The Basic Concepts of Ratios

Once the basic arithmetic operations are learnt there are other concepts that have to be learnt. It is better to learn the ratio of similar objects. This concept can be very helpful in solving the mathematical problems. One needs ratio help to understand the term better. The math ratio problems are fun to work with. Once many problems are solved the concept will become clearer. It is better to get help with ratios as this concept has to be learnt clearly so that other concepts like proportion have to be learnt. Both are related to each other. So, one has to understand the previous concept well. Only then proportion can be understood well.

The ratios in math define relationship between objects which are of similar kind. If there are 5 yellow balls in a basket and 4 white balls in a basket then the ratio is nothing but 5:4. So, the notation is obtained by using colon between the numbers. So, the relationship between the yellow balls in the basket and the white balls in the basket is defined and helps us to understand better. The standard size of the television can also be expressed using this term.  The term has a history as well. But it is very difficult to tell exactly when the term became common. But it is a very important in the mathematical world. Proportion can also be learnt using this term. So, it is better to learn the term very well. It is basically written in the form of a fraction. Fraction can also be written as a decimal. The vice-versa can also be done. So, to understand the basic understanding of fractions can also be very helpful. This term is also used with percentages. If both the numerator and the denominator are multiplied by the same number the value of the fraction does not change. It means the value of the fraction remains the same even after the multiplication. This concept can be very helpful. Having problem with real numbers keep reading my upcoming posts, i will try to help you.

The fractions can also be reduced to smaller ones. This can be done by dividing both the numerator and denominator with common factors. This will help in reducing the fraction to a smaller fraction. This can be done only if there are common factors for both the numerator and the denominator. Both the numerator and denominator must be divided by the same number. They should be divided till the fraction becomes smallest.

Basis Vectors

Introduction to basis vectors:

Definition

Consider the non-zero vectors v1, v2 …, vn in the vector space V over the field F. We call them basis vectors if they satisfy the following two conditions:

The vectors v1, v2 …, vn are linearly independent
Any vector v in the vector space V can be written as a linear combination of the basis vectors. In other words, there exists scalars a1, a2…, an not all zero such that v = a1v1 + a2v2 +…+ anvn I like to share this What are Vectors with you all through my article.


On satisfying the above two conditions we call the set B = {v1, v2 …, vn } as the basis of the vector space V and the elements of B are called the basis vectors.

Example for Basis Vectors

Let V = R3 and consider the vectors:

E1 = (1, 0, 0)

E2 = (0, 1, 0)

E3 = (0, 0, 1)

We will show that the above vectors form the basis of R3.

Let a, b, c be real numbers so that aE1 + bE2 + cE3 = 0

Then a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) = (0, 0, 0)

Implies (a, b, c) = (0, 0, 0) which means a = b = c = 0

Hence they are linearly independent vectors.

Also any vector v = (x, y, z) in V can be written as a linear combination of E1, E2, E3 as follows: v = xE1 + yE2 + zE3

Hence {E1, E2, E3} are the basis vectors for R3.

Note: We call the above set of basis vectors as the standard basis of R3. It can be extended to Rn where the standard basis vectors are E1, E2…, En.

Remember:  A vector space V is said to be of finite dimension or finitely generated if there exists a finite number of basis vectors in V else it is referred to as infinite dimensional space. Please express your views of this topic math problems 8th grade by commenting on blog.

Exercise to Basis Vectors:

Show that the infinite set S = {1, x, x2, …, xn,...} forms the basis vectors of the vector space F[x] of polynomials over the field F.

P Trend Statistics

Introduction to p trend statistics:

In statistics, the letter “p” stands for probability. p trend statistics is highly used for  normal distribution for calculating  the normal distribution value.

A normal distribution with the variables mean and standard deviation can be converted into a standard normal distribution  is obtained by  converting the normal distribution with the mean and standard deviation by performing the change of scale and origin

For calculating the normal distribution the formula must change to z-scale or z-term from the x –scale. The formula is given as,

` Z=(X-mu)/sigma`

Where `mu` - mean

`sigma` - Standard deviation.

How to Find P Trend Statistics

This example will clearly explain how to calculate p trend statistics.

P (0
= 0.5-0.4772

(    P (0
P (0
P (0
This is the normal distribution value for P (0
Example Problems for P Trend Statistics:

Problem 1:

Find the probability value of P (0
Solution

Given that  P (0
On splitting the above equation the value remains the same.

P (0
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P (0
On adding the values we get the final result as

P (0
Problem 2

Find the probability value of P (-1.15
Solution:

Given that P (-1.15
By using property of normal distribution, we get

P (-1.15
Now applying the symmetry property,

P (-1.15
= 0.3749 + 0.4559

P (-1.15
Problem 3:

Find the probability value of P (-1.55
Solution:

Given that P (-1.55
By using property of normal distribution, we get

P (-1.55
Now applying the symmetry property,

P (-1.55
= 0.4398 + 0.3944

P (-1.55

Practice Set Solutions

Introduction to Practice Set Solutions

A set of elements or information or numbers are referred as data set. Data set can hold any types of data. Data can be represented as tables and graphs. In tables data are stored as rows and columns. Data sets are classified into sequential data set and partitioned data set. In sequential data set data are stored as consecutively. In partitioned data set store the address, so that it is possible to access the data easily. Let us practice the data set solutions in statistics.

Practice Statistics Data Set Solutions - Example

Ex: Find the mean, median, mode, and range for the data set: {20, 27, 15, 16, 42, 28, 20}.

Solution:First arrange the set in ascending order {15, 16, 20, 20, 27, 28, 42}

Mean:Mean or average is the sum of all the elements in the set divided by total number of elements in a set.

Mean     = Sum of elements in a set / Number of elements in a set

= {15, 16, 20, 20, 27, 28, 42}/ 7

= `168/7`  = 24

Therefore mean is 24

Median:Median is the middle value of the data set after the arrangement of ascending or descending order the data set.

Therefore median is 20.

Mode:If any of the value in data set is repeated two or more times then that is referred to as mode.

20 is repeated twice.

Hence, the mode of the given data set is 20.

Range:Difference between the maximum and minimum value of the data set is called the range.

Range        = Maximum value – Minimum value

= 42 – 15

= 27

Therefore range is 27.

Practice Probability Data Set Solutions - Example

Example: When a pair of balanced dice is rolled, and what are the probabilities of getting the sum (1) 12 (2) 12 or 11 (3) 11or 10. Find the solutions.

Solution:-The sample space S = {(1, 1), (1, 2) … (6, 6)}

Number of possible outcomes n(S) = 36

Let A be the event of getting sum 12, B be the event of getting the sum 11

and C be the event of getting sum 10.

A = {(6, 6)} n(A) = 1.

B = {(5, 6), (6, 5) n(B) = 2

C = {(4, 6), (5, 5), (6, 4)}n(C) = 3

(1) P (getting sum 12) = P(A) =`(n(A))/(n(S)) = 1/36`



P(12) = `1/36`

(2) P(12 or 11) = P(A or B) = P(A ∪ B)

= P(A) + P(B)          (A and B are mutually exclusive i.e. A∩B=φ)

= `1/36 + 2/36 = 3/36= 1/12`

P (12 or 11) = `1/12`

(3) P(11 or 10) = P(B or C) solving the above the problems also.

= P(B) + P(C) (B and C are mutually exclusive)

= `2/36 + 3/36 = 5/36`

P (11 or 10) = `5/36`

Data Set Solutions – Practice Problems

Solve these practice problems

Practice 1: Find the mean, median, mode, and range for the data set: {25, 32, 20, 21, 47, 33, 25}.

Ans: Mean – 29, Median – 25, Mode – 25, Range – 27

Practice 2: When a pair of balanced dice is rolled, and what are the probabilities of getting the sum (1) 10 (2) 10 or 9 (3) 9 or 8.

Ans: (1)  `1/12` , (2)  `7/36` , (3) `1/4`

Regression with Categorical Data

Introduction regression with categorical data:

In statistics, regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps us understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.

(Source: Wikipedia)

Regression with Categorical Data:

Definition regression with categorical data:

A categorical variable describe a exacting quality or characteristic. The data is divided into category and the information together is called categorical data

A categorical variable have a dimension scale consisting of a set of categories. For instance, political philosophy is often measured as liberal, moderate, or conservative. Diagnoses regarding breast cancer based on a mamma-gram use the categories normal, benign, probably benign, suspicious, and malignant.

The development of methods for categorical variables was stimulated by research studies in the social and biomedical sciences. Categorical scales are pervasive in the social sciences foe measuring attitudes and opinions. Categorical scales in biomedical sciences measure outcomes such as whether a medical treatment is successful. Understanding Prime Factors of 245 is always challenging for me but thanks to all math help websites to help me out.

Regression models involve the following variables:

The unknown parameters denoted as ß; this might be a scalar or a vector of length k.
The independent variable, X.
The dependent variable, Y.
A regression model relates Y to a function of X and ß.

Y ˜ f (X, ß)

Example Problem Regression with Categorical Data:

Probability of (X) is the probability that X is true. Probability of (X|Y) is the probability that X is true and given that Y is true.

Two hundred students of a class are classified according to the following 2 by 3 table. Where A, B, and C are mutually exclusive properties.

Status    Section A
Section B    Section C    Totals
Female    40      40    90    170
Male    50      20    20    90

Totals  90      60   110 260

Solution for comparing categorical data:

What is the probability that has chosen from the given table person as female?

P (F) = `170 / 260` = 65 %.

What is the probability that chosen person has section A?

P (A) = `90 / 260` = 35 %.

If chosen students are female, what is the probability that she has section B?

P (B|F) = `60 / 260` = 23 % = `(p(B and F)) / (p(F))` .

If chosen student has section C, what is the probability that the individual is a male?

P (M|C) = `20 / 110` = 18 %   = `(p(C and M))/ (p (C))` .

If chosen section has B or C, what is the probability that the student is a male?

P (M|B or C) = `20 /60` = 33 %.

Standard Deviation Practice Problems

Introduction to standard deviation practice problems:
The variance of a random variable or distribution is the expectation, or mean, of the deviation squared of that variable from its expected value or mean. ( Source - Wikipedia )

The standard deviation is nothing but the square root of its variance.

In this article of standard deviation practice problems, few example problems for finding standard deviation and several practice problems to find standard deviation are given.

Example Problems to Practice Standard Deviation:

Example 1:

Find the standard deviation for the given set of numbers:

{ 24, 20, 17, 31, 43 }

Solution:

Step 1: Mean

Mean   =  ` ( 24 + 20 + 17 + 31 + 43 ) / 5`

=  ` 135 / 5`

=  27

Step 2: Variance

Variance  =   `( (24-27)^2 + (29-27)^2 + (17- 27)^2 + (31-27)^2 + (43-27)^2 )/5`

=   `( (-3)^2 + (2)^2 + (- 10)^2 + (4)^2 + (16)^2 )/5`

=   `(9+4+100+16+256)/5`

=   `430/5`

=  86

Step 3:Standard deviation

Standard deviation  =  `sqrt ( 86 )`

=  9.27

Example 2:

Find the standard deviation for the given set of numbers:

{ 7, 4, 5, 1, 3 }

Solution:

Step 1: Mean

Mean   =  ` ( 7 + 4 + 5 + 1 + 3 ) / 5`

=  ` 20 / 5`

=  4

Step 2: Variance

Variance  =   `( (7-4)^2 + (4-4)^2 + (5- 4)^2 + (1-4)^2 + (3-4)^2 )/5`

=   `( (3)^2 + (0)^2 + (1)^2 + (-3)^2 + (-1)^2 )/5`

=   `(9+0+1+9+1)/5`

=   `20/5`

=  4

Step 3:Standard deviation

Standard deviation  =  `sqrt ( 4 )`

=  2

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Practice Problems for Finding Standard Deviation:

1) Find the standard deviation for the set of numbers: { 45, 67, 89, 24 }

2) Find the standard deviation for the data set: { 44, 67, 87, 90, 42 }

3) Find the standard deviation for the set of numbers: { 39, 21, 57, 61, 73 }

4) Find the standard deviation for the data set: { 207, 137, 316, 590 }

5) Find the standard deviation for the set of numbers: { 64, 80, 38, 40, 41 }

6) Find the standard deviation for the data set: { 12, 21, 60, 41 }

7) Find the standard deviation for the set of numbers: { 56, 24, 78, 29, 88 }

8) Find the standard deviation for the data set: { 41, 60, 77, 11, 63 }

9) Find the standard deviation for the set of numbers: { 12, 204, 717, 231 }

10) Find the standard deviation for the data set: { 245, 356, 343, 326, 460 }

11) Find the standard deviation for the set of numbers: { 16, 34, 68, 19, 78 }

12) Find the standard deviation for the data set: { 25, 50, 45, 17, 77 }

13) Find the standard deviation for the set of numbers: { 4, 4, 7, 6, 14 }

14) Find the standard deviation for the data set: { 45, 10, 45, 156, 27 }

15) Find the standard deviation for the set of numbers: { 216, 334, 566, 149 }

Answer key:

1)  24.26        2)  20.39       3)  18.23       4)  172.44      5)  16.66     6)  18.55    7)  25.52     8)  22.80

9)  260.04     10)  68.86     11)  25.44     12)  21.02        13)  3.69          14)  51.38     15)  158.68

One to One Function Proof

Introduction to one to one function  proof:

A function from A to B is called one to one if f(a) = f(b) then a = b. Each element in the function B has paired with only one  element in the function B. One to one function is a function in which all the variables in the domain has an individual range. Each element in the domain has the correspondent element in the range. Here we will see about one to one function proof.

One to One Function Proof:

A function is known as one to one if every element in the range of the function corresponds with one and only element in the domain.

Proof- one to one function proof

Let us consider the function, f : N -> N, it is defined as f(n) = n2  where N indicates the positive integers.

Assume an integer a and b in which f(a) = f(b).

The function is f(n) = n2, substituting a and b,

we have a 2 = b 2  ,  where  a 2 - b 2 = 0.

a 2 - b 2 = 0  - >   (a + b) (a - b) = 0

where  a + b  = 0 ,  a - b = 0

hence we have  a + b = 0   - >  a =  - b

hence we prove for a - b = 0   - >  a = b

the function f(a)= f(b)

Hence proved.

Example Problem - One to One Function Proof

Example problems 1 - One to one function  proof

F{(1, 0), (2, 3) ,( 4, 5), (6, 7)} show that the function is one to one function.

Solution:

Domain of the given function is 1, 2, 4, 6.

Range of the given function is 0, 3, 5, 7.

Hence the each of the domain has corresponding range individually.

Hence the given set exhibits one to one function.

Example problems 2 - One to one function  proof

Which of the following functions is one to one function?

F1{(1, 8), (2, 9) ,( 4, 10), (6, 11)}

F2{(1, 0), (2, 3) ,( 4, 8), (6, 9)}

F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)}

Solution:

F1{(1, 8), (2, 9) ,( 4, 10), (6, 11)}

Domain of the given function1 is 1, 2, 4, 6.

Range of the given function1 is 8, 9, 10, 11.

Each element in the domain corresponds with the values of range.Understanding composite function is always challenging for me but thanks to all math help websites to help me out.

Hence the given function 1 is one to one function.

F2{(1, 0), (2, 3) ,( 4, 8), (6, 9)}

Domain of the given function2 is 1, 2, 4, 6.

Range of the given function 2 is 0, 3, 8, 9.

Each element in the domain corresponds with the values of range.

Hence the given function 2 is one to one function.

F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)}

Domain of the given function 3 is 1, 2, 4, 6.

Range of the given function 3 is 0, 0, 5, 7.

The element in the domain 1 and 2 has the same range.

Hence the given function 3 is not one to one function.

Answer: F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)} is not an one to one function.