Advanced business mathematics and statistics for entrepreneurs
ADVANCED BUSINESS MATHEMATICS AND STATISTICS by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – CSE & PGPSE STUDENTS (CSE & PGPSE are free online programmes open for all, free for all) mobile : 91+9414430763
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What are the measures of central tendency ? Mean = arithematic average (sum / number) Mode = the Variable which has highest frequency Median = the exact mid point of data. For example : 2,3,8,11,11 here Median = 8, mean = 7, Mode = 11
Formula of mean ? Add all the values and divide by number in the previous example : add all the values of 2,3,8,11,11 = 35 there are 5 values so divide 35 by 5 = 7 mean is denoted by Xbar
What is relation between mean, mode and median? mode=3median-2 mean in our example it should be : = 3*8 – 2 * 7 =10. but we have found 11.actually you will see, that the mode here should be 10 – as we discuss in later exercises
Can they be equal? Yes - in normal distribution, mean, mode and median are all equal. In normal distribution, we have 3 characteristics : 1. data are symmetrical 2. data are more in central values and less as we move apart 3. mean=mode=median most of statistical formula require normal distribution.
How to calculate median in bigger data : Formula :( N+1 )/ 2 n=number of data for example : 1,2,3,4,5,6,6,7 here we have 8 values , so (8+1)/2 = 4.5 so we should take mid value between 4 and 5, which is 4.5 answer
What types of data series are there ? There are many types of data series : individual data discrete series continuous series in continuous data series, there is no value which is not possible. (for example : 0 to 10, 10 to 20, 20 to 30) ....
What are the measures of dispersion? Dispersion = how the data is looking in comparison to mean. If data is wide apart from mean, there is high dispersion. If the data is just close to mean, there is very less dispersion. If data has more dispersion, there is less uniformity in the data. We have many tools to measure dispersion like range, variance etc.
Example of high and less dispersion of data : Low dispersion : 6,6,7,7,7,7,8,8,9, high dispersion : 1,4, 8, 19,20,50,60,80,100 you can see, the first data set has far more consistency and dispersion is less. Tools to measure dispersion are : range, standard deviation, variance, mean deviation etc. Range = highest – least value. In the first case range = 9-6 = 3, in 2 nd case range = 100-1 = 99
What is standard deviation? Here we find the difference between each value and mean. Then we square the difference and find the average. This is called variance. Square root of variance is called standard deviation. This gives us an estimate of dispersion of data.
Example of standard deviation? X has 5 values : 1,2,3,4,5 its total is 10. average = 15/5 = 3 now we take difference of each value : (1-3) = -2, (2-3)=-1, (3-3) = 0... so we get : -2,-1,0,1,2, now square them = 4,1,0,1,4 total =10 now find the mean=10/5 = 2 (this is variance) square root of 2 = 1.4 is the standard deviation.
What is regression? It is a simple tool to predict data. Regression assumes that there are at least two data sets, one is dependent on another. Example : if you say that demand is based on price, then we can have regression between price and demand. Price will be independent variable (called X), and demand will be dependent variable (called Y)
What is slope and intercept ? Simplest form of regression is linear regression (a straight line between dependent and independent variable). Here we need two things : slope and intercept. Slope is denoted by B and intercept is denoted by A. Formula of regression is : Y = A +BX 1. A is the point (value) of Y when X = 0 2. B denotes the rate of change in Y in response to change in X.
Example of intercept and slope ? If X change by 10% but Y changes by 20%, so slope = 20/10 = 2 if it is written that X,Y points are : (0,2), (2,4),(4,6),(6,8) ... here you can see that there is a linear relation between X and Y. (first digit is X and second digit is Y). Intercept is 2, because when X is 0, Y is 2.
What are the types of data ? 1. primary (which you collect yourself) 2. secondary (which is already collected for some other purpose, but you can also use it).
What are the various types of statistical analysis? 1. descriptive statistics : here you collect data and present it (for example data on market share) 2. inductive statistics : here you undertake statistical inferences and estimate for future 3. statistical decision theory : here you have to take decision about a situation based on statistics
What are the basic tools for statistical analysis ? 1. be clear about problem 2. formulate hypothesis 3. set significance level (how much accuracy do you want) 4. set sampling frame, research design & collect data 5. analyse data and draw inferences
What is hypothesis ? What do you want to test. We frame 2 hypothesis at least. One of them is null hypothesis and one is alternate thesis. Based on literature review & our own experiences, we frame some understanding on the subject. We have to frame null hypothesis which is opposite of this idea. Then we have to frame alternate hypothesis. We test out null hypothesis.
What is type I and type II error ? If we reject a null hypothesis, which is actually true, we are having type I error if we accept a null hypothesis which is actually false – we are having type II error. We have to set standards for both these errors. If you become liberal for type I error, then type II error will increase and vice versa.
What is alpha ά ? Type I error is called alpha
How do we test alpha ? We calculate P value. If P value is less than alpha, we reject null hypothesis if P value is more than alpha then we cant reject null hypothesis
What is p value ? It is actual calculation about what is the possibilitity of error. It is calculated to be compared with alpha. Alpha is determined in advance, but P value is actual observation.
How does statistics & econometrics help you in business decisions? You can test your decisions using data. You can also build models. There are various types of model : 1. physical, 2. geographic 3. schematic 4. analog 5. mathematical / statistical / econometrics based statistics and econometrics can help you in building the last types of models (5 th type)
What types of statistical analysis are possible ? 1. univariate (there is a single set of data) 2. bivariate (there are two sets of data) 3. multivariate (there are many sets of data)
What are univariate tools? Mean, mode, median, time series analysis, moving average analysis etc.
What are bivariate tools ? Correlation, regression, etc.
What are multivariate tools ? There are many like : conjoint analysis, multivariate regression etc. Here we have many variables : example : demand is dependent on
An biased die is tossed.Find the probability of getting a multiple of 3? The possible options are : 1 to 6. there are only 2 multiples of 3 : 3,6 so probability is (number of favourable outcomes ) / (total number of possibilities) = 2/6 = 1/3 answer
In a simultaneous throw of a pair of dice,find the probability of getting a total more than 7? We can have 36 possibilities (6*6) however, we need only those cases where the total is 8 or more. These are : (6,2),(6,3),(6,4),(6,5),(6,6),(5,3),(5,4),(5,5),(5,6),(4,4),(4,5),(4,6),(3,5),(3,6),(2,6) =15 answer = 15/36 = 5/12 answer
A bag contains 6 white and 4 black balls .Two balls are drawn at random .Find the probability that they are of the same colour? Both are white : 6/10*5/9 both are black = 4/10*3/9 add them : =42/90 or 7/15 or : 6c2/10C2*1/2 + 4c2/10c2 =21/45 = 7/15 answer
Two dice are thrown together.What is the probability that the sum of the number on the two faces is divisible by 4 or 6? The possibilities are : (1,3)(1,5) (2,2) (2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(6,2),(6,6) thus we are able to get 14 out of 36. so answer = 7/18 answer
Two cards are drawn at random from a pack of 52 cards What is the probability that either both are black or both are queens? Both are black = 26/52 * 25/51=25/102 both are queens : 4/52 * 3/51=3/663 both are black queens : 2/52*1/51 = 1/1326 now add them : (25/102 + 3/663 – 1/1326) =(325+6-1)/1326 =330/1326 or .25 answer
Two dices are tossed the probability that the total score is a prime number? Prime numbers are : 1,2,3,5,7,11 totals are : (1,2),(1,1),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),6,1),(6,5) =15/36 answer
Two dice are thrown simultaneously .what is the probability of getting two numbers whose product is even? If any one of the two numbers is an even number, the product will be even number. Thus we should pick up all those cases when both the numbers are odd numbers : (1,1),(1,3),(1,5),(3,1),(3,3),(3,5),(5,1),(5,3) (5,5) thus there are only 9 such cases. Remove them from 36, we get : 27 cases answer : 27/36 answer
In a lottery ,there are 10 prizes and 25 blanks.A lottery is drawn at random. what is the probability of getting a prize ? 10/(10+25) =10/35 or 2/7 answer
In a class ,30 % of the students offered English,20 % offered Hindi and 10 %offered Both.If a student is offered at random, what is the probability that he has offered English or Hindi? 30+20-10 = 40% or .4 answer
Two cards are drawn from a pack of 52 cards .What is the probability that either both are Red or both are Kings? Both are red ½ * 25/51 both are king = 4/52 + 3/51 now add both these answers =55/221
one card is drawn at random from a pack of 52 cards.What is the probability that the card drawn is a face card? Face cards are : Jack, queen, king total = 12 12/52 answer
A man and his wife appear in an interview for two vacancies in the same post.The probability of husband's selection is 1/7 and the probabililty of wife's selection is 1/5.What is the probabililty that only one of them is selected? Husband + not wife =1/7 * 4/5 = 4/35 wife + not husband =1/5 * 6/7 = 6/35 add = 10/35 answer
From a pack of 52 cards,one card is drawn at random.What is the probability that the card is a 10 or a spade? 4/52 + 13/52 – 1/52 =16/52 answer
A bag contains 4 white balls ,5 red and 6 blue balls .Three balls are drawn at random from the bag.What is the probability that all of them are red ? 5/15*4/14*3/13 or 5c2/15c2 = =2/91
A box contains 10 block and 10 white balls.What is the probability of drawing two balls of the same colour? Both are black : 10/20 * 9/19 =9/38 +both are white : 10/20 * 9/19 =9/38 or black : 10c2 / 20c2 +white : 10c2 / 20c 2 =90/190
A box contains 20 electricbulbs ,out of which 4 are defective, two bulbs are chosen at random from this box.What is the probability that at least one of these is defective ? In such questions (at least one type), it is better to reverse the question, solve it and deduct the answer from 1. So here we shall first calculate the probability of getting no defective bulb. Let us assume that no bulb is defective : 16/20 * 15/19 = 12/19 at least one is defective = 1 -12/19 = 7/19 answer
Two cards are drawn together from apack of 52 cards.What is the probability that one is a spade and one is a heart ? First is spade and 2 nd heart : 13/52 * 13/51 = 13/204 First is heart and 2 nd spade : 13/52 * 13/51 = 13/204 add them : 13/102 answer
The probability that a card drawn from a pack of 52 cards will be a diamond or a king? 13/52 + 4/52 – 1/52 =16/52
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