BUS 220: ELEMENTARY STATISTICS Chapter 6: Discrete Probability Distribution

2 What is a Probability Distribution? Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, two heads, and three heads. What is the probability distribution for the number of heads?

Probability Distribution of Number of Heads Observed in 3 Tosses of a Coin

Characteristics of a Probability Distribution 1. The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive events. 3. The list is exhaustive. The sum of the probabilities of the various events is equal to 1.

6 Random Variables Random variable - a quantity resulting from an experiment that, by chance, can assume different values.

7 Types of Random Variables Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement

8 Discrete Random Variables - Examples The number of students in a class. The number of children in a family. The number of cars entering a carwash in a hour. Number of home mortgages approved by Coastal Federal Bank last week.

9 Continuous Random Variables - Examples The distance students travel to class. The time it takes an executive to drive to work. The length of an afternoon nap. The length of time of a particular phone call.

10 Features of a Discrete Distribution The main features of a discrete probability distribution are: The sum of the probabilities of the various outcomes is The probability of a particular outcome is between 0 and The outcomes are mutually exclusive.

11 The Mean of a Probability Distribution Mean The mean is a typical value used to represent the central location of a probability distribution. The mean of a probability distribution is also referred to as its expected value.

12 The Variance, and Standard Deviation of a Discrete Probability Distribution Variance and Standard Deviation Measure the amount of spread in a distribution The computational steps are: 1. Subtract the mean from each value, and square this difference. 2. Multiply each squared difference by its probability. 3. Sum the resulting products to arrive at the variance. The standard deviation is found by taking the positive square root of the variance.

13 Mean, Variance, and Standard Deviation of a Discrete Probability Distribution - Example John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.

14 Mean of a Discrete Probability Distribution - Example

15 Variance and Standard Deviation of a Discrete Probability Distribution - Example

16 Binomial Probability Distribution Characteristics of a Binomial Probability Distribution There are only two possible outcomes on a particular trial of an experiment. The outcomes are mutually exclusive, The random variable is the result of counts. Each trial is independent of any other trial

17 Binomial Probability Formula

18 Binomial Probability - Example There are five flights daily from Pittsburgh via US Airways into the Bradford, Pennsylvania, Regional Airport. Suppose the probability that any flight arrives late is.20. What is the probability that none of the flights are late today?

Binomial – Shapes for Varying (n constant)

Binomial – Shapes for Varying n ( constant)

21 Cumulative Binomial Probability Distributions A study in June 2003 by the Illinois Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?

22 Binomial Distribution - Table Five percent of the worm gears produced by an automatic, high-speed Carter-Bell milling machine are defective. What is the probability that out of six gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six?

23 Poisson Probability Distribution The Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume. Assumptions of the Poisson Distribution (1) The probability is proportional to the length of the interval. (2) The intervals are independent.

24 Poisson Probability Distribution The Poisson distribution can be described mathematically using the formula:

25 Poisson Probability Distribution The mean number of successes can be determined in binomial situations by n, where n is the number of trials and the probability of a success. The variance of the Poisson distribution is also equal to n.

26 Assume baggage is rarely lost by Northwest Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags. Poisson Probability Distribution - Example

27 Poisson Probability Distribution - Table Assume baggage is rarely lost by Northwest Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with mean = 0.3, find the probability of not losing any bags

The Poissson probability distribution is always positively skewed and random variable has no specific upper limit

Continuous Probability Distributions Chapter 7

30 GOALS 1. Understand the difference between discrete and continuous distributions. 2. Compute the mean and the standard deviation for a uniform distribution. 3. Compute probabilities by using the uniform distribution. 4. List the characteristics of the normal probability distribution. 5. Define and calculate z values. 6. Determine the probability an observation is between two points on a normal probability distribution. 7. Determine the probability an observation is above (or below) a point on a normal probability distribution.

31 Characteristics of a Normal Probability Distribution 1. It is bell-shaped and has a single peak at the center of the distribution. 2. The total area under the curve is 1.00; half the area under the normal curve is to the right of this center point and the other half to the left of it. 3. It is symmetrical about the mean. The arithmetic mean, median, and mode are equal 4. It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it. 5. The location of a normal distribution is determined by the mean( ), the dispersion or spread of the distribution is determined by the standard deviation (σ).

32 The Normal Distribution - Graphically

33 The Normal Distribution - Families

34 The Standard Normal Probability Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z distribution. A z-value is the signed distance between a selected value, designated X, and the population mean, divided by the population standard deviation, σ. The formula is:

35 Areas Under the Normal Curve

36 The Normal Distribution – Example The weekly incomes of shift foremen in the glass industry follow the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the z value for the income, lets call it X, of a foreman who earns $1,100 per week? For a foreman who earns $900 per week?

37 The Empirical Rule About 68 percent of the area under the normal curve is within one standard deviation of the mean. About 95 percent is within two standard deviations of the mean. Practically all is within three standard deviations of the mean.

38 The Empirical Rule - Example As part of its quality assurance program, the Autolite Battery Company conducts tests on battery life. For a particular D-cell alkaline battery, the mean life is 19 hours. The useful life of the battery follows a normal distribution with a standard deviation of 1.2 hours. Answer the following questions. 1. About 68 percent of the batteries failed between what two values? 2. About 95 percent of the batteries failed between what two values? 3. Virtually all of the batteries failed between what two values?

39 Normal Distribution – Finding Probabilities In an earlier example we reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1,000 and a standard deviation of $100. What is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100?

40 Normal Distribution – Finding Probabilities

41 Finding Areas for Z Using Excel The Excel function =NORMDIST(x,Mean,Standard_dev,Cumu) =NORMDIST(1100,1000,100,true) generates area (probability) from Z=1 and below

42 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $790 and $1,000? Normal Distribution – Finding Probabilities (Example 2)

43 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Less than $790? Normal Distribution – Finding Probabilities (Example 3)

44 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $840 and $1,200? Normal Distribution – Finding Probabilities (Example 4)

45 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $1,150 and $1,250 Normal Distribution – Finding Probabilities (Example 5)

46 Using Z in Finding X Given Area - Example Layton Tire and Rubber Company wishes to set a minimum mileage guarantee on its new MX100 tire. Tests reveal the mean mileage is 67,900 with a standard deviation of 2,050 miles and that the distribution of miles follows the normal probability distribution. It wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced. What minimum guaranteed mileage should Layton announce?

47 Using Z in Finding X Given Area - Example

48 Using Z in Finding X Given Area - Excel

49 End of Chapter 7