A regular type of laminate is currently being used by a manufacturer of circuit boards. A special laminate has been developed to reduce warpage. The regular laminate will be used on one sample of specimens and the special laminate on another sample, and the amount of warpage will then be determined for each specimen. The manufacturer will then switch to the special laminate only if it can be demonstrated that the true average amount of warpage for that laminate is less than for the regular laminate. State the relevant hypotheses.

Null and alternative hypothesis:

Being μ1: the mean amount of warpage produced by the special laminate, and μ2: the mean amount of warpage produced by the regular laminate.

Step-by-step explanation:

The manufacturer is looking to test both laminates in order to see if there is a statistical significative improvement for using the special laminate.

The claim that is going to be tested, and then stated in the alternative hypothesis, is that the special laminate produce an amount of warpage that is significantly less than the warpage produced by the regular laminate.

The null hypothesis will state that there is no significant difference between the amount of warpage produced by both laminates.

We can express this null and alternative hypothesis as:

Being μ1: the mean amount of warpage produced by the special laminate, and μ2: the mean amount of warpage produced by the regular laminate.

This hypothesis should be tested with a two-sample t-test.

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

3 is greater then both of those numbers