Question: Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line. (Each pair of variables has a significant correlation.) Then use the regression equation to predict the value of y for each of the given x-values, if meaningful. The caloric content and the sodium content (in milligrams) for 6 beef hot dogs are shown in the table below.
Calories, $x$ 150 170 120 130 90 190
Sodium, $y$ 410 460 320 360 280 530
(a) $x = 160$ calories
(b) $x = 100$ calories
(c) $x = 140$ calories
(d) $x = 60$ calories
Find the regression equation.
$\hat{y} = \square x + (\square)$
(Round to three decimal places as needed.)
Choose the correct graph below.
A.
B.
C.
D.
(a) Predict the value of $y$ for $x = 160$. Choose the best prediction below.
A. 439.922
B. 185.922
1. **State the problem:** We are given data pairs of calories ($x$) and sodium content ($y$) for 6 beef hot dogs. We need to find the regression line equation $\hat{y} = bx + a$, plot the scatter plot with the regression line, and predict $y$ for given $x$ values.
2. **Given data:**
$$
x: 150, 170, 120, 130, 90, 190
$$
$$
y: 410, 460, 320, 360, 280, 530
$$
3. **Calculate means:**
$$\bar{x} = \frac{150 + 170 + 120 + 130 + 90 + 190}{6} = \frac{850}{6} = 141.667$$
$$\bar{y} = \frac{410 + 460 + 320 + 360 + 280 + 530}{6} = \frac{2360}{6} = 393.333$$
4. **Calculate slope $b$ of regression line:**
$$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$
Calculate numerator:
$$\sum (x_i - \bar{x})(y_i - \bar{y}) = (150-141.667)(410-393.333) + (170-141.667)(460-393.333) + (120-141.667)(320-393.333) + (130-141.667)(360-393.333) + (90-141.667)(280-393.333) + (190-141.667)(530-393.333)$$
$$= 8.333 \times 16.667 + 28.333 \times 66.667 + (-21.667) \times (-73.333) + (-11.667) \times (-33.333) + (-51.667) \times (-113.333) + 48.333 \times 136.667$$
$$= 138.889 + 1888.889 + 1588.889 + 388.889 + 5855.556 + 6608.333 = 16369.444$$
Calculate denominator:
$$\sum (x_i - \bar{x})^2 = (8.333)^2 + (28.333)^2 + (-21.667)^2 + (-11.667)^2 + (-51.667)^2 + (48.333)^2$$
$$= 69.444 + 802.778 + 469.444 + 136.111 + 2669.444 + 2336.111 = 6283.333$$
5. **Calculate slope:**
$$b = \frac{16369.444}{6283.333} = 2.606$$
6. **Calculate intercept $a$:**
$$a = \bar{y} - b \bar{x} = 393.333 - 2.606 \times 141.667 = 393.333 - 369.167 = 24.167$$
7. **Regression equation:**
$$\hat{y} = 2.606x + 24.167$$
8. **Predict $y$ for given $x$ values:**
(a) $x=160$:
$$\hat{y} = 2.606 \times 160 + 24.167 = 417.6 + 24.167 = 441.767$$
(b) $x=100$:
$$\hat{y} = 2.606 \times 100 + 24.167 = 260.6 + 24.167 = 284.767$$
(c) $x=140$:
$$\hat{y} = 2.606 \times 140 + 24.167 = 364.84 + 24.167 = 389.007$$
(d) $x=60$:
$$\hat{y} = 2.606 \times 60 + 24.167 = 156.36 + 24.167 = 180.527$$
9. **Choose the correct graph:** The data shows a positive correlation (as calories increase, sodium increases), so the graph with an upward-sloping scatterplot and fitted line is correct. This matches graph B or D. Since D mentions the line passing near points, **Graph D** is the best choice.
10. **Prediction for $x=160$:** The predicted $y$ is approximately $441.767$, closest to option A: 439.922.
**Final answers:**
- Regression equation: $$\hat{y} = 2.606x + 24.167$$
- Best graph: D
- Prediction for $x=160$: approximately 441.767 (option A: 439.922)