İ s t a n b u l K ü l t ü r U n i v e r s i t y Faculty of Engineering
MCB1007
Introduction to Probability and Statistics Final Exam
Fall 2014-2015
Number:
Name:
Department:
Section:
– You have 90 minutes to complete the exam. Please do not leave the examination room in the first 30 minutes of the exam. There are six questions, of varying credit (100 points total). Indicate clearly your final answer to each question. You are allowed to use a calculator. During the exam, please turn off your cell phone(s).
You cannot use the book or your notes. You have one page for “cheat-sheet” notes at the end of the exam papers. The answer key to this exam will be posted on Department of Mathematics and Computer Science board after the exam.
Good luck!
M. Fatih Uçar, PhD.
Emel Yavuz Duman, PhD.
Arzu Yemişçi, PhD.
Question 1. Question 4.
Question 2. Question 5.
Question 3. Question 6.
TOTAL
8+8 points A fair coin is tossed 120 times. Using normal approximation, find the probability of obtaining
(a) between 48 and 72 heads.
Answer.
(b) at least 75 heads.
Answer.
5+5+6 points
There are several types of trees in a forest. In a random sample of 240 trees, they have average height 35 mt with a standard deviation 7 mt. For the mean of the population sampled,
(a) Construct a 95% confidence interval.
Answer.
(b) What should be the minimum sample size, if error does not exceed 0.97?
Answer.
(c) With what degree of confidence could we say that the average height of trees are 35 ∓ 0.794?
Answer.
MCB1007 - Int. to Prob. and Statistics 2 Final Exam
14 points The following table shows the Mathematics exam scores of 57 student. Find the mean and median scores of the exam?
Score 21 ≤ x < 30 31 ≤ x < 40 41 ≤ x < 50 51 ≤ x < 60 61 ≤ x < 70 71 ≤ x < 80
Num. of Stud. 2 11 17 20 5 2
Answer.
5+5+8 points
A continuous random variable X has the following probability density function f(x) =
ke−|x2|, 0 < x < ∞, 0, elsewhere. (a) Find k and μ.
Answer.
(b) Find a lower bound for P (|x − μ| < 4) using Chebyshev’s theorem.
Answer.
MCB1007 - Int. to Prob. and Statistics 3 Final Exam
10+10 points (a) Eight students will select a topic for graduate thesis among topics A, B, C and D.
What is the probability that at most 3 students will select topic A?
Answer.
(b) Five pawns are placed at random on a 8 × 8 chessboard, no more than one pawn to a cell. What is the probability that no more than one pawn in the same row or column?
Answer.
16 points
Suppose that a pair fair dice is rolled together. LetX denotes the number of dice that come up prime number and Y denotes the number of dice that come up bigger than or equal to 5. Find the correlation coefficient of X and Y ?
Hint. In Statistics, correlation coefficient, denoted by ρ(x, y) is a measure of the linear relationship between two random variablesX and Y , giving a value −1 < ρ < 1, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation.
It is defined byρ(x, y) = √ Cov(X,Y )
V ar(X)V ar(Y ). Answer.
MCB1007 - Int. to Prob. and Statistics 4 Final Exam
