Question

1. select the correct answers by entering the data from the table in your calculator.| $x$ | $y$ ||-----|-----|| 0 | 90 || 1 | 82 || 2 | 69 || 3 | 63 || 4 | 47 || 5 | 41 || 6 | 29 || 7 | 22 |based on the scatter plot, the best model is ____a. cubicb. linearc. $-9.68$d. $y = -10.01x + 90.42$e. $10.42$f. $y = 0.089x^3 - 10.637x + 91.047$find the equation of the line of best fit: __use the equation to predict $y$ when $x = 10$. ____

Explanation:

Step1: Identify best model type

First, we check the rate of change of $y$:

• $\Delta y$ values: $82-90=-8$, $69-82=-13$, $63-69=-6$, $47-63=-16$, $41-47=-6$, $29-41=-12$, $22-29=-7$

The non-constant second differences (change of $\Delta y$) indicate a non-linear model, so cubic (a) is better than linear (b).

Step2: Calculate cubic regression

Using the data $(x,y)$: $(0,90),(1,82),(2,69),(3,63),(4,47),(5,41),(6,29),(7,22)$, the cubic regression equation is $y = 0.089x^3 - 10.637x + 91.047$ (matches option f).

Step3: Predict $y$ at $x=10$

Substitute $x=10$ into the cubic equation:

$$\begin{align*} y&=0.089(10)^3 - 10.637(10) + 91.047\\ &=0.089(1000) - 106.37 + 91.047\\ &=89 - 106.37 + 91.047 \end{align*}$$

Step4: Compute final predicted value

$$ y=89-106.37+91.047=73.677 $$

Answer:

1. Best model: a. cubic
2. Equation of best fit: f. $y = 0.089x^3 - 10.637x + 91.047$
3. Predicted $y$ when $x=10$: $73.68$ (rounded to two decimal places)