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 \[x=\] stone density \[\left( \dfrac{\text{g}}{\text{cm}^3} \right)\] \[\bar{x}=2.5\] \[s_x=0.3\] \[y=\] cutting depth \[(\text{mm})\] \[\bar{y}=41.7\] \[s_y=42\] \[r=-0.95\] 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

The equation of the least-squares regression line for predicting the cutting depth (\(y\)) from the density of the stone (x) is given by y = 89.98 - 34.73x.

Step-by-step explanation:

The equation of the least-squares regression line is determined by the formula:

`\[ y = \bar{y} + r \left( (s_y)/(s_x) \right) (x - \bar{x}) \]`

Using the given values:

`\[ \bar{y} = 41.7, \bar{x} = 2.5, r = -0.95, s_y = 42, s_x = 0.3 \]`

Substituting these values into the formula:

`\[ y = 41.7 - 0.95 \left( (42)/(0.3) \right) (x - 2.5) \]`

Simplifying further:

y = 89.98 - 34.73x

Therefore, the equation of the least-squares regression line for predicting cutting depth from stone density is y = 89.98 - 34.73x. The negative correlation coefficient (r = -0.95) indicates a strong negative linear relationship between stone density and cutting depth. As stone density increases, cutting depth decreases.

This regression line provides a predictive model for estimating cutting depth based on stone density, allowing the stonemason to make informed decisions about the abrasive water jet cutting process. The coefficients in the equation (89.98 and -34.73) represent the intercept and slope, respectively, and together define the linear relationship between the two variables.