Answer:
And we can find this probability using the complement rule:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:
Where 
and 
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the complement rule:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
Answer:
52 inches more deep
Step-by-step explanation:
because she already has 20 inches and then she needs 72 inches so 20-72 gives 52
A. 16 minutes.
b. around 3.5 tasks.
Answer:
(a) Three polynomials of degree 1 with real coefficients belong to the set 
, then:
(b) Three polynomials of degree 1 with real coefficients that hold the relation 
belong to the set 
. The relation between the coefficients is equivalent to 
, then:
Step-by-step explanation:
(a) Three polynomials of degree 1 with real coefficients belong to the set , then:
A vector space is any set whose elements hold the following axioms for any 
and 
and for any scalar 
and 
:
- There is the <em>zero element </em>such that:
- For all element
of the set, there is an element 
such that: 
Let's proof each of them for the first set. For the proof, I will define the polynomials 
, 
and 
and the scalar 
and 
.
and defining 
and 
, we obtain 
which is another polynomial that belongs to 
- A null polynomial is define as the one with all it coefficient being 0, therefore:
- Defining the inverse element in the addition as
, then 
![a[b(a_0+a_1x)] = ab (a_0+a_1x)\\a[ba_0+ba_1x] = aba_0+aba_1x\\\boxed{aba_0+aba_1x = aba_0+aba_1x}](https://tex.z-dn.net/?f=a%5Bb%28a_0%2Ba_1x%29%5D%20%3D%20ab%20%28a_0%2Ba_1x%29%5C%5Ca%5Bba_0%2Bba_1x%5D%20%3D%20aba_0%2Baba_1x%5C%5C%5Cboxed%7Baba_0%2Baba_1x%20%3D%20aba_0%2Baba_1x%7D)
![\boxed{a[(a_0+a_1x)+(b_0+b_1x)] = a(a_0+a_1x) + a(b_0+b_1x)}](https://tex.z-dn.net/?f=%5Cboxed%7Ba%5B%28a_0%2Ba_1x%29%2B%28b_0%2Bb_1x%29%5D%20%3D%20a%28a_0%2Ba_1x%29%20%2B%20a%28b_0%2Bb_1x%29%7D)
With this, we proof the set is a vector space with the usual polynomial addition and scalar multiplication operations.
(b) Three polynomials of degree 1 with real coefficients that hold the relation belong to the set . The relation between the coefficients is equivalent to , then:
Let's proof each of axioms for this set. For the proof, I will define again the polynomials , and and the scalar and . Again the relation between the coefficients holds
![[(a_0+a_1x) +( b_0+b_1x)]+(c_0+c_1x) = (a_0+a_1x) +[( b_0+b_1x)+(c_0+c_1x)]\\(a_0+b_0+c_0) + (a_1+b_1+c_1)x = (a_0+b_0+c_0) + (a_1+b_1+c_1)x](https://tex.z-dn.net/?f=%5B%28a_0%2Ba_1x%29%20%2B%28%20b_0%2Bb_1x%29%5D%2B%28c_0%2Bc_1x%29%20%3D%20%28a_0%2Ba_1x%29%20%2B%5B%28%20b_0%2Bb_1x%29%2B%28c_0%2Bc_1x%29%5D%5C%5C%28a_0%2Bb_0%2Bc_0%29%20%2B%20%28a_1%2Bb_1%2Bc_1%29x%20%3D%20%28a_0%2Bb_0%2Bc_0%29%20%2B%20%28a_1%2Bb_1%2Bc_1%29x)
and considering the coefficient relation and defining and , we have 
which is another element of the set since it is a degree one polynomial whose coefficient follow the given relation.
The proof of the other axioms can be done using the same logic as in (a) and checking that the relation between the coefficients is always the same.
