
- \( n \) = number of data points = 5
- \( \sum x \) = \( 1586 + 3444 + 5009 + 5572 + 7069 = 22780 \)
- \( \sum y \) = \( 30.65 + 31.43 + 44.52 + 37.20 + 57.41 = 201.21 \)
- \( \sum xy \) = \( 1586 \times 30.65 + 3444 \times 31.43 + 5009 \times 44.52 + 5572 \times 37.20 + 7069 \times 57.41 \)
Calculating \( \sum xy \):
\[ \begin{align*} 1586 \times 30.65 & = 48575.90 \ 3444 \times 31.43 & = 108297.92 \ 5009 \times 44.52 & = 222492.68 \ 5572 \times 37.20 & = 207946.40 \ 7069 \times 57.41 & = 405851.29 \ \end{align*} \]
Adding these together: \[ \sum xy = 48575.90 + 108297.92 + 222492.68 + 207946.40 + 405851.29 = 1093074.19 \]
- \( \sum x^2 \) = \( 1586^2 + 3444^2 + 5009^2 + 5572^2 + 7069^2 \)
Calculating \( \sum x^2 \):
\[ \begin{align*} 1586^2 & = 2510596 \ 3444^2 & = 11871536 \ 5009^2 & = 25090181 \ 5572^2 & = 31018384 \ 7069^2 & = 49972461 \ \end{align*} \]
Adding these together: \[ \sum x^2 = 2510596 + 11871536 + 25090181 + 31018384 + 49972461 = 113394158 \]
Now we can calculate the slope \( m \) and intercept \( b \):
The formulas are:
\[ m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]
\[ b = \frac{\sum y - m \sum x}{n} \]
Substituting the values into the formula for \( m \): \[ m = \frac{5 \times 1093074.19 - 22780 \times 201.21}{5 \times 113394158 - (22780)^2} \]
Calculating the numerator: \[ 5 \times 1093074.19 = 5465370.95 \] \[ 22780 \times 201.21 = 4589669.80 \] \[ \text{Numerator} = 5465370.95 - 4589669.80 = 875701.15 \]
Calculating the denominator: \[ 5 \times 113394158 = 566970790 \] \[ (22780)^2 = 518448484 \] \[ \text{Denominator} = 566970790 - 518448484 = 48422306 \]
Now calculating \( m \): \[ m = \frac{875701.15}{48422306} \approx 0.0181 \quad (\text{rounded to 4 decimal places}) \]
Now, calculating \( b \) with the value of \( m \): \[ b = \frac{201.21 - 0.0181 \times 22780}{5} \] Calculating the product: \[ 0.0181 \times 22780 \approx 412.858 \] Now calculating \( b \): \[ b = \frac{201.21 - 412.858}{5} = \frac{-211.648}{5} \approx -42.3296 \]
So, rounding \( m \) and \( b \) to the nearest thousandth gives: \[ m \approx 0.018 \quad \text{and} \quad b \approx -42.330 \]
Finally, the equation of the least squares regression line is: \[ y = 0.018x - 42.330 \]