Question

An investigation of the properties of bricks used to line aluminum smelter pots was published in an article. Six different commercial bricks were evaluated. The life span of a smelter pot depends on the porosity of the brick lining​ (the less​ porosity, the longer the life​ span); consequently, the researchers measured the apparent porosity of each brick​ specimen, as well as the mean pore diameter of each brick. See the table.

Brick	Apparent Porosity​ (%)	Mean Pore Diameter ​(micrometers)
A	18.6	12.0
B	18.3	9.8
C	16.2	7.3
D	6.9	5.2
E	17.4	10.9
F	20.4	16.8

a. Find the least squares line relating porosity​ (y) to mean pore diameter​ (x).

y=________+_________x ​(Round to the nearest thousandth as​ needed.)

b. Interpret the​ y-intercept of the line. Choose the correct answer below.

A.The​ y-intercept is β0. This value has no meaning because 0 is not in the observed range of the independent variable mean pore diameter.

B.There is not enough information to answer this question.

C.The​ y-intercept is β0. For each unit increase in mean pore​ diameter, the mean porosity is estimated to increase by β0.

D.The​ y-intercept is β0. The mean pore diameter is estimated to be β0 when the apparent porosity is zero.

c. Interpret the slope of the line. Choose the correct answer below.

A.The slope of the line is β1. This value has no meaning because 0 is not in the observed range of the independent variable mean pore diameter.

B. There is not enough information to answer this question.

C.The slope of the line is β1. The mean pore diameter is estimated to be β1 when the apparent porosity is zero.

D.The slope of the line is β1. For each unit increase in mean pore​ diameter, the mean porosity is estimated to increase by β1.

d. Predict the apparent percentage of porosity for a brick with a mean pore diameter of 10 micrometers.

y=________%​(Round to the nearest thousandth as​ needed.)

Answer 1

Solution:

Part a. Find the least squares line relating porosity​ (y) to mean pore diameter​ (x).

Regression equation is :

Thus we need to make following table:

Brick	x : Mean Pore Diameter ​(micrometers)	y: Apparent Porosity​ (%)	x^2	xy
A	12	18.6	144	223.2
B	9.8	18.3	96.04	179.34
C	7.3	16.2	53.29	118.26
D	5.2	6.9	27.04	35.88
E	10.9	17.4	118.81	189.66
F	16.8	20.4	282.24	342.72

Thus

Thus

and

where

Thus

Thus regression equation is :

Part b) Interpret the​ y-intercept of the line.

A.The​ y-intercept is β0. This value has no meaning because 0 is not in the observed range of the independent variable mean pore diameter.

Part c) Interpret the slope of the line.

D.The slope of the line is β1. For each unit increase in mean pore​ diameter, the mean porosity is estimated to increase by β1.

Part d) Predict the apparent percentage of porosity for a brick with a mean pore diameter of 10 micrometers.

That is put x = 10 in regression equation obtained in part a)