probability and statistics

Question

1.     There are 3 arrangements of the word DAD, namely DAD, ADD, and DDA. How many arrangements are there of the word ENDURINGLY?

2.     There are 13 men and 12 women in a ballroom dancing class. If 6 men and 6 women are chosen and paired off, how many pairings are possible?

3.     Suppose you are taking a multiple-choice test with 4 choices for each question. In answering a question on this test, the probability that you know the answer is 0.33. If you don’t know the answer, you choose one at random. What is the probability that you knew the answer to a question, given that you answered it correctly?

1.     A normal distribution which has a mean of 50 and standard deviation of 7 is taken into consideration.

68% of the distribution can be found between what two numbers?

a)      40 and 60

b)     0 and 43

c)      0 and 68

d)     43 and 57

2.     Consider the data X = (58,59,63,60,60,63,60,57,58,59). An unbiased estimation for population variance would be ____

Answer 1

1 had the 10 letter word been all different letters with no repetition, the answer would be 10!

but like DAD, ENDURINGLY has one letter repeat, the N. so the answer is 10!/2!

= 10x9x8x7x6x5x4x3 = 1,814,400 arrangements

3 4 questions. randomly guessing means you get it right 25% of the time. there's a 33% chance you know the answer. IF you get a question right, what is the probability you knew the answer?

Use Bayes Theorem

P(K/C) = P(C/K)P(K)/(P(C) K= you Knew the answer, C= correct answer

= (1)(.33)/(.25 + .33 - (.25)(.33)

= .33/..5008

= about .6589

= about.66 = 66% chance you knew the answer

Probability is somewhere between .33 and 1.00. (.33 + 1.00)/2 = about .66 = 66%

1 d) 43 and 57. 68% are within plus and minus one standard deviation from the mean. 50+7

empirical rule is 68-95-99.7

with 68% within one standard deviation, 50+7, 50-7, 43 to 57

95% within two standard deviations, 50+14, 50-14, 36 to 64

99.7% within three standard deviations, 50+21, 50-21, 29 to 71

2 the mean of 10 numbers summing to 597 = 597/10 = 59.7

population variance = sum of squared differences from the mean (59.7), divided by the number of data, 10

there's one 57, 2 58's, 2 59's, 3 60's and 2 63's

(57 - 59.7)^2 +2(58-59.7)^2 + 2(59-59.7)^2 + 3(60-59.7)^2 +2(63-59.7)^2 divided by 10

= 2.7^2 + 2(1.7)^2 + 2(.7)^2 + 3(.3)^2 + 2(3.3)^2 divided by 10

= 7.29 + 5.78 + .98 + .27 + 21.78 divided by 10

= 36.1/10 = 3.61 = the population variance

population standard deviation = sqr3.61 = 1.9

had this just been a sample, you would have divided by 9 instead of 10