Bias in Mathematical Terms

Introduction to bias in mathematical terms:

Bias in mathematical terms is applicable to the die or dies problem, biased dies are not give the probability of the equal number such as (2,2), example in standard die the number 3 will be appear in 6 chances (it means we have to roll the die 6 times )so the probability is 1/6.but in the biased die the number 3 will be appear 3 times in 6 rolls so the probability is 3/6=1/2


Types of bias- bias in mathematical terms:


In statistics the bias can be classified as many types

Selection bias
Systematic bias
Data-snooping bias
Statistical hypothesis testing
Sampling bias
Inductive bias
Notation bias
Publication bias
Infrastructure bias
Cognitive bias
Cognitive bias- bias in mathematical terms:

In this type only one bias we called as the conformation bias. I have recently faced lot of problem while learning Definition of Rational Number, But thank to online resources of math which helped me to learn myself easily on net.


Example problems - bias in mathematical terms:


Biased estimation is used to estimate the standard deviation of population it will be calculated by using the standard deviation of sample of size m

Solution:

Consider m=1

Here m is the standard deviation of sample size, so the standard deviation of sample size 1 will be always zero, so we cannot find the deviation for the single value, so we can tell the estimation is zero and the average of the standard deviation is zero, but in generally the population have some standard deviation value, so the above estimation will be the biased

Example 2:

Find the value of 13.39`xx` 3.45?

A) 461955

b) 461.955

c) 46.1955

D) 4619.55

Solution:

In the problem after decimal 2 digits are there (for both), therefore the product must contains 4 decimal places

1339`xx` 345=461955(here we did not consider decimal we consider only the digit)

Now take the decimal

46.1955

So the answer is c

(So here the biased question are based on the assumptions)

Online Geometry Book

Introduction:

Geometry is a construction of object according to our given measurement. Geometry books contains the collection of object  which could be square, rectangle, triangle, circle, parallelogram, cone, cylinder, cube, prism, quadrilateral. Basic geometry concepts are plane, points, rays, lines, line segment, type of lines and  angles. Main purpose of using geometry is finding area,  volume and surface of given object it may be two dimensional or three dimensional object.


Standard formulas for geometric figures:

Two dimensional objects:

Square:

Perimeter of square = 4 * side
Area of square = side * side
Rectangle:

Area of rectangle = Length x width
Perimeter of rectangle = 2(Length + Width)
Triangle

Area of triangle = `1/2` (Base* height)
Perimeter of triangle = (sum of three sides)
Parallelogram

Area of parallelogram = length × breadth
Circle problems:

Area of circle = ? r2 , where r is the radius.
Circumference of triangle = 2 * ? * r
Diameter = 2 * Radius
Three dimensional objects:

Three dimensional objects are: Cone, sphere, cylinder, hemisphere, prism

Volume of cylinder = ? * R2 * h
Volume of a right circular cone = `1/3`   * ?  * R3
Volume of a sphere  = `4/3` * ? * R3
Volume of a hemisphere = `2/3` * ? * R3

Example problems:


Example 1: Find the area and perimeter of square when the side length is 5cm?

Solution:

Area of square    = (side * side)

= 5 * 5

= 25cm^2

Perimeter of square = 4 * side

= 4 * 5 = 20 cm

Example 2: Find the area and perimeter of rectangle with length 4cm, width 3 cm?

Solution:

Area of rectangle = Length x width

= 4 * 3

= 12cm^2

Perimeter of rectangle = 2(Length + width)

= 2(4 + 3)

= 14cm

Example 3: Find the area of triangle base is 4cm,height is 2cm

Solution:

Area = `1/2` (4 * 2)

= `8/2`

= 4cm^2

Example 4: Find perimeter of triangle whose side lengths are 5cm, 5cm, and 8cm?

Solution:

Perimeter = (A + B + C) (Sum of three side lengths)

A = 5, B = 5, C = 8

= (A + B + C) (Sum of three sides)

= 5 + 5 + 8 = 18cm

Example 5: Find the area and circumference of the circle when the radius is 4cm?

Solution:

Area = `pi`

= 3.14 * 4 * 4

= 50.24cm^2

Circumference = 2 * ? * r

= 2 * 3.14 * 4

= 25.12cm

Example 6: Find the area of rhombus whose diagonals are 6cm and 9cm

Solution:

Area of rhombus =`1/2` (diagonal1 * diagonal2)

= `1/2`  ( 6 * 9)

= 27cm^2

Example 7: Find the volume of cylinder whose diameter of the base is 10 cm, height is 10 cm?

Solution:

Diameter of the base is 10 cm,

Radius = `10/2`

Radius(R) = 5cm

V = Area of the base × Height = `pi` * R * R * H

= 3.14 * 5 * 5 * 40

= 3140cm^2

Statistics Questions

Introduction :

“Statistic is the numerical statement of facts, capable of analysis and interpretation.The science of statistics is the study of the principles and the methods applied in collecting, presenting, analysis and interpreting the numerical data in any field of inquiry.”  It’s also denoted as “statistics is the science of counting”.

Statistics question - Example problems:

Example for statistic question 1:

Find the mean deviation of the mean for the following data: 15, 17, 10, 13, 7, 18, 9, 6, 14, 11

Solution :

Let the mean of the given data be x1. Then,

i=1?n|X1=x| / n

=120 / 10

=12. [n=10]

The values of (x-x1 ) are 3,5,-2,1,-5,6,-3,-6,2,-1.

Therefore, the values are |x-x1| is 3,5,2,1,5,6,3,6,2,1

Mean Deviation (x1)=i=1?n|x-x1| / n   = 34/10=3.4

Hence Mean Deviation (x1)=3.4.

Understanding descriptive statistics definition is always challenging for me but thanks to all math help websites to help me out.


Practice problem for statistic question:

Statistics question word problem 1: Six boys have heights in cm, of 171, 169, 168, 166, 165. Five girls have heights in cm, 161, 160, 157, 157, 155. Draw up a frequency distribution table of the thirty possible difference in height obtained when each boy is paired with any girl. Calculate the mean of these differences and their standard deviation.

Answer:  mean 9.5 cm, Standard deviation 3.01 cm.

Acre of Land

Acre of land - Introduction:

The acre is a unit of area in a numeral of systems, with the regal and United States normal systems.Generally the used acres nowadays are the worldwide acre and, in the U.S., the survey acre. The general use of the acre is to determine tracts of land.

One acre is equal to

1 acre = 4,840 square yards

1 acre = 43,560 square feet

Acre of Land - Examples:

Acre of land - Example 1:

To build colleges in 25 acres how much of the square yards and the square feet are there in land square yards and square feet.

Solution:

Step 1:

x acre          = x * 4,840 square yards

x acre          = x * 43,560 square feet

Step 2:

25 acres = 25 * 4840 square yards = 121 000 square yards

25 acres = 25 * 43560 square feet = 1089 000 square feet

Answer:

Therefore 121 000 square yards (or) 1089000 square feet’s land are needed to build a college.

Acre of land - Example 2:

To build church in 15 acres how much of the square yards and the square feet are there in land square yards and square feet.

Solution:

Step 1:

x acre          = x * 4,840 square yards

x acre          = x * 43,560 square feet

Step 2:

15 acres = 15 * 4840 square yards = 72 600 square yards

15 acres = 15 * 43560 square feet = 653 400 square feet

Answer:

Therefore 72 600 square yards (or) 653 400 square feet’s land are needed to build a church.

Acre of Land - more Examples:

Acre of land - Example 1:

To build schools in 10 acres how much of the square yards and the square feet are there in land square yards and square feet.

Solution:

Step 1:

x acre          = x * 4,840 square yards

x acre          = x * 43,560 square feet

Step 2:

10 acres = 10 * 4840 square yards = 48400 square yards

10 acres = 10 * 43560 square feet = 435600 square feet

Answer:

Therefore 48400 square yards (or) 435600 square feet’s land are needed to build a school.

Acre of land - Example 2:

To build temple in 5 acres how much of the square yards and the square feet are there in land square yards and square feet.

Solution:

Step 1:

x acre          = x * 4,840 square yards

x acre          = x * 43,560 square feet

Step 2:

5 acres = 5 * 4840 square yards = 24 200 square yards

5 acres = 5 * 43560 square feet = 217 800 square feet

Answer:

Therefore 24 200 square yards (or) 217 800 square feet’s land are needed to build a temple.

Different Situation

Introduction :

In math, different situation questions are nothing but the questions involving the different situations. The situations may be past, current or future. In general, we can observe the different situation based questions via word problems. In this article different situation, we are going to discuss few questions based on different situations.

I like to share this Adding Mixed Fractions with Different Denominators with you all through my article.

Example Problems for Different Situation Questions:

The example problems for different situation questions are as follows:

Example 1:

A library consists of 600 books. Mike bought 80 more books for the library.  At present, how many books are there in the library?

Solution:

Number of books already in the library = 600

Number of books that Mike bought   =  80

Number of books in the library          = 600 + 80

= 680

Therefore, there are 680 books in the library.

Example 2:

Chris sold 1200 meatballs on Monday. He sold 600 more meatballs on Tuesday than on Monday. On Tuesday, how many meatballs did Chris sell?

Solution:

Number of meatballs on Monday = 1200

Steve sold 600 more meatballs on Tuesday

Number of meatballs on Tuesday = 1200 + 600

= 1800

Example 3:

There are 750 seats in a theater. At present, 500 seats are occupied. Calculate the percentage of seats that are occupied.

Solution:

Total number of seats   =  750

Occupied seats   =  500

Percentage of seats occupied  =  `<< 500 / 750>>`   x 100

= `<< 50/75>>`   x 100

=  66.67


Practice Problems for Different Situation Questions:

The practice problems for different situation questions are as follows:

1) A library consists of 2000 books. Milton bought 120 more books for the library.  At present, how many books are there in the library?

Answer: 2120 books

2) Kalvin sold 1310 meatballs on Monday. He sold 100 more meatballs on Tuesday than on Monday. On Tuesday, how many meatballs did Kalvin sell?

Answer: 1410 meatballs.

3) There are 450 seats in a theatre. At present, 410 seats are occupied. Calculate the percentage of seats that are occupied.

Answer: 91.1 percent.

Symbol for less than or Equal To

Introduction :

In math, the inequality shows the major task. In arithmetic it has many signs like greater than, less than, greater than or equal to and less than or equal to. In this list of symbols less than equal to symbol is `<=` . This less than equal to symbol specifies that, the term which is left hand side of the inequality is less than or equal to the term which is right of the inequality. In a number line it is mentioned as the closed dot on a number line and the arrow mark which is pointed to the left side of the dotted number.

Symbols and Rules:

Symbol – Symbol for less than or equal to.

`<=` This symbol is the specification of less than or equal to sign.
Rules – Symbol for less than or equal to:

When multiply the inequality term less than equal to sign by negative sign then the negative sign becomes positive and the positive sign becomes negative and the less than equal to sign becomes greater than equal to symbol.
When divide the inequality term less than sign by negative sign then the negative sign becomes positive and the positive sign becomes negative and the less than equal to symbol becomes greater than equal to sign.

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Example Problems – less than Equal to Symbol:

Example 1 – Symbol for less than or equal to:

Find the value of x from the inequality 19a+17`<=` 23

Solution:

Given, 19a+17`<=` 23

Now add the above inequality by -17 on both sides.

19a+17`<=` 23

-17 `<=` -17

----------------------

19a `<=` 6

Now divide by 19 on both sides so, a`<=` 6/19

Example 2 – Symbol for less than or equal to:

Find the value of y from the inequality -5b+83 `>=` 3.

Solution:

Given, -5b+83 `>=` 3.

Now add -83 on both sides

-5b+83`>=` 3

-83  `gt=` -83

---------------

-5b     `>=` -80

Now multiply the above term -5b `>=` -80 by negative sign.

- (-5b `>=` -80) is 5b `<=` 80.

The answer is b`<=` 16.

Example 3 – Symbol for less than or equal to:

Find the value of y from the inequality -m+6 `>=` 4.

Solution:

Given, -m+6 `>=` 4.

Now add -6 on both sides

-m+6 `>=` 4

-6 `>=` -6

---------------

-m      `>=` -2

Now multiply the above inequality -m `>=` -2 by negative sign.

- (-m `>=` -2) becomes m`<=` 2.

m`<=` 2 is the answer.

Example 4 – Symbol for less than or equal to:

Find the value of y from the inequality -n+5 `>=` 5.

Solution:

Given, -n+5 `>=` 5.

Now add -5 on both sides

-n+5 `>=` 5

-5  `>=` -5

---------------

-n      `>=` -0

Now multiply the above inequality -n`>=` -0 by negative sign.

- (-n `>=` -0) is n `<=` 0.

n`<=` 0 is the simplified form of given inequality.

Five Steps of Problem Solving

Introduction to five steps of problem solving:

Here we are going to see the article as five steps of problem solving method.In problem solving concepts we have to do follwing steps.

First understand the problem
Find what should be calculated
Use require formula
Form the equation and plug the data.
Simplify and get the result
let us see the five steps of problem solving.

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Factoring- Five Steps of Problem Solving:

Factor x^2+8x+6=0

Solution:

Step 1:

it is in the standard form of the quadratic equation Ax2+bx+c=0

Step 2:

here A=1, B=8 and c=6, the formula for the quadratic equation is

x= `(-b+- sqrt((b^2-4ac)))/ (2a) `

Step 3:

plug a, b and c values in the above formula

So it will be look like that below,

x= `(-6+- sqrt((6^2-4xx1 xx 8)))/ (2xx1) `X= `"(-6+- sqrt((36-32)))/ (2)`

Step 4:

if we solve these equation means we get the two root valueX= `"(-6+ sqrt(4))/ (2)`X=`"(-6- sqrt(4))/ (2)`

Step 5:

so the answer is x=-2 and x=-4

Geometry- five steps of problem solving:

Find the area and perimeter of the rectangle whose length is 6cm and width is 8cm?

Solution:

Step 1:

Length of the rectangle is 6cm and width is 8cm

Step2:

Formula for area of the rectangle is ==>  length* width

Step3:

Plug the values of length and width in the above formula
Area of the rectangle => 6x8=48cm2

Step 4:

Perimeter of the rectangle formula is 2(length + breath)

Step5:

Plug the values of length and width in the above formula

Perimeter = 2(6+8) =2(14) =28cm

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Fractions - Five Steps of Problem Solving

Add `5/8+3/2`

Solution:

Step 1:

Here the denominators of the fraction are different such as 8 and 2 so we take the greatest common factor of 8 and 2

Step 2:

Here the greatest common factor is 8 it will be divided by the denominator of each fraction (8 is divided by 8 so we get 1) so we have to multiply both the numerator and denominator by 1 so we get the faction like that` (5xx1)/ (8xx1) =5/8`

Step3:

Similarly we do that in the second fraction the denominator of that fraction is 2 so 8 will be divided by 2 means we get the value as 4

Step 4:

Multiply 4 on both the numerator and denominator so the second fraction looks like that below

`(3xx4)/ (2xx4) =12/8`

Step 5:

So the two fractions have the same denominator value now we do the operation as addition

`5/8+12/8=17/8`