Find the z-score corresponding to the given area. Remember, z is distributed as the standard normal distribution with mean µ = 0 and standard deviation σ = 1. (12 pts.)a. The area to the left of z is 15%.b. The area to the right of z is 65%.c. The area to the left of z is 10%.d. The area to the right of z is 5%e. The area between –z and z is 95%. (Hint: draw a picture and figure out the area to the left of the –z)f. The area between –z and z is 99%

a)

b)

c)

d)  

e)  

f)  

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".  

Part a

Let Z the random variable that represent the variable of a population, and for this case we know the distribution for Z is given by:

Where and

We want 15% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.15,0,1)"

And we got

Part b

We want 65% of the area in the right and that implies 100-65=35% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.35,0,1)"

And we got

Part c

We want 10% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.10,0,1)"

And we got

Part d

We want 5% of the area in the right and that implies 100-5=95% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.95,0,1)"

And we got

Part e

For this case we waant two values that accumulates 95% of the area in the middle. That means we need to have 100-95 % = 5% of the area in the tails and since the distribution is symmetric we need to have 5/2 = 2.5% of the area on each tail. So we can use the following codes to find the two values:

"=NORM.INV(0.025,0,1)"

"=NORM.INV(1-0.025,0,1)"

Part f

For this case we waant two values that accumulates 99% of the area in the middle. That means we need to have 100-99 % = 1% of the area in the tails and since the distribution is symmetric we need to have 1/2 = 0.5% of the area on each tail. So we can use the following codes to find the two values:

"=NORM.INV(0.005,0,1)"

"=NORM.INV(1-0.005,0,1)"

1. f(x)=x²+10x+16

use the formula to find the vertex = (-b/2a, f(-b/2a)) , here in the above equation a=1(as, a> 0 the parabola is open upward), b=10. by putting the values.

-b/2a = -10/2(1) = -5

f(-b/2a)= f(-5)= (-5)²+10(-5)+16= -9

so, vertex = (-5, -9)

now, find y- intercept put x=0 in the above equation. f(0)= 0+0+16, we get point (0,16).

now find x-intercept put y=0 in the above equation. 0= x²+10x+16

x²+10x+16=0 ⇒x²+8x+2x+16=0 ⇒x(x+8)+2(x+8)=0 ⇒(x+8)(x+2)=0 ⇒x=-8 , x=-2

from vertex, y-intercept and x-intercept you can easily plot the graph of given parabolic equation. the graph is attached below.

2. f(x)=−(x−3)(x+1)

by multiplying the factors, the general form is f(x)= -x²+2x+3.

use the formula to find the vertex = (-b/2a, f(-b/2a)) , here in the above equation a=-1(as, a< 0 the parabola is open downward), b=2. by putting the values.

-b/2a = -2/2(-1) = 1

f(-b/2a)= f(1)=-(1)²+2(1)+3= 4

so, vertex = (1, 4)

now, find y- intercept put x=0 in the above equation. f(0)= 0+0+3, we get point (0, 3).

now find x-intercept put y=0 in the above equation. 0= -x²+2x+3.

-x²+2x+3=0 the factor form is already given in the question so, ⇒-(x-3)(x+1)=0 ⇒x=3 , x=-1

from vertex, y-intercept and x-intercept you can easily plot the graph of given parabolic equation. the graph is attached below.

3. f(x)= −x²+4

use the formula to find the vertex = (-b/2a, f(-b/2a)) , here in the above equation a=-1(as, a< 0 the parabola is open downward), b=0. by putting the values.

-b/2a = -0/2(-1) = 0

f(-b/2a)= f(0)= −(0)²+4 =4

so, vertex = (0, 4)

now, find y- intercept put x=0 in the above equation. f(0)= −(0)²+4, we get point (0, 4).

now find x-intercept put y=0 in the above equation. 0= −x²+4

−x²+4=0 ⇒-(x²-4)=0 ⇒ -(x-2)(x+2)=0 ⇒x=2 , x=-2

from vertex, y-intercept and x-intercept you can easily plot the graph of given parabolic equation. the graph is attached below.

4. f(x)=2x²+16x+30

use the formula to find the vertex = (-b/2a, f(-b/2a)) , here in the above equation a=2(as, a> 0 the parabola is open upward), b=16. by putting the values.

-b/2a = -16/2(2) = -4

f(-b/2a)= f(-4)= 2(-4)²+16(-4)+30 = -2

so, vertex = (-4, -2)

now, find y- intercept put x=0 in the above equation. f(0)= 0+0+30, we get point (0, 30).

now find x-intercept put y=0 in the above equation. 0=2x²+16x+30

2x²+16x+30=0 ⇒2(x²+8x+15)=0 ⇒x²+8x+15=0 ⇒x²+5x+3x+15=0 ⇒x(x+5)+3(x+5)=0 ⇒(x+5)(x+3)=0 ⇒x=-5 , x= -3

from vertex, y-intercept and x-intercept you can easily plot the graph of given parabolic equation. the graph is attached below.

5. y=(x+2)²+4

the general form of parabola is y=a(x-h)²+k , where vertex = (h,k)

if a> 0 parabola is opened upward.

if a< 0 parabola is opened downward.

compare the given equation with general form of parabola.

-h=2 ⇒h=-2

k=4

so, vertex= (-2, 4)

as, a=1 which is greater than 0 so parabola is opened upward and the graph has minimum.

hope this : )