Lecture Notes for
Statistical Methods for Business I BMGT 211
Chapters 5 and 6
Professor Ahmadi, Ph.D.
Department of Management
Revised May1, 2004 Professor Ahmadi’s Lecture Notes Page 2 Chapter 5 Formulas
Required Conditions for a Discrete Probability Function
f(x) > 0
f(x) = 1
Discrete Uniform Probability Function
f(x) = 1/n where n = the number of values the random variable may assume
Expected Value of a Discrete Random Variable
E(x) = µ = (x f(x))
Variance of a Discrete Random Variable
2 2 Variance (x) = = (x - µ) f(x
Number of Experimental Outcomes Providing Exactly x Successes in n Trials
n n! = x x!(n - x)! where n! = n (n - 1) (n - 2) . . . (2)(1) (Remember: 0! = 1)
Binomial Probability Function
n! f(x) = p x (1 - p) n – x where x = 0 ,1, 2, ..., n x!(n - x)!
The Mean of a Binomial Distribution
µ = n p
The Variance of a Binomial Distribution
2 = n p (1 - p)
Professor Ahmadi’s Lecture Notes Page 3 Chapter 5
Discrete Probability Distributions
The manager of the university bookstore has kept records of the number of diskettes sold per day. She provided the following information regarding diskettes sales for a period of 60 days:
Number of Number Diskettes Sold of Days 0 6 1 9 2 12 3 18 4 12 5 3 a. Identify the random variable b. Is the random variable discrete or continuous? c. Develop a probability distribution for the above data. d. Is the above a proper probability distribution?
e. Develop a cumulative probability distribution.
f. Determine the expected number of daily sales of diskettes.
g. Determine the variance and the standard deviation.
h. If each diskette yields a net profit of 50 cents, what are the expected yearly profits from the sales of diskettes?
Professor Ahmadi’s Lecture Notes Page 4 Chapter 5
Introduction to Binomial Distribution
A production process has been producing 10% defective items. A random sample of four items is selected from the production process. a. What is the probability that the first 3 selected items are non-defective and the last item is defective?
b. If a sample of 4 items is selected, how many outcomes contain exactly 3 non-defective items?
c. What is the probability that a random sample of 4 contains exactly 3 non-defective items?
d. Determine the probability distribution for the number of non-defective items in a sample of four.
e. Determine the expected number (mean) of non-defectives in a sample of four.
f. Find the standard deviation for the number of non-defectives.
Professor Ahmadi’s Lecture Notes Page 5 Chapter 5
POISSON PROBABILITY DISTRIBUTION
During the registration period, students consult their advisor for course selection. A particular advisor noted that during each half hour an average of eight students came to see him for advising. a. What is the probability that during a half hour period exactly four students will consult him?
b. What is the probability that during a half hour period less than three students will consult him?
c. What is the probability that during an hour period ten students will consult him?
d. What is the probability that during an hour and fifteen minute period thirty students will consult him?
Professor Ahmadi’s Lecture Notes Page 6 Chapter 6 Formulas
Uniform Probability Density Function for a Random Variable x:
1 for a x b b - a f(x) = 0 elsewhere
Mean and Variance of a Uniform Continuous Probability Distribution:
a + b (b - a)2 = 2 2 12
The Z Transformation Formula:
(x - ) z =
Solving for x using the Z transformation formula: x Z
Professor Ahmadi’s Lecture Notes Page 7 Chapter 6
Continuous Probability Distributions
I. - The Uniform Distribution
The driving time for an individual from her home to her work is uniformly distributed between 300 to 480 seconds. a. Give a mathematical expression for the probability density function.
b. Compute the probability that the driving time will be less than or equal to 435 seconds.
c. Determine the probability that the driving time will be exactly 400 seconds.
d. Determine the expected driving time. e. Determine the standard deviation of the driving time.
Professor Ahmadi’s Lecture Notes Page 8 Chapter 6
II. - The Normal Distribution
1. Given that Z is the standard normal random variable, give the probabilities associated with the following:
a. P(Z < - 2.09) = ?
b. P(Z > -0.95) = ?
c. P(-2.55 < Z < -2.33) = ?
2. Z is a standard normal variable. Find the value of Z in the following:
a. The area between -Z and zero is 0.4929. Z = ?
b. The area to the right of Z is 0.0192.Z = ?
c. The area between -Z and Z is 0.668. Z = ?
Professor Ahmadi’s Lecture Notes Page 9 3. The weight of certain items produced is normally distributed with a mean weight of 60 ounces and a standard deviation of 8 ounces.
a. What percentage of the items will weigh between 50.4 and 72 ounces?
b. What percentage of the items will weigh between 42 and 52 ounces?
c. What percentage of the items will weigh at least 74.4 ounces?
d. What are the minimum and the maximum weights of the middle 60% of the items?
Professor Ahmadi’s Lecture Notes Page 10 4. Sun Love grapefruit growers have determined that the diameter of their grapefruits is normally distributed with a mean of 4.5 inches and a standard deviation of 0.3 inches. (You can find the step-by-step solution to this problem in my workbook.)
a. What is the probability that a randomly selected grapefruit will have a diameter of at least 4.14 inches?
b. What percentage of grapefruits has a diameter between 4.8 to 5.04 inches?
c Sun Love packs their largest grapefruits in a special package called "Super Pack." If 5% of all their grapefruits are packed in "Super Packs," what is the smallest diameter of the grapefruits, which are in the "Super Packs?"
d In this year's harvest, there were 111,500 grapefruits, which had a diameter over 5.01 inches. How many grapefruits has Sun Love harvested this year?
Professor Ahmadi’s Lecture Notes Page 11 5. In grading eggs, 30% are marked small, 45% are marked medium, 15% are marked large, and the rest are marked extra-large. If the average weight of the eggs is normally distributed with a mean of 3.2 ounces and a standard deviation of 0.6 ounces:
a What are the smallest and the largest weights of the medium size eggs?
b What is the weight of the smallest egg, which will be in the extra-large category?
Professor Ahmadi’s Lecture Notes Page 12