Hi,Re TY, good news is that You will not be expected to do this "long-hand"

that's why there are charts, Excel functions, calculators like the TI83, etc

to find either the z-value given the P-value or the P-value given the z-value



for ex: using Excel function:  NORMSINV(.9625) = 1.7805 

and P(-2.00 <_ z <_ +2.00)= NORMSDIST(2)-NORMSDIST(-2) = .9544



Important to Understand z -values as they relate to the Standard Normal curve:

Below:  z = 0, z = ± 1, z= ±2 , z= ±3 are plotted.  

Note: z = 0 , 50% of the area under the curve is to the left and 50%  to the right





This particular 'chart' below shows the amount of area under the Normal distribution curve between z = 0 and the z given 

(half the total area to the left of the z-value)

that is to say one would add .50 to each entry for total area under the curve to the left of the z value.

Highlighted + .50 is .9625 and P(z<+a)=0.9625, a = 1.78  (the 8 coming from the .08 column)

As to P(-2.00 <_ z <_ +2.00) = .4772 + .4772 = .9544

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09

0.0	0.0000	0.0040	0.0080	0.0120	0.0160	0.0199	0.0239	0.0279	0.0319	0.0359

0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636	0.0675	0.0714	0.0753

0.2	0.0793	0.0832	0.0871	0.0910	0.0948	0.0987	0.1026	0.1064	0.1103	0.1141

0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406	0.1443	0.1480	0.1517

0.4	0.1554	0.1591	0.1628	0.1664	0.1700	0.1736	0.1772	0.1808	0.1844	0.1879

0.5	0.1915	0.1950	0.1985	0.2019	0.2054	0.2088	0.2123	0.2157	0.2190	0.2224

0.6	0.2257	0.2291	0.2324	0.2357	0.2389	0.2422	0.2454	0.2486	0.2517	0.2549

0.7	0.2580	0.2611	0.2642	0.2673	0.2704	0.2734	0.2764	0.2794	0.2823	0.2852

0.8	0.2881	0.2910	0.2939	0.2967	0.2995	0.3023	0.3051	0.3078	0.3106	0.3133

0.9	0.3159	0.3186	0.3212	0.3238	0.3264	0.3289	0.3315	0.3340	0.3365	0.3389

1.0	0.3413	0.3438	0.3461	0.3485	0.3508	0.3531	0.3554	0.3577	0.3599	0.3621

1.1	0.3643	0.3665	0.3686	0.3708	0.3729	0.3749	0.3770	0.3790	0.3810	0.3830

1.2	0.3849	0.3869	0.3888	0.3907	0.3925	0.3944	0.3962	0.3980	0.3997	0.4015

1.3	0.4032	0.4049	0.4066	0.4082	0.4099	0.4115	0.4131	0.4147	0.4162	0.4177

1.4	0.4192	0.4207	0.4222	0.4236	0.4251	0.4265	0.4279	0.4292	0.4306	0.4319

1.5	0.4332	0.4345	0.4357	0.4370	0.4382	0.4394	0.4406	0.4418	0.4429	0.4441

1.6	0.4452	0.4463	0.4474	0.4484	0.4495	0.4505	0.4515	0.4525	0.4535	0.4545

1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608	0.4616		0.4633

1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686	0.4693	0.4699	0.4706

1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.4750	0.4756	0.4761	0.4767

2.0		0.4778	0.4783	0.4788	0.4793	0.4798	0.4803	0.4808	0.4812	0.4817

 

Hi,

the sum of two numbers&  is 19 and the sum of their squares is 221

Question states*** 

  x + y = 19  0r  

x^2 + y^2 = 221

 x^2 + (19-x)^2 = 221

 x^2 + 361-38x + x^2 = 221

  2x^2 - 38x  + 140 = 0

   x^2 - 19x + 70 = 0

factoring

(x - 14)(x-5) = 0

(x - 14)= 0,  x = 14   and y = 5

0r(x-5) = 0,  x =  5   and y = 14

numbers are 5 and 14

 

Hi,

The supplement of an angleis  10 degrees more than 5 times its complement

Question states***

 (180-x) = 5(90-x) + 10   || Solve for x

 

Hi,Two buses leave at the same time in opposite directions. 

One bus is traveling at 57 mph and the other at 60 mph. 



Note: both will travel the same amount of time

 , t = 5.5hr  

 

Hi, 

Standard Form of an Equation of an Ellipse is  

where Pt(h,k) is the center. (a variable positioned to correspond with major axis)

 a and b  are the respective vertices distances from center

(0, -1) and (12, -1), C(6,-1) and minor axis of length 6. b = 3 

 





the vertex form of a Parabola opening up(a>0) or down(a<0),  

where(h,k) is the vertex  and  x = h  is the Line of Symmetry

  x = -3  is the Line of Symmetry

 

Hi,

3x+2y+z=7, 5x+5y+4z=3, 3x+2y+3z=1



3x+2y+z=7

3x+2y+3z=1  || Subtract 2nd from 1st

  -2z = 6

    z = -3



3x+ 2y = 10    ||multiply 1st EQ by 5 and subtract from 3 times the 2nd EQ

5x + 5y = 15

       5y = -5

        y = -1  and x = 4

and...Using 3x+2y+3z=1

 

 

 

Hi,

Frank has 50 coins, all quaters and dimes. 

If he has q quarters, how many dimes does he have? 

 

Hi,

In a primary care clinic, 25% of all patients who have appointments fail to show up. 

10 scheduled appointments:

P(1/2 show up) = 

P(all show) = 

week: 100 appointments. Would expect  = 25 to not show Up

 

Hi,

10 gallons cost $45 but has a 10% discount

With a 10% discount, the sale price is 90% of the original price

 90% of $45 is  = $40.50   

 

Hi,

   | Cross Multiplying to solve

   5x = 8000

    x = 1600

 

Hi,

The measure of the smallest angle of a triangle is 1/4 the measure of the largest angle. 

The measure of the second angle is 45 degrees less than the measure of the largest angle.

Question states***

  x/4 + (x-45) + x = 180°

       (9/4)x = 225

            x = 225(4/9)= 100°, the Largest angle. Smallest 25° and other is 55°

and...

 

 

Hi,

Note: vertex form of a Parabola opening up(a>0) or down(a<0),  

where(h,k) is the vertex  and  x = h  is the Line of Symmetry

The standard form is , where  the focus is (h,k + p)

 ,  V(0,0), 4p = 12,  p = 3, F(0,3) and directrix is y = -3

 

Hi,

One angle in a triangle has a measure that is three times as large as the smallest angle. 

The measure of the third angle is 45 degrees more than that of the smallest angle. 

smallest angle:  x

2nd angle:  3x

3rd angle: x + 45

Question states***

  x + 3x + (x+ 45) = 180°  

        5x = 135°  

         x = 27° , smallest angle.  largest is 81° 

and (3rd angle is 72°  )

 27°  + 81° + 72°   = 180°  

 

Hi,

Using the standard slope-intercept form for an equation of a line y = mx + b

  where m is the slope and b the y-intercept.

slope -5/6 and y-intercept (0, 4)

    and (6, -1) also on this Line   (x = 6 ⇒ y = -1)

Plot (x,y) ordered pairs:  (0,4) and (6, -1) and connect with Line

 

Hi, this makes more CENTS :)

Sam has five more dimes than quarters and four fewer nickels than quarters

Quarters: x

Dimes: x+5

Nickels: x - 4

Question states***total of $4   0r 470 CENTS

CENTS makes sense

  25x + 10(x+5) + 5(x-4) = 470¢

     40x + 50 - 20 = 470

         40x = 440

           x = 11, number of quarters.   16 Dimes and 7 Nickels

and

  = 470¢   0r $4.70

 

Hi, this makes more CENTS :)

Sam has five more dimes than quarters and four fewer nickels than quarters

Quarters: x

Dimes: x+5

Nickels: x - 4

Question states***total of $4   0r 470 CENTS

CENTS makes sense

  25x + 10(x+5) + 5(x-4) = 470¢

     40x + 50 - 20 = 470

         40x = 440

           x = 11, number of quarters.   16 Dimes and 7 Nickels

and

  = 470¢   0r $4.70

 

Hi,

 x + (x+2) + (x+4) = 75, 3x = 69, x = 23  |||   23, 25, 27 the 3 consecutive odd integers

"PEMDAS", "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction".

2(4x-3)-8=4+2x

 8x - 6 - 8 = 4 + 2x,  6x = 18,  x = 18/6  0r  x = 3

2/3(3x+18)=5x-9 

  2x + 12 = 5x - 3,  15 = 3x,  x = 5 

 

Hi,

An adult ticket for the fair  is 7 dollars more than a childs ticket which cost 5 dollars

childs ticket:  $5

adult ticket : a = $5 + $7 = $12

Cost of two adults and three children:

    = $39 

 

Hi,

Combining Like Terms and Isolating the variable on one side of the EQ

-4x + 10 = -5x + 18   |||adding 5x and -10 to both sides of the EQ

      x = 8

By isolating the variable...now we can read  

 

Hi,

The oldest girl is three times older than the youngest age . 

The middle girl is 3 years older than the youngest.

youngest:  x

middle girl: x+ 3

oldest girl: 3x

Question states***The sum of their age is 28 

  x + (x+3) + 3x = 28

     5x = 25

      x = 5, the age of the youngest.  The middle girl is 8 



and (oldest is 15)

 

Hi,

two more than 6 times a number is less than 11 

Question states***

  6x+ 2 < 11

    6x < 9

     x < 9/6    0r  x < 3/2

 

Hi,

how to solve this problem

Combining Like Terms and isolating the variable on one side of the EQ 

7x-9x+10=-4x+6

 -2x + 10 = -4x + 6   ||adding 4x and -10 to both sides of the EQ

    2x = -4

     x = -2

 

Hi,

a+b=39 and 

a-b=11   || Adding the two EQs to eliminate b

  2a = 50

   a = 25   and b = 14    

 

Hi,

three consecutive integers , & 

 such that four times the square of the third, less than three times the square of the first, 

minus 41, is twice the square of the second

Question states***

 4(x+2)^2 - 3x^2 - 41 = 2(x+1)^2

 4(x^2 + 4x + 4) - 3x^2 - 41 = 2(x^2 + 2x+ 1)

 4x^2 + 16x + 16 - 3x^2 - 41 = 2x^2 + 4x + 2

                0 = x^2 - 12x + 27

                0 = (x-9)(x-3)

                  x = 9  0r x = 3

Integers are 9, 10, 11

0r  3,4,5

 

Hi,

four times a certain number increased by 6 is equal to 94

Question states***

 4n + 6 = 94

   4n = 88

    n = 22

 

Hi,

the sum of three numbers is 75. 

The second number is 5 more than 4 times the first, and the third is 2 times the first.

first number:  x

second number: 4x+5

third number: 2x

Question states*** sum of these numbers is 75

 x + (4x+5) + 2x = 75

         7x = 70

          x = 10, first number.  Second number is 45.

and...(third is 20)

 

Hi,

Question states***regular triangle has a perimeter of 12x + 3

regular triangle is an equilateral triangle (a triangle in which all three sides are equal) 

  

expression for the length of one of its sides is:  4x+1

 

Hi,