To find the least squares regression line for the given data, we need to calculate the slope (m) and the y-intercept (b) of the line \(y = mx + b\).
Given Data
Here's the data provided:
| Running Time (x) | Explosions (y) | |-------------------|----------------| | 82 | 5 | | 86 | 7 | | 87 | 18 | | 90 | 21 | | 94 | 27 | | 95 | 28 |
Step 1: Calculate the necessary sums
We need to compute the following:
- \(n\) = number of data points
- \(\sum x\)
- \(\sum y\)
- \(\sum xy\)
- \(\sum x^2\)
Where:
- \(n = 6\) (the total number of movies)
Let's calculate these sums:
\[ \sum x = 82 + 86 + 87 + 90 + 94 + 95 = 534 \]
\[ \sum y = 5 + 7 + 18 + 21 + 27 + 28 = 106 \]
\[ \sum xy = (82 \cdot 5) + (86 \cdot 7) + (87 \cdot 18) + (90 \cdot 21) + (94 \cdot 27) + (95 \cdot 28) \]
Calculating each term:
- \(82 \cdot 5 = 410\)
- \(86 \cdot 7 = 602\)
- \(87 \cdot 18 = 1566\)
- \(90 \cdot 21 = 1890\)
- \(94 \cdot 27 = 2538\)
- \(95 \cdot 28 = 2660\)
Now adding these results together: \[ \sum xy = 410 + 602 + 1566 + 1890 + 2538 + 2660 = 10566 \]
\[ \sum x^2 = 82^2 + 86^2 + 87^2 + 90^2 + 94^2 + 95^2 \]
Calculating each \(x^2\):
- \(82^2 = 6724\)
- \(86^2 = 7396\)
- \(87^2 = 7569\)
- \(90^2 = 8100\)
- \(94^2 = 8836\)
- \(95^2 = 9025\)
Now adding these: \[ \sum x^2 = 6724 + 7396 + 7569 + 8100 + 8836 + 9025 = 49650 \]
Step 2: Compute the slope (m) and intercept (b)
The formulas for the slope and intercept of the least squares regression line are as follows:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y) - m(\sum x)}{n} \]
Substituting the values: \[ m = \frac{6(10566) - (534)(106)}{6(49650) - (534)^2} \]
Calculating \(m\): \[ m = \frac{63456 - 56604}{297900 - 284556} \] \[ m = \frac{6849}{13344} \approx 0.513 \]
Now, for \(b\):
\[ b = \frac{106 - (0.513)(534)}{6} \] \[ b = \frac{106 - 273.882}{6} \] \[ b = \frac{-167.882}{6} \approx -27.980 \]
Final equation
So, rounding the values to the nearest thousandth:
\[ y \approx 0.513x - 27.980 \]
In conclusion, the least squares regression line is:
\[ y = 0.513x - 27.980 \]