Young's modulus is a quantitative measure of stiffness of an elastic material. suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 gpa and 1.6 gpa, respectively (values given in the article "influence of material properties variability on springback and thinning in sheet stamping processes: a stochastic analysis" (intl. j. of advanced manuf. tech., 2010: 117– (a) if x is the sample mean young's modulus for a random sample of n = 64 sheets, where is the sampling distribution of x centered, and what is the standard deviation of the x distribution?

0.2 GPa

Step-by-step explanation:

Data provided:

Mean value = 70 GPa

Standard deviation, σ = 1.6 GPa

Now,

a) sample, n = 64

From central limit theorem,

The sample of n = 64 will have the same mean i.e 70 GPa

and,

The standard deviation

σ' =

on substituting the respective values, we get

σ' =

or

σ' = 0.2 GPa

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!