Explain how to perform a two-sample z-test for the difference between two population means using independent samples with σ1 and σ2 known.

Z= (X1`-X2`)- (u1-u2)/   (here s1=σ1 and s2= σ2)

Step-by-step explanation:

Let  X1` be the mean of the first random sample of size n1 from a normal population with a mean u1 and known standard deviation σ1.Let  X2` be the mean of the second random sample of size n2 from another normal population with a mean u2 and known standard deviation σ2. Then the sampling distribution of the difference X1`-X2` is normally distributed with a mean of u1-u2 and a standard deviation of

.  (here s1=σ1 and s2= σ2)  In other words the variable

Z= (X1`-X2`)- (u1-u2)/   (here s1=σ1 and s2= σ2)

is exactly standard normal no matter how small sample sizes are . Hence it is used as a test statistic for testing hypotheses about the difference between two population means.

The procedure is stated below.

1) Formulate the null and alternative hypotheses.

2) Decide on significance level ∝

3) Use the test statistic Z= (X1`-X2`)- (u1-u2)/   (here s1=σ1 and s2= σ2)

4) Find the rejection region

5)Compute the value of Z from the sample data

6) Rehect H0 if Z falls in the critical region, accept H0 , otherwise.

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.