normal standard deviation

Anonymous
timer Asked: Dec 26th, 2018
account_balance_wallet $30

Question description

Suppose that we are told that the weights of 5-year old children in a

particular region of the world are normally distributed with a mean of 16 KGs and

standard deviation of 5 KGs.


a. Approximately what proportion of 5-year old children are heavier than 20 KGs?

b.What proportion of 5-year old children are between 20 and 22 KGs?

c. What weight corresponds to the point where 20% of all 5-year old children are greater than this weight?


. . . . . . . . . . . . SLIDES BY John Loucks St. Edward’s Univ. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 1 Chapter 3, Part A Discrete Probability Distributions Introduction to probability Random Variables Discrete Probability Distributions Binomial Probability Distribution Poisson Probability Distribution .40 .30 .20 .10 0 1 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2 3 4 Slide 2 Uncertainties Managers often base their decisions on an analysis of uncertainties such as the following: What are the chances that sales will decrease if we increase prices? What is the likelihood a new assembly method will increase productivity? What are the odds that a new investment will be profitable? © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 3 Probability Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near zero indicates an event is quite unlikely to occur. A probability near one indicates an event is almost certain to occur. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 4 Probability as a Numerical Measure of the Likelihood of Occurrence Increasing Likelihood of Occurrence Probability: 0 The event is very unlikely to occur. .5 The occurrence of the event is just as likely as it is unlikely. 1 The event is almost certain to occur. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 5 Statistical Experiments In statistics, the notion of an experiment differs somewhat from that of an experiment in the physical sciences. In statistical experiments, probability determines outcomes. Even though the experiment is repeated in exactly the same way, an entirely different outcome may occur. For this reason, statistical experiments are sometimes called random experiments. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 6 Assigning Probabilities ◼ Basic Requirements for Assigning Probabilities 1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. 0 < P(Ei) < 1 for all i Where: Ei is the ith experimental outcome and P(Ei) is its probability 2. The sum of the probabilities for all experimental outcomes must equal 1. P(E1) + P(E2) + . . . + P(En) = 1 Where: n is the number of experimental outcomes © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 7 1. Random Variables A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 Random Variables Examples of Random Variables The first, second, and fourth variables above are discrete, while the third one is continuous. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 9 Random Variables Examples of Random Variables Question Family size Type Random Variable x x = Number of dependents in family reported on tax return Discrete Distance from x = Distance in miles from home to store home to the store site Continuous Own dog or cat Discrete x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 10 Example: JSL Appliances Discrete random variable with a finite number of values Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Example: JSL Appliances Discrete random variable with an infinite sequence of values Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . . We can count the customers arriving, but there is no finite upper limit on the number that might arrive. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 2. Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Discrete Probability Distributions The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: f(x) > 0 f(x) = 1 For a discrete probability distribution we calculate the probability of being less than some value x, i.e. P(X < x), by simply summing up the probabilities of the values less than x. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 14 Example: DiCarlo Motors, Inc. Using past data on daily car sales, … a tabular representation of the probability distribution for car sales was developed. x 0 1 2 3 4 5 f(x) .18 .39 .24 .14 .04 .01 1.00 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Example: DiCarlo Motors, Inc. Graphical Representation of the Probability Distribution Probability .50 .40 .30 .20 .10 0 1 2 3 4 5 Values of Random Variable x (car sales) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 Example: DiCarlo Motors, Inc. The probability distribution provides the following information. • There is a 0.18 probability that no cars will be sold during a day. • The most probable sales volume is 1, with f(1) = 0.39. • There is a 0.05 probability of an outstanding sales day with four or five cars being sold. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Discrete Uniform Probability Distribution The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. The discrete uniform probability function is f(x) = 1/n the values of the random variable are equally likely where: n = the number of values the random variable may assume © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 Expected Value and Variance The expected value, or mean, of a random variable is a measure of its central location. E(x) =  = xf(x) The variance summarizes the variability in the values of a random variable. Var(x) =  2 = (x - )2f(x) The standard deviation, , is defined as the positive square root of the variance. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Example: DiCarlo Motors, Inc. Expected Value of a Discrete Random Variable x 0 1 2 3 4 5 f(x) xf(x) .18 .00 .39 .39 .24 .48 .14 .42 .04 .16 .01 .05 E(x) = 1.50 expected number of cars sold in a day © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 20 Example: DiCarlo Motors, Inc. Variance and Standard Deviation of a Discrete Random Variable x x- (x - )2 0 1 2 3 4 5 .18 -1.5 2.25 .4050 .39 -0.5 0.25 .0975 .24 0.5 0.25 .0600 .14 1.5 2.25 .3150 .04 2.5 6.25 .2500 .01 3.5 12.25 .1225 Variance of daily sales =  2 = 1.2500 f(x) (x - )2f(x) cars squared Standard deviation of daily sales = 1.118 cars © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 21 3. Binomial Probability Distribution Four Properties of a Binomial Experiment 1. The experiment consists of a sequence of n identical trials. 2. Two outcomes, success and failure, are possible on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. stationarity assumption 4. The trials are independent. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 22 Binomial Probability Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 23 Binomial Probability Distribution Binomial Probability Function n! f (x) = p x (1 − p)( n − x ) x !(n − x )! where: f(x) = the probability of x successes in n trials p = the probability of success on any one trial n = the number of trials x = number of successes in n trials n! =n( n - 1)( n - 2) . . . (2)(1) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 24 Binomial Probability Distribution Binomial Probability Function n! f (x) = p x (1 − p)( n − x ) x !(n − x )! Number of experimental outcomes providing exactly x successes in n trials Probability of a particular sequence of trial outcomes with x successes in n trials © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 25 Example: Nastke Clothing Store Binomial Probability Distribution The store manager estimates that the probability of a customer making a purchase is 0.30. What is the probability that 2 of the next 3 customers entering the store make a purchase? Let: p = .30 (success), n = 3, x = 2 n! f ( x) = p x (1 − p ) (n − x ) x !( n − x )! 3! f (2) = (0.3)2 (0.7)1 = 3(.09)(.7) = .189 2!(3 − 2)! © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 26 Binomial Probability Distribution TABLE 3.6 PROBABILITY DISTRIBUTION FOR THE NUMBER OF CUSTOMERS MAKING A PURCHASE X f (x) 0 0.343 1 0.441 2 0.189 3 0.027 Total 1.000 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 27 Binomial Probability Distribution Expected Value E(x) =  = np Variance Var(x) =  2 = np(1 − p) Standard Deviation s = np(1- p) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 28 Example: Nastke Clothing Store Binomial Probability Distribution • Expected Value E(x) =  = 3(.3) = .9 customers out of 3 • Variance Var(x) =  2 = 3(.3)(.7) = .63 • Standard Deviation  = 3(.3)(.7) = .794 customers © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 29 Poisson Probability Distribution Poisson Probability Distribution A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ). © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 30 Poisson Probability Distribution Poisson Probability Distribution Examples of Poisson distributed random variables: the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour Bell Labs used the Poisson distribution to model the arrival of phone calls. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 31 Poisson Probability Distribution Poisson Probability Distribution Two Properties of a Poisson Experiment 1. The probability of an occurrence is the same for any two intervals of equal length. 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 32 Poisson Probability Distribution Poisson Probability Function 𝜇 𝑥 𝑒 −𝜇 𝑓 𝑥 = 𝑥! where: x = the number of occurrences in an interval f(x) = the probability of x occurrences in an interval  = mean number of occurrences in an interval e = 2.71828 x! = x(x – 1)(x – 2) . . . (2)(1) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 33 Poisson Probability Distribution Poisson Probability Function Since there is no stated upper limit for the number of occurrences, the probability function f(x) is applicable for values x = 0, 1, 2, … without limit. In practical applications, x will eventually become large enough so that f(x) is approximately zero and the probability of any larger values of x becomes negligible. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 34 Poisson Probability Distribution Example: Mercy Hospital Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 35 Poisson Probability Distribution Poisson Probability Distribution Example: Mercy Hospital  = 6/hour = 3/half-hour, x = 4 𝑓 4 = 34 (2.71828)−3 4! Using the probability function = .1680 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 36 Chapter 3, Part B Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f (x) f (x) Exponential Uniform f (x) Normal x x x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 37 Continuous Random Variables Examples of continuous random variables include the following: • The flight time of an airplane traveling from Chicago to New York • The lifetime of the picture tube in a new television set • The drilling depth required to reach oil in an offshore drilling operation © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 38 Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. For a continuous probability distribution we calculate the probability of being less than some value x, i.e. P(X < x), by calculating the area under the curve to the left of x. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 39 Continuous Probability Distributions The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2. f (x) f (x) Exponential Uniform f (x) x1 x 2 Normal x x1 x1 x2 xx12 x2 x x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 40 Normal Probability Distribution The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 41 Normal Probability Distribution It has been used in a wide variety of applications: Heights of people Test scores Amounts of rainfall Scientific measurements © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 42 Normal Probability Distribution Normal Probability Density Function f (x) = 1 s 2p e -(x-m )2 /2s 2 where:  = mean  = standard deviation  = 3.14159 e = 2.71828 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 43 Normal Probability Distribution Characteristics The distribution is symmetric, and is bell-shaped. x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 44 Normal Probability Distribution Characteristics The entire family of normal probability distributions is defined by its mean  and its standard deviation  . Standard Deviation  Mean  x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 45 Normal Probability Distribution Characteristics The highest point on the normal curve is at the mean, which is also the median and mode. x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 46 Normal Probability Distribution Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 47 Normal Probability Distribution Characteristics The standard deviation determines the width of the curve: larger values result in wider, flatter curves.  = 15  = 25 x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 48 Normal Probability Distribution Characteristics Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). .5 .5 x © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 49 Normal Probability Distribution Characteristics 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean. 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 50 Normal Probability Distribution Characteristics 99.72% 95.44% 68.26%  – 3  – 1  – 2   + 3  + 1  + 2 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. x Slide 51 Standard Normal Probability Distribution -Use probability tables that have been calculated on a computer. - Only one special Normal distribution, N(0, 1), has been tabulated. - If we want to calculate probabilities from different Normal distributions we convert the probability to one involving the standard Normal distribution. → standardization A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 52 Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. =1 z 0 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 53 Standard Normal Probability Distribution Converting to the Standard Normal Distribution z= x−  We can think of z as a measure of the number of standard deviations x is from . © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 54 Example: Pep Zone Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 55 Example: Pep Zone Standard Normal Probability Distribution The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P(x > 20). © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 56 Example: Pep Zone Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. z = (x - )/ = (20 - 15)/6 = .83 Step 2: Find the area under the standard normal curve between the mean and z = .83. see next slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 57 Example: Pep Zone Probability Table for the Standard Normal Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . .5 .1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 .7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 . . . . . . . . . . . P(0 < z < .83) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 58 Example: Pep Zone Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z = .83. P(z > .83) = .5 – P(0 < z < .83) = .5- .2967 = .2033 Probability of a stockout P(x > 20) © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 59 Example: Pep Zone Solving for the Stockout Probability Area = .5 - .2967 Area = .2967 = .2033 0 .83 z © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 60 End of Chapter 3 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 61

Tutor Answer

Tutorlamb
School: Rice University

Attached.

1

Running Head: NORMAL STANDARD D...

flag Report DMCA
Review

Anonymous
Super fast turn around time. Very impressed and will use Studypool again if I need a quick job. Communication was a bit tough but not unbearable.

Similar Questions
Hot Questions
Related Tags

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors