Math / Data & Statistics

QuestionCompute the least-squares regression line for predicting $y$ from $x$ given the following summary statistics. Round the slope and $y$ -intercept to at least four decimal places. $\begin{array}{ll} \bar{x}=12.5 & s_{x}=2.3 \\ s_{y}=13,000 & r=0.40 \end{array}$
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Regression line equation: $\hat{y}=$ $\square$

Studdy Solution

STEP 1

1. We are given the summary statistics for a dataset: $\bar{x} = 12.5$ , $s_x = 2.3$ , $s_y = 13,000$ , and $r = 0.40$ .
2. The formula for the least-squares regression line is $\hat{y} = b_0 + b_1x$ , where $b_1$ is the slope and $b_0$ is the y-intercept.
3. The slope $b_1$ is calculated using the formula $b_1 = r \cdot \frac{s_y}{s_x}$ .
4. The y-intercept $b_0$ is calculated using the formula $b_0 = \bar{y} - b_1 \cdot \bar{x}$ .

STEP 2

1. Calculate the slope $b_1$ .
2. Calculate the y-intercept $b_0$ .
3. Write the equation of the regression line.

STEP 3

Calculate the slope $b_1$ using the formula:
$b_1 = r \cdot \frac{s_y}{s_x}$
Substitute the given values:
$b_1 = 0.40 \cdot \frac{13,000}{2.3}$
$b_1 = 0.40 \cdot 5652.1739$
$b_1 \approx 2260.8696$

STEP 4

Calculate the y-intercept $b_0$ using the formula:
$b_0 = \bar{y} - b_1 \cdot \bar{x}$
Substitute the given values and the calculated slope:
$b_0 = 30.2 - 2260.8696 \cdot 12.5$
$b_0 = 30.2 - 28260.8700$
$b_0 \approx -28230.6700$

STEP 5

Write the equation of the regression line:
$\hat{y} = -28230.6700 + 2260.8696x$
The least-squares regression line is:
$\hat{y} = -28230.6700 + 2260.8696x$

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