Question

A stonemason wants to look at the relationship between the density of stones she cuts and the depth to which her abrasive water jet cuts them. The data show a linear pattern with the summary statistics shown below: mean standard deviation x=x=x, equals stone density \left( \dfrac{\text{g}}{\text{cm}^3} \right)( cm 3 g ​ )left parenthesis, start fraction, start text, g, end text, divided by, start text, c, m, end text, cubed, end fraction, right parenthesis \bar{x}=2.5 x ˉ =2.5x, with, \bar, on top, equals, 2, point, 5 s_x=0.3s x ​ =0.3s, start subscript, x, end subscript, equals, 0, point, 3 y=y=y, equals cutting depth (\text{mm})(mm)left parenthesis, start text, m, m, end text, right parenthesis \bar{y}=41.7 y ˉ ​ =41.7y, with, \bar, on top, equals, 41, point, 7 s_y=42s y ​ =42s, start subscript, y, end subscript, equals, 42 r=-0.95r=−0.95r, equals, minus, 0, point, 95 Find the equation of the least-squares regression line for predicting the cutting depth from the density of the stone. Round your entries to the nearest hundredth.

Answer 1

Answer: y=374.2 -133 x

​Step-by-step explanation: I just did the khan test

Least-squares regression equation

The equation for the least-squares regression line for predicting yyy from xxx is of the form:

\hat{y}=a+bx

y

^

​

=a+bxy, with, hat, on top, equals, a, plus, b, x,

where aaa is the yyy-intercept and bbb is the slope.

Hint #22 / 4

Finding the slope

We can determine the slope as follows:

b=r\left(\dfrac{s_y}{s_x}\right)b=r(

s

x

​

s

y

​

​

)b, equals, r, left parenthesis, start fraction, s, start subscript, y, end subscript, divided by, s, start subscript, x, end subscript, end fraction, right parenthesis

In our case,

b=-0.95\left(\dfrac{42}{0.3}\right)=-133b=−0.95(

0.3

42

​

)=−133b, equals, minus, 0, point, 95, left parenthesis, start fraction, 42, divided by, 0, point, 3, end fraction, right parenthesis, equals, minus, 133

Hint #33 / 4

Finding the yyy-intercept

Because the regression line passes through the point (\bar x, \bar y)(

x

ˉ

,

y

ˉ

​

)left parenthesis, x, with, \bar, on top, comma, y, with, \bar, on top, right parenthesis, we can find the yyy-intercept as follows:

a=\bar y-b\bar xa=

y

ˉ

​

−b

x

ˉ

a, equals, y, with, \bar, on top, minus, b, x, with, \bar, on top

In our case,

a=41.7 +133 (2.5)=374.2a=41.7+133(2.5)=374.2