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Problem 11

# The following table gives the projected state subsidies (in millions of dollars) to the Massachusetts Bay Transit Authority (MBTA) over a 5-yr period. $$\begin{array}{lccccc} \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Subsidy, } \boldsymbol{y} & 20 & 24 & 26 & 28 & 32 \\ \hline \end{array}$$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the state subsidy to the MBTA for the eighth year $$(x=8)$$.

Expert verified
The equation of the least-squares line for the given data is $$y = 2.8x + 17.6$$. The estimated state subsidy for the MBTA for the eighth year (x = 8) is $40 million. See the step by step solution ### Step by step solution ## Unlock all solutions Get unlimited access to millions of textbook solutions with Vaia Premium Over 22 million students worldwide already upgrade their learning with Vaia! ## Step 1: Calculate the necessary sums Given the table, we can calculate the sums needed: $$\sum{x} = 1 + 2 + 3 + 4 + 5 = 15$$ $$\sum{y} = 20 + 24 + 26 + 28 + 32 = 130$$ $$\sum{xy} = (1)(20) + (2)(24) + (3)(26) + (4)(28) + (5)(32) = 20 + 48 + 78 + 112 + 160 = 418$$ $$\sum{x^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$$ ## Step 2: Calculate the slope (m) and the y-intercept (b) Now that we have the sums, let's calculate the slope and y-intercept: $$n = 5$$ (total number of years) $$m = \frac{n \sum{xy} - \sum{x}\sum{y}}{n \sum{x^2} - (\sum{x})^2} = \frac{5(418) - (15)(130)}{5(55) - (15)^2} = \frac{2090 - 1950}{275 - 225} = \frac{140}{50} = 2.8$$ $$b = \frac{\sum y - m \sum x}{n} = \frac{130 - (2.8)(15)}{5} = \frac{130 - 42}{5} = \frac{88}{5} = 17.6$$ ## Step 3: Create the equation for the least-squares line Now that we have the slope and y-intercept, we can create the equation for the least-squares line: $$y = 2.8x + 17.6$$ ## Step 4: Estimate the state subsidy for the eighth year To estimate the state subsidy for the eighth year, plug in x = 8 into the equation: $$y = 2.8(8) + 17.6 = 22.4 + 17.6 = 40$$ Therefore, the estimated state subsidy for the MBTA for the eighth year (x = 8) is$40 million.

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