Standard Deviation word problem in Probability Theory

In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. standard deviation formula is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation.We can use Standard deviation calculator to make it easy.

In simple terms, it shows how much variation there is from the "average" (mean). It may be thought of as the average difference of the scores from the mean of distribution, how far they are away from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.Let's see an example from probability problems
Question

A volunteer ambulance service handles 0 to 5 service calls on any given day .The probability distribution for the number of service calls is as follows :

Number of calls : 0 probability 0.08,1-0.16,2-0.27,3-0.20,4-0.15and 5-0.14

a)What is the expected number of service calls(2 decimals)?

b)What is the variance in the number of service calls(to 2 decimals)?

c)What is the standard deviation (to 2 decimals)?

Answer

a)

Expected number= 0(0.08)+1(0.16)+2(0.27)+3(0.20)+4(0.15)+5(0.14)

=0+0.16+0.54+0.60+0.60+0.70

=2.6


b)

Var(x) = E(x²)-[E(x)]²

=8.94-(2.60)²

=8.94+6.76

=2.18

c)

Standard deviation = v2.18 = 1.48

How to solve a inequality

An inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality).these points covered when linear equations explained

• The notation a < b means that a is less than b.
• The notation a > b means that a is greater than b.
• The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

In each statement above, a is not equal to b. These relations are known as strict inequalities. Let's see an example on this from 9th grade math

Question:-

Solve the inequality

13 ≤ 3y+4 < style="font-weight: bold; color: rgb(51, 51, 255);">

Answer:-


We have to find the range of y.

So we isolate y.

13-4 ≤ 3y+4-4 < 18-4

9 ≤ 3y < 14

Noe divide each term by 3.

we get 3 ≤ y < 14/3 is the answer

similarly we can get more free answers to math problems

Problem on Distance Formula

The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.

Lets see an example on this from college algebra tutorial


Find the distance between(-3,-2)and (1,7)

here we have

x1=-3 ,y1=-2 and x2=1 and y2=7


We know the distance formula

D= √(x₂-x₁)²+(y₂-y₁)²

x₂ and y₂ are the x,y coordinates for one point
x₁and y₁ are the x,y coordinates for the second point
d is the distance between the two points
one of the Discrete math Tutor is solving the above problem here

D =√(-3-1)²+(-2-7)²

= √ (-4)² +(-9)²

= √ 16 + 81

= 9.848

So the distance is 9.85 approximately

Question on Calculating Simple Interest

In simple term the word Simple Interest refers just the amount of money paid on a loan. It's the easiest type of interest or most basic type of interest. In order to understand how various types of transactions work, it helps to have a complete understand the terms used.

For example: you may pay interest on a car loan, and it is very much necessary and important to understand how interest works. Better yet, your bank may be paying you interest on your deposits – and you can maximize your earnings by knowing more about interest.

Here S.I refers to simple interest.
Formula for S.I =

P x T x R
  100

Question: If 11,000,00 dollars is invested. Part of the 11,000,00 is invested 4% and the rest at 7%.
If the investment earns $888.00 per year,how much was invested at each rate?

Answer :

Principal time rate of interest simple interest earned
   x      1            4%              0.04x
1000-x    1            7%             .07(11000-x)

Simple interest = PTR
Where P is the Principal. T is the time and R is the rate of interest
SI = 680
Therefore

0.04x + 0.07(11000-x) = 680
0.04x + 770 - 0.07x   = 680

     -0.03x + 770   = 680
     -0.03x         = 680 - 770
     ------           ----------  
     -0.03            -0.03
          x         = 90/0.03
          x         = 3,000

x = 3,000
so, 3,000 is invested at 4% and 8,000 is the principal invested at 7%

A problem on Logarithms

Topic:-Logarithms

The logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.
For example, the logarithm of 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10³ = 1000, so log₁₀1000 = 3, and 2⁵ = 32, so log₂32 = 5.
Let's see a problem on logarithms.

Question:-

solve 6^x+5=3^x-4

Answer:-

Given 6x+5=3x-4

log 6 x+5=log 3x-4

(x+5)log6 = (x-4)log3

xlog6+5log6 = xlog3-4log3

xlog6-xlog3= -4 log3-5log6

x[log6-log3]= -[4log3+5log6]

xlog 6/2 = -[log 34+log65]

xlog3= -log34*65

     -log(34*65)
x=  -----------------
       log3

x=-log3(34*65)

For more help on this ,you can reply me.

problem on logarithms

Topic:-logarithms

To solve logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.
Example logarithm problems : 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10³ = 1000, so log₁₀1000 = 3, and 2⁵ = 32, so log₂32 = 5.
The logarithm of x to the base b is written log_b(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

This math help is having a example

Question:-

         y        
solve log2---- = 4          
         3

Answer:-

         y        
here  log2---- = 4          
         3

log rule:-

logba = x

bx=a

So,We get
             y
   24= ---
             3

            y
  2*2*2*2= ---
            3
           y
     16 = ---
           3

    16*3=y

    So y=48 is the Answer.

For more help on this Please reply me

Calculating Area by Integration

Integration is an important concept in calculus where for a given relations and functions, x is a real variable and for the interval, integral will be calculated. Here is one such example.

Topic : Finding an area of the given function by Integration
Definite and indefinite integrals are the two special integrals.

Problem : Given function y = x² - 5x - 6. Find the area by Integration when function is enclosed by x - axis at the interval (0,4)

Solution :
The area is given by the formula

area = _a∫^b f(x) dx = _a∫^b y dx

given y = x² - 5x - 6
enclosed by x - axis, x = 0 and x = 4
Therefore,

Area = _{x = 4}∫^{x = 0} y dx

= ₀∫⁴ (x² - 5x - 6) dx

= [x³/3 - 5x²/2 - 6x _x=0]^x=4

= 4³/3 - 5(4)²/2 - 6(4)

= 64/3 - 80/2 - 24

= (128 - 240 - 144)/6

= (128 - 384)/6

= -256/6


So area is approximately 42.5 squnits

If you have any queries related to the above problem or the concept please feel free to contact calculus help.

Question on Finding Volume of the Solid, Using Shell Method for setup and Evaluating the integral

Here is a Post on Finding Volume of a Solid by shell Method set up and Evaluating the integral.

Topic : Volume of a Solid

On Integration also one can find out volume of any solid.

Question : Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis
y = 2x, y = 4, x = 0

Solution :

We know that the volume of the solid formed by rotating the area between the curves of f(x)
and g(x) and the lines x = a and x = b about the y-axis is given by







The graph of the line “ y = 2x” and “y = 4” between the two axes is denoted by the shaded region in the graph drawn below:






















So, the volume of the shaded region when revolved around y-axis is given by:





















Hence, the volume of the solid generated by revolving the curves “ y = 2x” and “y = 4” around y axis in the interval [0,2] is “16π /3”.


If you have any queries do write to our calculus help.

Question on Solving a Problem using Proportion Method

Topic : Proportion Method

Question : Fuji apples cost $2.10 for 3 lb. Us a proportion to find how much it would cost to buy 7 lb. By proportion method :

Solution :
2.10/3 = ?/7
0.7 x 7=$ ?
$ 4.9 will be the cost of 7 Fuji Apples$

Topic : Parallelogram

Question : If ABCD is a parallelogram with equal diagonals, then prove that ABCD is a rectangle.

Solution :

Given ABCD is a parallelogram AC = BD

To prove : ABCD is a rectangle

Proof : In ∆ABD and ∆BCA

AD = BC (opposite sides of a parallelogram are equal)

BD = AC (given)

AB = AB

Therefore by SSS theorem of congruency

∆ADB congruent to ∆BCA

So ∟DAB = ∟CBA (corresponding angles of congruent triangles)

Now ∟DAB + ∟CBA = 180º (sum of consecutive angles of a parallelogram is 180º)

So ∟DAB =∟CBA = 180º/2 = 90º

If one of the angle of a parallelogram is a right angle then the parallelogram is a rectangle.

Hence ABCD is a rectangle.

Question on Triangles Congruency

Topic : Congruency of Triangle

Question to Find Length of the Graph of a Function

Topic : Functions and Graphical Representation

Question : Find the arc length of the graph of the function over the indicated interval.
y = 2x^(3/2) + 3 Length of the arc on [0,8]

Solution :
We know that the arc length of curve formed the curve of f(x) in the interval [a, b] is given by

Convex and Non-Convex Polygons

Topic : Polygons

Question : Draw the shape of convex and non-convex polygons.

Solution :
Polygons are as shown below,

Dimensions of Rectangle when Value of Diagonal is Given

Topic : Dimension of Rectangle

Question : Find the dimensions of rectangle RECT if the diagonals intersect at a 60º angle and RC = 18√5 inches. Leave your answer in simplest radical form.

Solution :

Given RC = 18√5 inches

Since RECT is a rectangle both diagonals are equal and bisect eachother

we have OE = OC = OT = OR = 1/2(18√5) = 9√5inches

Consider triangle OEC as
∟EOC = 60º and OE = OC

It is an equilateral triangle. Hence EC = OE = OC = 9√5

To find TC :

Consider triangle OCT; using sine rule

9√5 / Sin 30º = TC / Sin 120º

TC = (Sin 120º * 9√5 ) / Sin 30º

= [(√3/2) * 9√5] /1/2

= 9√5 * √3

= 9√15 inches

So the sides TC = RE = 9√15 inches

Hence the dimensions of RECT are 9√15 inches , 9√5 inches

Question to Find the Unknown Value of a Angles

Topic : Similar Triangles

Question :

If two triangles with vertex A and ECD are similar to each other. If ∟A + ∟ECD = 80º , then find ∟B + ∟D ?

Solution :

∟∆BAC ~ ∆ECD
So , ∟A = ∟EDC
∟B = ∟DEC
∟ACB = ∟D

AB = AC (as ∟BAC is an isosceles triangle)
DE = DC(as ∟ECD is an isosceles triangle)
So, AB = ∟B = ACB
And ∟DEC = ∟ECD

Now ∟A + ∟ECD = 80º (given)
So, ∟A = ∟ECD = 80º/2 = 40º (as ∟A = ∟ECD)

In ∆ABC, ∟A + ∟B + ∟ACB = 180º
40º + ∟B + ∟B = 180º (as ∟A = 40º, ∟B = ∟ACB)
2 ∟B = 180º - 40º
∟B = 140º / 2
∟B =70º

Similarly, in ∆ECD, ∟D =70º
So, ∟B + ∟D = 70º + 70º =140º

Question on Profit and Loss

Topic : Profit and Loss

Question : Vimmi buys a sewing machine for Rs 3725 and spends Rs 275 on repairs. She then sells it for Rs 5000. Find her profit percent or loss percent.

Solution :
C.P = Rs 3725
Expenses = Rs 275
So C.P = Rs 3725 + Rs 275 = Rs 4000
S.P = Rs 5000
Profit = S.P – C.P
Profit = Rs 5000 – Rs 4000
Profit = Rs 1000
Profit Percent = (Profit Amt x 100 ) / C.P
Profit Percent = (1000 x 100) / 4000
Profit Percent = 100 / 4
Profit Percent = 25 %

Question on Factorisation using Identities

Topic : Factorisation using identities

Problem : Factorise 49p^2 – 36

Solution :
We know that (a + b)^ 2 = a^2 + 2ab + b^2 …………… (I)
(a – b) 2= a^2 – 2ab + b62………….. (II)
(a + b) (a – b) = a^2 – b^2………… (III)
Let us use these identities for factorisation.

There are two terms; both are squares and the second is negative. The expression is of the form (a^2 – b^2). Identity III is applicable here;
49 p^2– 36 = (7p)^ 2 – (6)^2
= (7p – 6) (7p + 6) (required factorisation)

Question on Equation of Line

Topic : Equation of a line l

Question : Find the Equation of a line l that goes through (12,1) and has a slop of ¼ ?

Solution :

We have Standard Form for “Point Slope”

y – y1 = m (x - x1)

Given (x1,y1) = (12,1)
m = ¼

Substituting in Standard Form

y -1 = ¼ (x - 12)
y = x/4 -3 + 1
y = x/4 -2

Hence the Equation of Line l is y = x/4 – 2

Hope the Answer will be helpful for you.

Word Problem

Topic : Age Word Problem

Question : When Juan divides each of five consecutive integers by his age, the sum of the five remainders he gets is 32. when nicki, who is several years older divides each of a diferrent set of five consecutive integrs by her age, the sum of the five remainders is also 32. what is the sum of the ages of juan and nicki?

Answer :

The answer I got is
1)First person,s Age16----- consecutive integers 14,15,16, 17,18 remainders are 14, 15, 0, 1,2 total 32
2)Second person’s Age27----- consecutive integers are 26,27,28,29,30
Remainders are 26, 0, 1, 2, 3 total 32
sum of the ages= 43
Method I had adopted
Remainders can be
r, r+1, r+2, r+3,r+4, total 32= 5r+10 implies r is not an integer , hence ruled out.
r, r+1,r+2, r+3,0 total32= 4r+ 6 implies r is not an integer ruled out
r, r+1, r+2, 0,1 total 32 = 3r+4implies r= not an integer
r,r+1,0, 1,2 total 32= 2r +4 r= 14 implies possibility
r,0, 1,2,3 total 32= r+6 , r= 26 possible
0,r,r+1,r+2, r+3, total 32= 4r+ 6 implies r not an integer
Case1 r=14
This implies remainders 14,15,0,1,2 -------total 32 consecutive integers are 14,15,16,17,18
Evidently 16 meets the requirement. That is the age of person1 is 16
Case2 r= 26,0,1,2,3, consecutive integers are 26,27,28,29,30
Evidently 27 meets the requirement,Age of second person is 27

Hope the Entire Explanation did the needful.