Normal distribution for statistical analysis
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This document provides a summary of key concepts related to normal distribution for statistical analysis, including definitions of frequency distribution and important probability distributions like normal, binomial, chi square, student T, and F distribution. It also lists the assumptions of normal and binomial distributions and provides examples of probability calculations for scenarios involving dice, cards, samples, and more.
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Normal distribution for statistical analysis
- 1. NORMAL DISTRIBUTION FOR STATISTICAL ANALYSIS 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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- 3. WHAT IS A FREQUENCY DISTRIBUTION ? It is also called probability distribution, it is a bundle of all the samples from population
- 4. What are the important distributions ? 1. normal 2. binomial 3. chi square 4 student T 5. f distribution
- 5. What are the assumptions of normal distribution ? 1. mean = median = mode (or almost equal) 2. data is bell shaped \i
- 6. What are the assumptions of binomial distribution ? There are two possible outcomes -
- 7. A coin is tossed 8 times, what is the probability that there are 4 heads? It is a case of binomial distribution formula is : NCrP^rQ^(n-r) N = total number = 8 C = combination r= selection = 4 P=probability of r Q = (1-P) =8C4*(.5)^4*(.5)^4 (8!/(4!*4!)) * .0625*.0625 =.2734
- 8. Mean = 120, standard deviation = 40, what is the probability that X is more than 120 & upto 150 ? Here we have basic formula : X – mean / standard deviation = (120 – 120 ) /40 = 0 = (150 – 120) / 40 = .75 thus the value is between 0 and .75 .5+.37 = 87% answer
- 9. In a sample of 500 workers mean salary is 500, std. Dev. = 48 , find the workers having salary more than 600? =(600 – 500) / 48 =2.08 table value of 2.08 = .4812 thus .5 + .4812 = .98 number of workers above 600 = 1-.98 = .02 multiply this to 500 = 10 answer
- 10. 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
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 15. 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
- 16. 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
- 17. 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
- 18. 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
- 19. 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
- 20. 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
- 21. 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
- 22. 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
- 23. 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
- 24. 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
- 25. 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
- 26. 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
- 27. 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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