Question: Use the regression equation to predict the value of $y$ for $x = -2.4$. Assume that the variables $x$ and $y$ have a significant correlation.
$x$ $-5$ $-3$ $4$ $1$ $-1$ $-2$ $0$ $2$ $3$ $-4$
$y$ $11$ $6$ $-6$ $-1$ $3$ $4$ $1$ $-4$ $-5$ $8$
A. $5.282$
B. $-3.704$
C. $0.066$
D. $-3.766$
1. **State the problem:** We want to find the predicted value of $y$ when $x = -2.4$ using the regression equation from the given data.
2. **Calculate the means:**
$$\bar{x} = \frac{-5 + (-3) + 4 + 1 + (-1) + (-2) + 0 + 2 + 3 + (-4)}{10} = \frac{-5}{10} = -0.5$$
$$\bar{y} = \frac{11 + 6 + (-6) + (-1) + 3 + 4 + 1 + (-4) + (-5) + 8}{10} = \frac{17}{10} = 1.7$$
3. **Calculate the slope $b$ of the regression line:**
$$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$
Calculate numerator:
$$\sum (x_i - \bar{x})(y_i - \bar{y}) = (-5 + 0.5)(11 - 1.7) + (-3 + 0.5)(6 - 1.7) + \ldots + (-4 + 0.5)(8 - 1.7)$$
Calculations:
$(-4.5)(9.3) = -41.85$
$(-2.5)(4.3) = -10.75$
$(4.5)(-7.7) = -34.65$
$(1.5)(-2.7) = -4.05$
$(-0.5)(1.3) = -0.65$
$(-1.5)(2.3) = -3.45$
$(0.5)(-0.7) = -0.35$
$(2.5)(-5.7) = -14.25$
$(3.5)(-6.7) = -23.45$
$(-3.5)(6.3) = -22.05$
Sum numerator = -41.85 - 10.75 - 34.65 - 4.05 - 0.65 - 3.45 - 0.35 - 14.25 - 23.45 - 22.05 = -155.5$
Calculate denominator:
$$\sum (x_i - \bar{x})^2 = (-4.5)^2 + (-2.5)^2 + 4.5^2 + 1.5^2 + (-0.5)^2 + (-1.5)^2 + 0.5^2 + 2.5^2 + 3.5^2 + (-3.5)^2$$
$$= 20.25 + 6.25 + 20.25 + 2.25 + 0.25 + 2.25 + 0.25 + 6.25 + 12.25 + 12.25 = 82.5$$
4. **Calculate slope:**
$$b = \frac{-155.5}{82.5} = -1.8848$$
5. **Calculate intercept $a$:**
$$a = \bar{y} - b \bar{x} = 1.7 - (-1.8848)(-0.5) = 1.7 - 0.9424 = 0.7576$$
6. **Regression equation:**
$$y = a + bx = 0.7576 - 1.8848x$$
7. **Predict $y$ for $x = -2.4$:**
$$y = 0.7576 - 1.8848(-2.4) = 0.7576 + 4.5235 = 5.2811 \approx 5.282$$
**Final answer:** $\boxed{5.282}$ which corresponds to choice A.