Chapter 6: Probability Distributions. Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?


 Joy Jones
 1 years ago
 Views:
Transcription
1 Chapter 6: Probability Distributions Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities? 1
2 Learning Objectives 1. Random variable 2. Probability distributions for discrete random variables 3. Mean of a probability distribution 4. Summarizing the spread of a probability distribution 5. Probability distribution for continuous random variables 2
3 Learning Objective 1: Randomness The numerical values that a variable assumes are the result of some random phenomenon: Selecting a random sample for a population or Performing a randomized experiment 3
4 Learning Objective 1: Random Variable A random variable is a numerical measurement of the outcome of a random phenomenon. 4
5 Learning Objective 1: Random Variable Use letters near the end of the alphabet, such as x, to symbolize Variables A particular value of the random variable Use a capital letter, such as X, to refer to the random variable itself. Example: Flip a coin three times X=number of heads in the 3 flips; defines the random variable x=2; represents a possible value of the random variable 5
6 Learning Objective 2: Probability Distribution The probability distribution of a random variable specifies its possible values and their probabilities. Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes 6
7 Learning Objective 2: Probability Distribution of a Discrete Random Variable A discrete random variable X has separate values (such as 0,1,2, ) as its possible outcomes Its probability distribution assigns a probability P(x) to each possible value x: For each x, the probability P(x) falls between 0 and 1 The sum of the probabilities for all the possible x values equals 1 7
8 Learning Objective 2: Example What is the estimated probability of at least three home runs? P(3)+P(4)+P(5)= =0.17 8
9 Learning Objective 3: The Mean of a Discrete Probability Distribution The mean of a probability distribution for a discrete random variable is x p(x) where the sum is taken over all possible values of x. The mean of a probability distribution is denoted by the parameter, µ. The mean is a weighted average; values of x that are more likely receive greater weight P(x) 9
10 Learning Objective 3: Expected Value of X The mean of a probability distribution of a random variable X is also called the expected value of X. The expected value reflects not what we ll observe in a single observation, but rather that we expect for the average in a long run of observations. It is not unusual for the expected value of a random variable to equal a number that is NOT a possible outcome. 10
11 Learning Objective 3: Example Find the mean of this probability distribution. The mean: x p(x) = 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) =
12 Learning Objective 4: The Standard Deviation of a Probability Distribution The standard deviation of a probability distribution, denoted by the parameter, σ, measures its spread. Larger values of σ correspond to greater spread. Roughly, σ describes how far the random variable falls, on the average, from the mean of its distribution 12
13 Learning Objective 5: Continuous Random Variable A continuous random variable has an infinite continuum of possible values in an interval. Examples are: time, age and size measures such as height and weight. Continuous variables are measured in a discrete manner because of rounding. 13
14 Learning Objective 5: Probability Distribution of a Continuous Random Variable A continuous random variable has possible values that form an interval. Its probability distribution is specified by a curve. Each interval has probability between 0 and 1. The interval containing all possible values has probability equal to 1. 14
15 Chapter 6: Probability Distributions Section 6.2: How Can We Find Probabilities for BellShaped Distributions? 15
16 Learning Objectives 1. Normal Distribution Rule for normal distributions 3. ZScores and the Standard Normal Distribution 4. The Standard Normal Table: Finding Probabilities 5. Using the TIcalculator: find probabilities 16
17 Learning Objectives 6. Using the Standard Normal Table in Reverse 7. Using the TIcalculator: find zscores 8. Probabilities for Normally Distributed Random Variables 9. Percentiles for Normally Distributed Random Variables 10. Using Zscores to Compare Distributions 17
18 Learning Objective 1: Normal Distribution The normal distribution is symmetric, bellshaped and characterized by its mean µ and standard deviation. The normal distribution is the most important distribution in statistics Many distributions have an approximate normal distribution Approximates many discrete distributions well when there are a large number of possible outcomes Many statistical methods use it even when the data are not bell shaped 18
19 Learning Objective 1: Normal Distribution Normal distributions are Bell shaped Symmetric around the mean The mean ( ) and the standard deviation ( ) completely describe the density curve Increasing/decreasing moves the curve along the horizontal axis Increasing/decreasing controls the spread of the curve 19
20 Learning Objective 1: Normal Distribution Within what interval do almost all of the men s heights fall? Women s height? 20
21 Learning Objective 2: Rule for Any Normal Curve 68% of the observations fall within one standard deviation of the mean 95% of the observations fall within two standard deviations of the mean 99.7% of the observations fall within three standard deviations of the mean 21
22 Learning Objective 2: Example : % Rule Heights of adult women can be approximated by a normal distribution = 65 inches; =3.5 inches Rule for women s heights 68% are between 61.5 and 68.5 inches [ µ = ] 95% are between 58 and 72 inches [ µ 2 = 65 2(3.5) = 65 7 ] 99.7% are between 54.5 and 75.5 inches [ µ 3 = 65 3(3.5) = ] 22
23 Learning Objective 2: Example : % Rule What proportion of women are less than 69 inches tall?? = 84% 16% 68% (by Rule)? (height values) 23
24 Learning Objective 3: ZScores and the Standard Normal Distribution The zscore for a value x of a random variable is the number of standard deviations that x falls from the mean z x A negative (positive) zscore indicates that the value is below (above) the mean zscores can be used to calculate the probabilities of a normal random variable using the normal tables in the back of the book 24
25 Learning Objective 3: ZScores and the Standard Normal Distribution A standard normal distribution has mean µ=0 and standard deviation σ=1 When a random variable has a normal distribution and its values are converted to z scores by subtracting the mean and dividing by the standard deviation, the zscores have the standard normal distribution. 25
26 Learning Objective 4: Table A: Standard Normal Probabilities Table A enables us to find normal probabilities It tabulates the normal cumulative probabilities falling below the point +z To use the table: Find the corresponding zscore Look up the closest standardized score (z) in the table. First column gives z to the first decimal place First row gives the second decimal place of z The corresponding probability found in the body of the table gives the probability of falling below the zscore 26
27 Learning Objective 4: Example: Using Table A Find the probability that a normal random variable takes a value less than 1.43 standard deviations above µ; P(z<1.43)=.9236 TI Calculator = Normcdf(1e99,1.43,0,1)=
28 Learning Objective 4: Example: Using Table A Find the probability that a normal random variable takes a value greater than 1.43 standard deviations above µ: P(z>1.43)= =.0764 TI Calculator = Normcdf(1.43,1e99,0,1)=
29 Learning Objective 4: Example: Find the probability that a normal random variable assumes a value within 1.43 standard deviations of µ Probability below 1.43σ =.9236 Probability below 1.43σ =.0764 ( ) P(1.43<z<1.43) = =.8472 TI Calculator = Normcdf(1.43,1.43,0,1)=
30 Learning Objective 5: Using the TI Calculator To calculate the cumulative probability 2 nd DISTR; 2:normalcdf(lower bound, upper bound,mean,sd) Use 1E99 for negative infinity and 1E99 for positive infinity 30
31 Learning Objective 5: Find Probabilities Using TI Calculator Find probability to the left of P(z<1.64)=normcdf(1e99,1.64,0,1)=.0505 Find probability to the right of 1.56 P(z>1.56)=normcdf(1.56,1e99,0,1)=.0594 Find probability between .50 and 2.25 P(.5<z<2.25)=normcdf(.5,2.25,0,1)=
32 Learning Objective 6: How Can We Find the Value of z for a Certain Cumulative Probability? To solve some of our problems, we will need to find the value of z that corresponds to a certain normal cumulative probability To do so, we use Table A in reverse Rather than finding z using the first column (value of z up to one decimal) and the first row (second decimal of z) Find the probability in the body of the table The zscore is given by the corresponding values in the first column and row 32
33 Learning Objective 6: How Can We Find the Value of z for a Certain Cumulative Probability? Example: Find the value of z for a cumulative probability of Look up the cumulative probability of in the body of Table A. A cumulative probability of corresponds to z = Thus, the probability that a normal random variable falls at least 1.96 standard deviations below the mean is
34 Learning Objective 6: How Can We Find the Value of z for a Certain Cumulative Probability? Example: Find the value of z for a cumulative probability of Look up the cumulative probability of in the body of Table A. A cumulative probability of corresponds to z = Thus, the probability that a normal random variable takes a value no more than 1.96 standard deviations above the mean is
35 Learning Objective 7: Using the TI Calculator to Find ZScores for a Given Probability 2 nd DISTR 3:invNorm; Enter invnorm(percentile,mean,sd) Percentile is the probability under the curve from negative infinity to the zscore Enter 35
36 Learning Objective 7: Examples The probability that a standard normal random variable assumes a value that is z is What is z? Invnorm(.975,0,1)=1.96 The probability that a standard normal random variable assumes a value that is > z is What is z? Invnorm(.975,0,1)=1.96 The probability that a standard normal random variable assumes a value that is z is What is z? Invnorm(1.881,0,1)=1.18 The probability that a standard normal random variable assumes a value that is < z is What is z? Invnorm(.119,0,1)=
37 Learning Objective 7: Example Find the zscore z such that the probability within z standard deviations of the mean is Invnorm(.75,0,1)=.67 Invnorm(.25,0,1)= .67 Probability = P(.67<Z<.67)=.5 37
38 Learning Objective 8: Finding Probabilities for Normally Distributed Random Variables 1. State the problem in terms of the observed random variable X, i.e., P(X<x) 2. Standardize X to restate the problem in terms of a standard normal variable Z P(X x) P Z z x 3. Draw a picture to show the desired probability under the standard normal curve 4. Find the area under the standard normal curve using Table A 38
39 Learning Objective 8: P(X<x) Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure less than 100? P(X<100) = P Z P(z 1.00) Normcdf(1E99,100,120,20)= % of adults have systolic blood pressure less than
40 Learning Objective 8: P(X>x) Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure greater than 100? P(X>100) = 1 P(X<100) P(X>100)= =.8413 Normcdf(100,1e99,120,20)=.8413 P Z % of adults have systolic blood pressure greater than 100 P(Z 1.00)
41 Learning Objective 8: P(X>x) Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure greater than 133? P(X>133) = 1 P(X<133) P(X>133)= =.2578 Normcdf(133,1E99,120,20)=.2578 P Z % of adults have systolic blood pressure greater than 133 P(Z.65)
42 Learning Objective 8: P(a<X<b) Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure between 100 and 133? P(100<X<133) = P(X<133)P(X<100) P Z P Z Normcdf(100,133,120,20)= % of adults have systolic blood pressure between 100 and P(Z.65) P(Z 1.00)
43 Learning Objective 9: Find X Value Given Area to Left Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What is the 1 st quartile? P(X<x)=.25, find x: Look up.25 in the body of Table A to find z= Solve equation to find x: Check: x z 120 ( 0.67)* P(X<106.6) P(Z<0.67)=0.25 TI Calculator = Invnorm(.25,120,20)=
44 Learning Objective 9: Find X Value Given Area to Right Adult systolic blood pressure is normally distributed with µ = 120 and σ = % of adults have systolic blood pressure above what level? P(X>x)=.10, find x. P(X>x)=1P(X<x) Look up 10.1=0.9 in the body of Table A to find z=1.28 Solve equation to find x: x z 120 (1.28)* Check: P(X>145.6) =P(Z>1.28)=0.10 TI Calculator = Invnorm(.9,120,20)=
45 Learning Objective 10: Using Zscores to Compare Distributions Zscores can be used to compare observations from different normal distributions Example: You score 650 on the SAT which has =500 and =100 and 30 on the ACT which has =21.0 and =4.7. On which test did you perform better? Compare zscores SAT: z ACT: z Since your zscore is greater for the ACT, you performed better on this exam 45
46 Chapter 6: Probability Distributions Section 6.3: How Can We Find Probabilities When Each Observation Has Two Possible Outcomes? 46
47 Learning Objectives 1. The Binomial Distribution 2. Conditions for a Binomial Distribution 3. Probabilities for a Binomial Distribution 4. Factorials 5. Examples using Binomial Distribution 6. Do the Binomial Conditions Apply? 7. Mean and Standard Deviation of the Binomial Distribution 8. Normal Approximation to the Binomial 47
48 Learning Objective 1: The Binomial Distribution Each observation is binary: it has one of two possible outcomes. Examples: Accept, or decline an offer from a bank for a credit card. Have, or do not have, health insurance. Vote yes or no on a referendum. 48
49 Learning Objective 2: Conditions for the Binomial Distribution Each of n trials has two possible outcomes: success or failure. Each trial has the same probability of success, denoted by p. The n trials are independent. The binomial random variable X is the number of successes in the n trials. 49
50 Learning Objective 3: Probabilities for a Binomial Distribution Denote the probability of success on a trial by p. For n independent trials, the probability of x successes equals: P(x) n! x!(n  x)! px (1 p) n x, x 0,1,2,...,n 50
51 Learning Objective 4: Factorials Rules for factorials: n!=n*(n1)*(n2) 2*1 1!=1 0!=1 For example, 4!=4*3*2*1=24 51
52 Learning Objective 5: Example: Finding Binomial Probabilities John Doe claims to possess ESP. An experiment is conducted: A person in one room picks one of the integers 1, 2, 3, 4, 5 at random. In another room, John Doe identifies the number he believes was picked. Three trials are performed for the experiment. Doe got the correct answer twice. 52
53 Learning Objective 5: Example 1 If John Doe does not actually have ESP and is actually guessing the number, what is the probability that he d make a correct guess on two of the three trials? The three ways John Doe could make two correct guesses in three trials are: SSF, SFS, and FSS. Each of these has probability: (0.2)2(0.8)= The total probability of two correct guesses is 3(0.032)=
54 Learning Objective 5: Example 1 The probability of exactly 2 correct guesses is the binomial probability with n = 3 trials, x = 2 correct guesses and p = 0.2 probability of a correct guess. P(2) 3! 2!1! (0.2)2 (0.8) 1 3(0.04)(0.8) nd Vars 0:binampdf(n,p,x) Binampdf(3,.2,2)=
55 Learning Objective 5: Binomial Example employees, 50% Female None of the 10 employees chosen for management training were female. The probability that no females are chosen is: P(0) 10! 0!10! (0.50)0 (0.50) Binompdf(10,.5,0)= E4 It is very unlikely (one chance in a thousand) that none of the 10 selected for management training would be female if the employees were chosen randomly 55
56 Learning Objective 6: Do the Binomial Conditions Apply? Before using the binomial distribution, check that its three conditions apply: Binary data (success or failure). The same probability of success for each trial (denoted by p). Independent trials. 56
57 Learning Objective 6: Do the Binomial Conditions Apply to Example 2? The data are binary (male, female). If employees are selected randomly, the probability of selecting a female on a given trial is With random sampling of 10 employees from a large population, outcomes for one trial does not depend on the outcome of another trial 57
58 Learning Objective 7: Binomial Mean and Standard Deviation The binomial probability distribution for n trials with probability p of success on each trial has mean µ and standard deviation σ given by: np, np(1 p) 58
59 Learning Objective 7: Example: Racial Profiling? Data: 262 police car stops in Philadelphia in of the drivers stopped were AfricanAmerican. In 1997, Philadelphia s population was 42.2% AfricanAmerican. Does the number of AfricanAmericans stopped suggest possible bias, being higher than we would expect (other things being equal, such as the rate of violating traffic laws)? 59
60 Learning Objective 7: Example: Racial Profiling? Assume: 262 car stops represent n = 262 trials. Successive police car stops are independent. P(driver is AfricanAmerican) is p = Calculate the mean and standard deviation of this binomial distribution: 262(0.422) (0.422) (0.578) 8 60
61 Learning Objective 7: Example: Racial Profiling? Recall: Empirical Rule When a distribution is bellshaped, close to 100% of the observations fall within 3 standard deviations of the mean. u (8) (8)
62 Learning Objective 7: Example: Racial Profiling? If there is no racial profiling, we would not be surprised if between about 87 and 135 of the 262 drivers stopped were AfricanAmerican. The actual number stopped (207) is well above these values. The number of AfricanAmericans stopped is too high, even taking into account random variation. Limitation of the analysis: Different people do different amounts of driving, so we don t really know that 42.2% of the potential stops were African American. 62
63 Learning Objective 8: Approximating the Binomial Distribution with the Normal Distribution The binomial distribution can be well approximated by the normal distribution when the expected number of successes, np, and the expected number of failures, n(1p) are both at least
Probability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More informationUnit 16 Normal Distributions
Unit 16 Normal Distributions Objectives: To obtain relative frequencies (probabilities) and percentiles with a population having a normal distribution While there are many different types of distributions
More informationStatistics 100 Binomial and Normal Random Variables
Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random
More informationThe Normal Curve. The Normal Curve and The Sampling Distribution
Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will get observations
More informationCh5: Discrete Probability Distributions Section 51: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 51: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationStats on the TI 83 and TI 84 Calculator
Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and
More informationUniversity of California, Los Angeles Department of Statistics. Normal distribution
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Normal distribution The normal distribution is the most important distribution. It describes
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationUsing Your TINSpire Calculator: Normal Distributions Dr. Laura Schultz Statistics I
Using Your TINSpire Calculator: Normal Distributions Dr. Laura Schultz Statistics I Always start by drawing a sketch of the normal distribution that you are working with. Shade in the relevant area (probability),
More information4: Probability. What is probability? Random variables (RVs)
4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.23, 2.1 Normal distribution Math 283 / October
More informationCopyright 2013 by Laura Schultz. All rights reserved. Page 1 of 6
Using Your TINSpire Calculator: Binomial Probability Distributions Dr. Laura Schultz Statistics I This handout describes how to use the binompdf and binomcdf commands to work with binomial probability
More information13.2 Measures of Central Tendency
13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers
More informationChapter 4 The Standard Deviation as a Ruler and the Normal Model
Chapter 4 The Standard Deviation as a Ruler and the Normal Model The standard deviation is the most common measure of variation; it plays a crucial role in how we look at data. Z scores measure standard
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationLecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000
Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event
More informationPROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful.
PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the pvalue of the test
More information3. Continuous Random Variables
3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0.
Ch. 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 1 Find Areas Under the Standard Normal Curve 1) Find the area under the standard normal
More informationChapter 6 Continuous Probability Distributions
Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationChapter 2. The Normal Distribution
Chapter 2 The Normal Distribution Lesson 21 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve
More informationYou flip a fair coin four times, what is the probability that you obtain three heads.
Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.
More informationThe basics of probability theory. Distribution of variables, some important distributions
The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a
More informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 68 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationKey Concept. Density Curve
MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal
More informationUnit 8: Normal Calculations
Unit 8: Normal Calculations Summary of Video In this video, we continue the discussion of normal curves that was begun in Unit 7. Recall that a normal curve is bellshaped and completely characterized
More informationSampling Distribution of a Sample Proportion
Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given
More informationDEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS QM 120. Continuous Probability Distribution
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Dr. Mohammad Zainal Continuous Probability Distribution 2 When a RV x is discrete,
More informationthe number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match?
Poisson Random Variables (Rees: 6.8 6.14) Examples: What is the distribution of: the number of organisms in the squares of a haemocytometer? the number of hits on a web site in one hour? the number of
More informationDef: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.
Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationThe normal approximation to the binomial
The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There
More informationEach exam covers lectures from since the previous exam and up to the exam date.
Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,
More informationProbability. Distribution. Outline
7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The
More informationStudy 6.3, #87(83) 93(89),97(93)
GOALS: 1. Understand that area under a normal curve represents probabilities and percentages. 2. Find probabilities (percentages) associated with a normally distributed variable using SNC. 3. Find probabilities
More informationChapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs
Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)
More informationChapter 6: Continuous Probability Distributions
Chapter 6: Continuous Probability Distributions Chapter 5 dealt with probability distributions arising from discrete random variables. Mostly that chapter focused on the binomial experiment. There are
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationNormal Probability Distribution
Normal Probability Distribution The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use
More informationContinuous Distributions, Mainly the Normal Distribution
Continuous Distributions, Mainly the Normal Distribution 1 Continuous Random Variables STA 281 Fall 2011 Discrete distributions place probability on specific numbers. A Bin(n,p) distribution, for example,
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationSTT 315 Practice Problems II for Sections
STT 315 Practice Problems II for Sections 4.14.8 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Classify the following random
More informationThe normal approximation to the binomial
The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very
More informationHISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
More informationChapter 3 Normal Distribution
Chapter 3 Normal Distribution Density curve A density curve is an idealized histogram, a mathematical model; the curve tells you what values the quantity can take and how likely they are. Example Height
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationComputing Binomial Probabilities
The Binomial Model The binomial probability distribution is a discrete probability distribution function Useful in many situations where you have numerical variables that are counts or whole numbers Classic
More informationChapter 5: Normal Probability Distributions  Solutions
Chapter 5: Normal Probability Distributions  Solutions Note: All areas and zscores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in reallife applications that they have been given their own names.
More informationChapter 3: Data Description Numerical Methods
Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationNormal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.
Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e (y µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationA frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes
A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that
More informationx Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 31 Example 31: Solution
Chapter 3 umerical Descriptive Measures 3.1 Measures of Central Tendency for Ungrouped Data 3. Measures of Dispersion for Ungrouped Data 3.3 Mean, Variance, and Standard Deviation for Grouped Data 3.4
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1342 (Elementary Statistics) Test 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the indicated probability. 1) If you flip a coin
More information1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700
Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,
More informationProbability Distributions
CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,
More informationThe Normal Distribution
The Normal Distribution Continuous Distributions A continuous random variable is a variable whose possible values form some interval of numbers. Typically, a continuous variable involves a measurement
More informationSTAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable
More informationSection 5 Part 2. Probability Distributions for Discrete Random Variables
Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 41/42 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationSTATISTICS FOR PSYCH MATH REVIEW GUIDE
STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.
More informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.18.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
More informationSample Exam #1 Elementary Statistics
Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables
More informationStatistics GCSE Higher Revision Sheet
Statistics GCSE Higher Revision Sheet This document attempts to sum up the contents of the Higher Tier Statistics GCSE. There is one exam, two hours long. A calculator is allowed. It is worth 75% of the
More informationWeek 3&4: Z tables and the Sampling Distribution of X
Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
More informationCharacteristics of Binomial Distributions
Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation
More informationUnit 29 ChiSquare GoodnessofFit Test
Unit 29 ChiSquare GoodnessofFit Test Objectives: To perform the chisquare hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni
More information( ) ( ) Central Tendency. Central Tendency
1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution
More informationEXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!
STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.
More informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationProbability distributions
Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.142.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,
More informationProbability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationChapter 3 Descriptive Statistics: Numerical Measures. Learning objectives
Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the
More informationWe will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:
MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having
More informationMath 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5
ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has
More informationgiven that among year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20...
Probability  Chapter 5 given that among 1214 year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20... 5.6 What percentage of boys in this age range
More information. 58 58 60 62 64 66 68 70 72 74 76 78 Father s height (inches)
PEARSON S FATHERSON DATA The following scatter diagram shows the heights of 1,0 fathers and their fullgrown sons, in England, circa 1900 There is one dot for each fatherson pair Heights of fathers and
More informationDiscrete Random Variables and their Probability Distributions
CHAPTER 5 Discrete Random Variables and their Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Discrete Random Variable
More informationCHAPTER 7: THE CENTRAL LIMIT THEOREM
CHAPTER 7: THE CENTRAL LIMIT THEOREM Exercise 1. Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews
More informationLecture 2: Discrete Distributions, Normal Distributions. Chapter 1
Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:304:30, Wed 45 Bring a calculator, and copy Tables
More information
An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476
Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationPractice Questions Chapter 4 & 5
Practice Questions Chapter 4 & 5 Use the following to answer questions 13: Ignoring twins and other multiple births, assume babies born at a hospital are independent events with the probability that a
More information, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0
Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after SimeonDenis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will
More information6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.
6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous
More informationChapter 15 Binomial Distribution Properties
Chapter 15 Binomial Distribution Properties Two possible outcomes (success and failure) A fixed number of experiments (trials) The probability of success, denoted by p, is the same on every trial The trials
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 (b) 1
Unit 2 Review Name Use the given frequency distribution to find the (a) class width. (b) class midpoints of the first class. (c) class boundaries of the first class. 1) Miles (per day) 12 9 34 22 56
More informationGCSE HIGHER Statistics Key Facts
GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information
More information