Review of Binomial Concepts A single experiment, with probability of success p, is repeated n independent times X = number of successes the discrete random variable

Probability distribution for X when n=10 and p=0.5 Cumulative distribution for X when n=10 and p=0.5 P(X = k) P(X  k)

P(X = k): BINOMDIST(k, n, p, 0) P(X  k): BINOMDIST(k, n, p, 1) Excel Commands for Binomial Probabilities X is a binomial random variable with binomial parameters n and p Important Formulas for Binomial Random Variable E[X] = n * p  X = sqrt [ n * p * (1 – p) ]

Example for Binomial Distribution A study by one automobile manufacturer indicated that one out of every four new cars required repairs under the company’s new-car warranty, with an average cost of $50 per repair. a)For 100 new cars, what is the expected cost of repairs? What is the standard deviation? b)Provide a reasonable estimate of the most it might cost the manufacturer for a specified group of 100 cars. (Hint: Ensure a 95% probability of not exceeding amount.)

Example for Binomial Distribution Suppose you are in charge of hiring new undergraduate accounting majors for Coopers and Young (C&Y), an accounting firm headquartered in Chicago. This year your goal is to hire 20 graduates. On the average, about 40% of the people you make offers to will accept. Unfortunately, offers have to go out simultaneously this year, so you plan to make more than 20 offers. a)How many offers should you make so that your expected number of hires is 20? b)How many offers should you make if you want to have an 80% chance of hiring at least 20 people? c)Find a 95% probability interval if you make 50 offers.

Normal Example Otis Elevator in Bloomington, Indiana, reported that the number of hours lost per week last year due to employees’ illnesses was approximately normally distributed, with a mean of 60 hours and a standard deviation of 15 hours. Determine, for a given week, the following probabilities: a)The number of hours lost will exceed 85 hours. b)The number of hours lost will be between 45 and 55 hours.

Normal Example A mail-order company has estimated one-year orders for a popular item to be normally distributed with a mean of 180,000 units and a standard deviation of 15,000 units. a)What is the probability of selling all of the stock on hand if inventory equals 200,000 units? b)What inventory should the company have on hand if they want the probability of running out of stock to be 5%?

The Normal Distribution (preview) The normal distribution is related to a particular type of continuous random variable (as opposed to “discrete random variable”) It is the “bell-shaped” curve that you may have heard about It is used widely in statistics and appears in lots of practical problems It has an expected value (or “mean”) and a standard deviation just like other random variables We’ll discuss it in detail shortly, but a little background first….

Continuous Random Variables a random variable takes a probability event and assigns a number, or value, to it a discrete random variable deals with a finite number of events and numbers Recall A continuous random variable can take on any value within a specified interval of numbers There are an infinite number of events and an infinite number of values

Examples of Continuous Random Variables X = the precise time that a train arrives, when it is scheduled to arrive at 8:00 PM X = the residual chemical levels in the blood 24 hours after taking a specific medication X = the precise amount of soda placed in a bottle by a soda filling machine (or should it be “pop”?) X = the amount of energy used in a home during one hour

Continuous or Discrete? Or Both? Sometimes a discrete random variable has so many possible outcomes (still finite, however) that we consider it to be continuous Sometimes a continuous random variable is actually discrete when we measure it X = salary of an office manager X = price of a therm of natural gas in January X = grades of a student on the SAT X = length of a manufactured part (limits of our measurement techniques)

How to Talk about Continuous Random Variables Since a continuous random variable has so many possible outcomes, we do not consider things like: P(X = 1.0) P(X = ) P(X = -0.5) Instead, we consider: P(X  1.0) P(X  ) P(-0.5  X  ) equalities (no), intervals (yes) ( In fact, P(X = k) = 0 for a continuous random variable. Again, this is “equal to.” )

Example: A Delivery Truck Suppose the warehouse says that the delivery truck will arrive sometime “between 8 AM and noon.” What is the probability that the truck will arrive between 9 AM and 10 AM? The total area of the red block equals 1.0, the total probability. So P(9  X  10) = 0.25 Probability density function (distribution); similar to probability mass function in discrete random variables

The Normal Distribution Bell-shaped Horizontal axis represents values of the random variable Expected value, or mean, is  Symmetrical about  Total blue area is 1.0 Vertical axis is not very important

The Normal Distribution (cont’d) P(  -   X   +  ) = “Probability that X is within 1 std dev of its mean is 68.26%”  is the standard deviation of the normal random variable

The Normal Distribution (cont’d) P(  - 2   X   + 2  ) = “Probability that X is within 2 std dev of its mean is 95.44%”

The Normal Distribution (cont’d) P(  - 3   X   + 3  ) = “Probability that X is within 3 std dev of its mean is 99.73%”

The Normal Distribution (summary) A continuous random variable X is “normally distributed with expectation/mean  and standard deviation  ” if its probability distribution is 1.Bell-shaped 2.Symmetrical about the value  3.The following are true: i.P(  -   X   +  ) = ii.P(  - 2   X   + 2  ) = iii.P(  - 3   X   + 3  ) = We say X is N( ,  2 )

Excel’s Normal Functions If X is N( ,  2 ): P(X  k) = NORMDIST(k, , , 1) NORMINV( prob, ,  ) gives the value of k so that P(X  k) = prob “normal inverse”

The Standardized Normal Distribution Z is N( 0, 1 2 ) The standardized normal X is N( ,  2 ) 1.For comparison of several different normal distributions 2.For calculations without a computer

Formula relating standard Z and any given X: Z = ( X -  ) /  “Given the value for X, the corresponding value for Z is given by the above formula.” ClassYour Score MeanStd Dev Compare your performance in all courses…

The standardized normal is also useful for hand calculations of normal probabilities… Suppose X is normally distributed with mean 20 and standard deviation 4.3. What is P(X  23)? What is P(X  16.5)? Z = ( X -  ) /  (23 – 20) / 4.3 = (16.5 – 20) / 4.3 = P(Z  0.698) = ? P(Z  ) = ? (See Excel) … = … =

The Normal Approximation to the Binomial Both distributions have the same shape…

Suppose X is a binomial variable with n = 900 and p = 0.35 E[X] =  = np = 315  = sqrt( np(1-p) ) = P(X  300) = BINOMDIST(300, 900, 0.35, 1) = P(X  300)  P(Y  300) = NORMDIST(300, 315, 14.31, 1) = P(X = 300) = BINOMDIST(300, 900, 0.35, 0) = P(X = 300)  P(299.5  Y  300.5) = NORMDIST(300.5, 315, 14.31, 1) – NORMDIST(300.5, 315, 14.31, 1) = Let Y be the normal variable with  = 315 and  = The second is called the “continuity correction”