Solved on Dec 07, 2023

Compute the regression line equation for a dataset with $\bar{x}=9$ , $s_{x}=1$ , $\bar{y}=682$ , $s_{y}=51$ , $r=-0.71$ . Round $a$ and $b$ to two decimal places.

STEP 1

Assumptions
1. The given dataset has a mean of $x$ values denoted by $\bar{x}=9$ .
2. The standard deviation of $x$ values is given by $s_{x}=1$ .
3. The mean of $y$ values is denoted by $\bar{y}=682$ .
4. The standard deviation of $y$ values is given by $s_{y}=51$ .
5. The correlation coefficient between $x$ and $y$ values is given by $r=-0.71$ .
6. The regression line we are looking for has the form $\hat{y} = ax + b$ , where $a$ is the slope and $b$ is the y-intercept.
7. The slope $a$ of the regression line can be calculated using the formula $a = r \cdot \frac{s_{y}}{s_{x}}$ .
8. The y-intercept $b$ of the regression line can be calculated using the formula $b = \bar{y} - a\bar{x}$ .

STEP 2

Calculate the slope $a$ of the regression line using the formula $a = r \cdot \frac{s_{y}}{s_{x}}$ .
$a = -0.71 \cdot \frac{51}{1}$

STEP 3

Perform the multiplication to find the slope $a$ .
$a = -0.71 \cdot 51$

STEP 4

Calculate the value of the slope $a$ .
$a = -36.21$

STEP 5

Round the slope $a$ to two decimal places.
$a \approx -36.21$

STEP 6

Now, calculate the y-intercept $b$ of the regression line using the formula $b = \bar{y} - a\bar{x}$ .
$b = 682 - (-36.21 \cdot 9)$

STEP 7

Perform the multiplication and subtraction to find the y-intercept $b$ .
$b = 682 + 36.21 \cdot 9$

STEP 8

Calculate the value of the y-intercept $b$ .
$b = 682 + 325.89$

STEP 9

Calculate the sum to find the y-intercept $b$ .
$b = 1007.89$

STEP 10

Round the y-intercept $b$ to two decimal places.
$b \approx 1007.89$

STEP 11

Write down the equation of the regression line with the calculated slope $a$ and y-intercept $b$ .
$\hat{y} = -36.21x + 1007.89$
The regression line is $\hat{y} = -36.21x + 1007.89$ .

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Compute the regression line equation for a dataset with $\bar{x}=9$ , $s_{x}=1$ , $\bar{y}=682$ , $s_{y}=51$ , $r=-0.71$ . Round $a$ and $b$ to two decimal places.

STEP 1

STEP 2

Calculate the slope $a$ of the regression line using the formula $a = r \cdot \frac{s_{y}}{s_{x}}$ .
$a = -0.71 \cdot \frac{51}{1}$

STEP 3

Perform the multiplication to find the slope $a$ .
$a = -0.71 \cdot 51$

STEP 4

Calculate the value of the slope $a$ .
$a = -36.21$

STEP 5

Round the slope $a$ to two decimal places.
$a \approx -36.21$

STEP 6

Now, calculate the y-intercept $b$ of the regression line using the formula $b = \bar{y} - a\bar{x}$ .
$b = 682 - (-36.21 \cdot 9)$

STEP 7

Perform the multiplication and subtraction to find the y-intercept $b$ .
$b = 682 + 36.21 \cdot 9$

STEP 8

Calculate the value of the y-intercept $b$ .
$b = 682 + 325.89$

STEP 9

Calculate the sum to find the y-intercept $b$ .
$b = 1007.89$

STEP 10

Round the y-intercept $b$ to two decimal places.
$b \approx 1007.89$

STEP 11

Write down the equation of the regression line with the calculated slope $a$ and y-intercept $b$ .
$\hat{y} = -36.21x + 1007.89$
The regression line is $\hat{y} = -36.21x + 1007.89$ .

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Compute the regression line equation for a dataset with xˉ=9\bar{x}=9xˉ=9, sx=1s_{x}=1sx​=1, yˉ=682\bar{y}=682yˉ​=682, sy=51s_{y}=51sy​=51, r=−0.71r=-0.71r=−0.71. Round aaa and bbb to two decimal places.

STEP 1

STEP 2

Calculate the slope aaa of the regression line using the formula a=r⋅sysxa = r \cdot \frac{s_{y}}{s_{x}}a=r⋅sx​sy​​.a=−0.71⋅511a = -0.71 \cdot \frac{51}{1}a=−0.71⋅151​

STEP 3

Perform the multiplication to find the slope aaa.a=−0.71⋅51a = -0.71 \cdot 51a=−0.71⋅51

STEP 4

Calculate the value of the slope aaa.a=−36.21a = -36.21a=−36.21

STEP 5

Round the slope aaa to two decimal places.a≈−36.21a \approx -36.21a≈−36.21

STEP 6

Now, calculate the y-intercept bbb of the regression line using the formula b=yˉ−axˉb = \bar{y} - a\bar{x}b=yˉ​−axˉ.b=682−(−36.21⋅9)b = 682 - (-36.21 \cdot 9)b=682−(−36.21⋅9)

STEP 7

Perform the multiplication and subtraction to find the y-intercept bbb.b=682+36.21⋅9b = 682 + 36.21 \cdot 9b=682+36.21⋅9

STEP 8

Calculate the value of the y-intercept bbb.b=682+325.89b = 682 + 325.89b=682+325.89

STEP 9

Calculate the sum to find the y-intercept bbb.b=1007.89b = 1007.89b=1007.89

STEP 10

Round the y-intercept bbb to two decimal places.b≈1007.89b \approx 1007.89b≈1007.89

STEP 11

Write down the equation of the regression line with the calculated slope aaa and y-intercept bbb.y^=−36.21x+1007.89\hat{y} = -36.21x + 1007.89y^​=−36.21x+1007.89The regression line is y^=−36.21x+1007.89\hat{y} = -36.21x + 1007.89y^​=−36.21x+1007.89.

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Compute the regression line equation for a dataset with $\bar{x}=9$ , $s_{x}=1$ , $\bar{y}=682$ , $s_{y}=51$ , $r=-0.71$ . Round $a$ and $b$ to two decimal places.

Calculate the slope $a$ of the regression line using the formula $a = r \cdot \frac{s_{y}}{s_{x}}$ .
$a = -0.71 \cdot \frac{51}{1}$

Perform the multiplication to find the slope $a$ .
$a = -0.71 \cdot 51$

Calculate the value of the slope $a$ .
$a = -36.21$

Round the slope $a$ to two decimal places.
$a \approx -36.21$

Now, calculate the y-intercept $b$ of the regression line using the formula $b = \bar{y} - a\bar{x}$ .
$b = 682 - (-36.21 \cdot 9)$

Perform the multiplication and subtraction to find the y-intercept $b$ .
$b = 682 + 36.21 \cdot 9$

Calculate the value of the y-intercept $b$ .
$b = 682 + 325.89$

Calculate the sum to find the y-intercept $b$ .
$b = 1007.89$

Round the y-intercept $b$ to two decimal places.
$b \approx 1007.89$

Write down the equation of the regression line with the calculated slope $a$ and y-intercept $b$ .
$\hat{y} = -36.21x + 1007.89$
The regression line is $\hat{y} = -36.21x + 1007.89$ .