Question

Thank you in advance!

1. From past experience, at the end of the semester, 3 out of 25 students come and thank Professor Brown for a wonderful course. Now at the end of this semester, after grades have been posted, 8 students come to his office.

(First introduce X random variable, then state what is the distribution of X)

1. What is the probability that of those 8 students, 5 are there to say thank you?
2. What is the probability that at least 3 students are there to say thank you?
3. What is the mean value and SD of the number of students that come to thank Professor Kerobyan?

2.The Online Quiz from Applied Statistics consists of 4 questions. Statistics show that there is a 78% chance that the student will answer to any one of Exam problems correctly. If the number of attempts for each question is unlimited, find the probabilities

(First introduce X random variable, then state what is the distribution of X)

1. The student will correctly answer the first question after the 10^th attempt
2. The student will correctly answer three questions after 10 total attempts
3. What is the average number and SD of attempts up to when the student answers all the questions correctly?

Answer 1

1. P ( a student will thank Professor) = 3/25 =0.12

8 students com eto his office, n=8

Let X be the number of students who will thank professor

X~ Binomial ( 8, 0.12)

a) P( X=5) = 8C5 (0.12)5 (1-0.12)3 = 0.00095

b) P( X >=3) = 1- P( X <3) = 1- 0.9392 = 0.0608

c) Mean = np = 8* 0.12 =0.96

Standard deviation = = = 0.9191

2.  P ( a anwer is correct) = 0.78 = p

Let X be the number of attempts before  answering r questions correctly

a) X ~ Negative binomial ( r=1, p= 0.78)

P( X = 10) = (1-0.78)10 (0.78) = 2.071 * 10-7

b) X~ Negative Binomial ( r=3, p=0.78)

P( X=10) = 10C2 (0.78)3(1-0.78)8 = 0.000117

c) X~ Negative Binomial ( r=4, p= 0.78)

Mean = r/p = 4/0.78 =5.13

Standard deviation= = =0.60133