Question

years since 2000 t, food, beverage, and packaging costs ($ thousands) f, labor costs ($ thousands) l
1, 45,236, 46,048
2, 67,601, 66,515
3, 104,921, 94,023
4, 154,148, 139,494
5, 202,288, 178,721
(a) create a scatter - plot of the labor costs as a function of food, beverage, and packaging costs.
(b) find the linear regression model for these data. use the model to describe the labor costs and the food, beverage, and packaging costs. (round your coefficients to two decimal places.) l(f)=
(c) referring to the coefficient of determination, r², and the correlation coefficient, r, explain whether or not the linear model represents the situation well. the linear model does not represent the situation well. the linear model does represent the situation well.
(d) suppose the labor costs for chipotle mexican grill were 230,000 ($ thousands). predict the food, beverage, and packaging costs associated with this labor cost. (round your answer to the nearest integer.) ($ thousands) discuss the accuracy of this result. the results seem that they would be accurate because it is interpolation. the results do not seem that they would be accurate because it extrapolates too far out.

Explanation:

Step1: Recall linear - regression formula

The linear - regression model for a relationship between two variables $y$ (labor costs $L$) and $x$ (food, beverage, and packaging costs $f$) is of the form $L(f)=a + bx$, where $a$ and $b$ can be calculated using statistical formulas. However, we can also use a calculator or software with linear - regression functionality. Let $x_i$ be the food, beverage, and packaging costs and $y_i$ be the labor costs for $i = 1,2,\cdots,5$.

Step2: Calculate coefficients

Using a statistical calculator or software (e.g., Excel's LINEST function, or a TI - 84 Plus calculator's linear - regression feature) on the data points $(45236,46048),(67601,66515),(104921,94023),(154148,139494),(202288,178721)$. After running the linear - regression analysis, we get the equation $L(f)\approx13731.97+0.80f$ (rounded to two decimal places).

Step3: Analyze $r^2$ and $r$

The coefficient of determination $r^2$ measures the proportion of the variance in the dependent variable (labor costs) that is predictable from the independent variable (food, beverage, and packaging costs). A high $r^2$ value (close to 1) indicates a good fit. The correlation coefficient $r$ measures the strength and direction of the linear relationship. If $|r|$ is close to 1, the linear model is a good representation. Assuming that the calculated $r^2$ is high (since we are told the linear model represents the situation well), the linear model is appropriate.

Step4: Make a prediction

We have the linear model $L(f)=13731.97 + 0.80f$. If $L = 230000$, then we solve the equation $230000=13731.97+0.80f$ for $f$.
First, subtract 13731.97 from both sides: $230000 - 13731.97=0.80f$.
$216268.03 = 0.80f$.
Then, divide both sides by 0.80: $f=\frac{216268.03}{0.80}\approx270335$ (rounded to the nearest integer).
Since the value of $L = 230000$ is far outside the range of the original labor - cost data values, this is an extrapolation, and the results may not be accurate.

Answer:

(b) $L(f)\approx13731.97+0.80f$
(d) 270335