: For each conjecture below, you are to describe in words what the Null Hypothesis and Alternative Hypothesis are. Consider the decision that you have to make based upon your conjectures. Explain in words or with a chart what the Type I and Type II errors mean in context. Finally, describe the ramifications of making these errors within the context of the problem and describe which of the 2 errors are worse (in your opinion).

1. It is believed that a new drug can cure a cold.
2. The teacher will never check homework today.
3. In a big store like Wal-Mart, they will never catch me if I shoplift.
4. If I try to learn to ski, I will end up hurting myself.

Determine the matrices associated with ProjΠ and ReflΠ where Π : 2x + 5y – z = 0

Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x − ∂M/∂y)/(M − N)
is a function of x + y. That is, let f(t) be a function such that

f(x + y) = (∂N/∂x − ∂M/∂y)/(M − N)
Assume that you can solve the differential equation
M dx + N dy = 0

by multiplying by an integrating factor μ that makes it exact and that it can also be
written as a function of x + y, μ = g(x + y) for some function g(t). Give a method
for finding this integrating factor μ, and use it to find the general solution to the
differential equation

(3 + y + xy)dx + (3 + x + xy)dy = 0.

Please answer all parts of the question. Please show all work and all steps.

1a.) Show that the solutions of x' = arc tan (x) + t cannot have maxima

1b.) Find the value of a such that the existence and uniqueness theorem applies to the ivp x' = (3/2)((|x|)^(1/3)), x(0) = a.

1c.) Find the limits, as t approaches both positive infinity and negative infinity, of the solution Φ(t) of the ivp x' = (x+2)(1-x^4), x(0) = 0

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Solve the initial value problem

dy/dx + H(x)y = e^5x ; y(0) = 2

where

H(x) = −1 0 ≤ x ≤ 3
1 3 < x

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold:

(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).

(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).

(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).

(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}

(Discrete Math) A routing transit number (RTN) is a bank code that appears in the bottom of checks. The most common form of an RTN has nine digits, where the last digit is a check digit. If d1d2 . . . d9 is a valid RTN, the congruence 3(d1 + d4 + d7) + 7(d2 + d5 + d8) + (d3 + d6 + d9) ≡ 0 (mod 10) must hold.

(a) Show that the check digit of the RTN can detect all single errors.

(b) Determine which transposition errors an RTN check digit can catch and which ones it cannot catch.

y''' - 7y'' + 15y' - 9y = 8e^(x) - 9x

A) Find the fundamental set of solutions of the reduced equation. (Hint: 3 is a root of the characteristic polynomial.)

B) Find a particular solution of the given equation.

C) Find the general solution of the given equation.

(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively.

(b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively.

Find all possible orders of elements in the group Z4 × Z5 × Z10. For each possible order, give an example of an element of that order, and prove that no other orders are possible.

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1.

Prove that Φ is an isomorphism is and only if the group G is abelian.

Find all solutions of the equation x^{5}=2

What is the minimum and maximum number of solutions that we can expect to see in any given system of nonlinear equations? In your own words, what is the meaning of extraneous solutions? When solving a system of nonlinear equations, is it possible to always use the Addition Method? Explain your reasoning in complete sentences. PLEASE TYPE, DO NOT WRITE IT DOWN and Check your punctuation and proofreading.