1. **State the problem:** We want to find the least squares regression line equation $y = mx + b$ for the data points given: $(136,163)$, $(159,223)$, $(179,236)$, $(192,397)$, and $(267,430)$. 2. **Formula and explanation:** The slope $m$ of the regression line is given by $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ where $n$ is the number of points. The intercept $b$ is $$b = \frac{\sum y - m \sum x}{n}$$ 3. **Calculate sums:** - $n = 5$ - $\sum x = 136 + 159 + 179 + 192 + 267 = 933$ - $\sum y = 163 + 223 + 236 + 397 + 430 = 1449$ - $\sum xy = 136\times163 + 159\times223 + 179\times236 + 192\times397 + 267\times430$ $$= 22168 + 35457 + 42244 + 76144 + 114810 = 290823$$ - $\sum x^2 = 136^2 + 159^2 + 179^2 + 192^2 + 267^2$ $$= 18496 + 25281 + 32041 + 36864 + 71289 = 183971$$ 4. **Calculate slope $m$:** $$m = \frac{5 \times 290823 - 933 \times 1449}{5 \times 183971 - 933^2} = \frac{1454115 - 1351917}{919855 - 870249} = \frac{102198}{49506}$$ 5. **Simplify slope:** $$m = \frac{\cancel{102198}}{\cancel{49506}} = 2.064 \text{ (rounded to 3 decimals)}$$ 6. **Calculate intercept $b$:** $$b = \frac{1449 - 2.064 \times 933}{5} = \frac{1449 - 1925.112}{5} = \frac{-476.112}{5} = -95.222$$ 7. **Final regression line equation:** $$y = 2.064x - 95.222$$ This means for each additional calorie, sales increase by about 2.064 units, starting from -95.222 when calories are zero (extrapolated).