1 Probability Distributions

Size: px
Start display at page:

Download "1 Probability Distributions"

Transcription

1 1 Probability Distributions In the chapter about descriptive statistics sample data were discussed, and tools introduced for describing the samples with numbers as well as with graphs. In this chapter models for the population will be introduced. One will see how the properties of a population can be described in mathematical terms. Later we will see how samples can be used to draw conclusions about those properties. That step is called statistical inference. Definition A random variable(rv) X is a variable whose value is determined by the outcome of a random experiment. As discussed for variables in samples, rvs can be categorical or numerical, and if they are numerical they can be either discrete or continuous. random variable categorical numerical discrete continuous In data description we observed that the proper methods depend on the type of the variable. This is similar for rvs. The choice of model depends on their type. The models for continuous rvs will be different than those for categorical or discrete rvs. All random variables are described by their distribution. Definition 1 The distribution of a random variable gives the values the random variable can have and the probabilities for these to occur. 1.1 Categorical Random Variables Definition The distribution of a categorical rv is a table giving all possible values (categories) of the rv and the associated probabilities. The distribution of a categorical rv can be shown in form of a bar graph. Example: The population investigated are the students of a selected college. The random variable of interest is the residence status, it can be either resident or nonresident, so it is categorical. The probability distribution is: resident status probability resident 0.73 nonresident 0.27 If a student is chosen randomly from this college, the probability for the student being a resident is Is x the random variable resident status then write P (x =resident)=

2 1.2 Numerical Random Variables Discrete Random Variables Remember: A discrete rv is a random variable whose possible values are isolated points along the number line. Example 1 1. number of teeth in a patient 2. number of houses in a certain block 3. number of heads when tossing 3 coins The probability distribution for a discrete rv, X, can be given as a formula, table, or graph that gives the possible values of X, and their corresponding probabilities, p(x). Example: Toss two unbiased coins and let X equal the number of heads observed. The simple events of this experiment are: coin1 coin 2 x P (X = x) H H 2 1/4 H T 1 1/4 T H 1 1/4 T T 0 1/4 So that we get the following distribution for X=number of heads observed: x P (X = x) 0 1/4 1 1/2 2 1/4 With the help of this distribution can calculate that P (X 1) = P (X = 0) + P (X = 1) = 1/4 + 1/2 = 3/4. Properties for discrete probability distributions: 0 P (X = x) 1 x P (X = x) = 1 Example 2 Consider the distribution of the variable X=number of vehicles owned per family. Suppose the following table gives the distribution of the variable x P (X = x) ? 2

3 What is the value of P (X = 4), if no family owns more than 4 vehicles? P (X = 4) = 1 ( ) = = 0.08, because the total of the probabilities must be 1. What is the probability that a family has more than 2 vehicles? P (X > 2) = P (X = 3) + P (X = 4) = = The expected value or population mean µ (mu) of a rv x is the value that you would expect to observe on average if the experiment is repeated over and over again. It is the center of the distribution. Definition: Let X be a discrete rv with probability distribution P (X = x). The population mean µ or expected value of X is given as µ = E(X) = x xp (X = x). Example: The expected value of the distribution of x=the number of heads observed tossing two coins is calculated by µ = = 1 Example 3 The mean µ of vehicles owned per family is µ = = 2.14 The standard deviation of a distribution measures the spread of the distribution. It denoted by the Greek letter. Let X be a discrete rv with probability distribution P (X = x). The standard deviation of the rv X is = (x µ) 2 P (X = x) x Example (continued): The population standard deviation of x=number of heads observed tossing two coins is calculated by = (0 1) (1 1) (2 1)2 1 4 = = 1 2 = 1 2 The alternative formula for the standard deviation is usually quicker to evaluate: = x2 P (x) µ 2 Example: Donations have been collected. Every person in a population has been asked for a donation. The following table gives the distribution of the donations given. 3

4 X $0 $10 $ $50 P (X = x) Interpretation: If one person is randomly selected from the population the probability the person donated $50 is equal to The mean of this distribution is µ = x xp (X = x) = = = 9.5 This population donated in average $9.5. Calculating the standard deviation. so that 2 = x(x µ) 2 P (X = x) = (0 9.5) (10 9.5) ( 9.5) (50 9.5) = = = 2 = The standard deviation of this distribution equals $ Using the alternative formula you do: and = = = x x 2 P (X = x) µ 2 = = = Continuous Random Variables Continuous data variables are described by histograms. For histograms the measurement scale is divided in class intervals and the area of the rectangles put above those intervals is proportional to the relative frequency of the data falling into this interval. The relative frequency can be interpreted as an estimate for the probability for falling into the associated interval. With this interpretation the histogram becomes an estimate of the probability distribution of the continuous random variable. A probability distribution of a continuous rv is a smooth curve, called a density curve if and only if 1. The total area under the curve is equal to The area under the curve and above any particular interval gives the probability of observing a value of x in the corresponding interval when an experimental unit is selected at random from the population. 4

5 We can calculate that the probability for falling in the interval [ 2; 0] equals Example: The density of a uniform distribution in an interval [0; 5] looks like this: Use the density function to calculate probabilities for a random variable x with a uniform distribution on [0; 5]: P (X 3) = area under the curve from to 3 = = 0.6 P (1 X 2) = area under the curve from 1 to 2 = = 0.2 P (X > 3.5) = area under the curve from 3.5 to = = 0.3 5

6 Remark: Since there is zero area under the curve above a single value, the definition implies for continuous random variables and numbers a and b: P (X = a) = 0 P (X a) = P (X < a) P (X b) = P (X > b) P (a < X < b) = P (a X b) This is generally not true for discrete random variables. How to choose a model for a given variable in a sample? The model (density function) should resemble the histogram for the given variable. Fortunately, many continuous data variables have bell shaped histograms. The normal probability distribution provides a good model for modelling this type of data Normal Probability Distribution The density function of a normal distribution is unimodal, mound shaped, and symmetric. There are many different normal distributions, they are distinguished from one another by their population mean µ and their population standard deviation. µ is the center of the distribution, right at the highest point of the density distribution function. At the values µ and µ + the density curve has turning points. Coming from the curve turns from a left to a right curve at µ and again into in a left curve at µ +. 6

7 The function describing the density curve for a given mean µ and a given standard deviation is f(x) = 1 (x µ) 2 2π e 2 2 Example: If the normal distribution is used as a model for a specific situation, the mean and the standard deviation have to be chosen for that situation. E.g. the height of students at a certain university follow a normal distribution with µ = 178 cm and = 8 cm. Given that we know the mean and the standard deviation of a normal distribution we can locate intervals telling us where most of the values of the population are located. The Rule In the Normal Distribution with mean µ and standard deviation : Approximately 68% of the observations fall within one standard deviation of the mean, within [µ, µ + ]. Approximately 95% of the observations fall within two standard deviations of the mean, within [µ 2, µ + 2]. Approximately 99.7% of the observations fall within three standard deviations of the mean, within [µ 3, µ + 3]. Example: Continuing the example above. This rule tells us to expect about 68% of the height of students at the university to fall within [178-8, ] =[170, 186] cm. 95% of the height of students at the university to fall within [178-2(8), (8)8] =[162, 194] cm. 99.7% of the height of students at the university to fall within [178-3(8), (8)] =[154, 2] cm. 7

8 Definition: The normal distribution with µ = 0 and = 1 is called the Standard Normal Distribution. In order to work with the normal distribution, we need to be able to calculate the following: 1. We must be able to use the normal distribution to compute probabilities, which are areas under the normal curve. 2. We must be able to describe extreme values in the distribution, such as the largest 5%, the smallest 1%, the most extreme 10% (which would include the largest 5% and the smallest 5%), that is we have to be able to calculate percentiles of any normal distribution. We first look how to compute these for a Standard Normal Distribution. Since the normal distribution is a continuous distribution the following holds for every normal distributed random variable X: P (X < z) = P (X z)= area under the curve from to z. The area under the curve of a normal distributed random variable is hard to calculate. There is no simple formula that can be used to calculate the area. Appendix Table A (in the text book) tabulates for standard normal distributed random variables for many different values of z the area under the curve from to z. These are values from the so called cumulative density function. From now on use Z to indicate a standard normal distributed random variable (µ = 0 and = 1). Using the table you find that, P (Z < 1.75) = P (z 1.75) = and 8

9 P (Z > 1.34) = 1 P (z 1.34) = = and Shaded area equals P ( 1 Z 1) = P (Z 1) P (Z 1) = = The shaded area equals

10 The first probability can be interpreted as meaning that, in a long sequence of observations from a Standard Normal distribution, about 4.01% of the observed values will be smaller than Try this for different values! Now we will look how to identify extreme values. Definition: For any particular number r between 0 and 1, the r th percentile x r of a distribution is a value such that the cumulative area from to x r is equal to r. If X is a random variable the r th percentile x r satisfies the following equation: P (X x r ) = r To determine the percentiles for a standard normal distribution (denote them by z r ), we can use Table A again. Suppose we want to describe the values that make up the smallest 2%. So we are looking for the 0.02 th percentile z 0.02, with P (Z z 0.02 ) = So look in the body of the Table A for the cumulative area The closest you will find 0.02 for z r = 2.05 This is the best approximation you can find from the table. The result is that the smallest 2% of the values of a standard normal distribution fall within the interval (, 2.05]. 10

11 Suppose now we are interested in the largest 5%. So we are looking for z, with P (Z > z ) = 0.05 In Table A we can only find areas to the left of a given value, the first step is to determine the area to the left of z : P (z z ) = = 0.95 That tells us that in fact z = z 0.95 the 0.95 th percentile. Checking the table we find values and , with 0.95 exactly in the middle, so we take the average of the corresponding numbers and get z 0.95 = = And now we are interested in the most extreme 5%. That means we are interested in the middle 95%. Since the normal distribution is the symmetric the most extreme 5 % can be split up in the lower 2.5% and the upper 2.5%. Symmetry about 0 implies that z = z In Table A we find z = 1.96, so that z = 1.96 We found the result, that the 5% most extreme values are outside the interval [ 1.96, 1.96]. Now remains the step to determine those areas for any normal distribution using the results of the standard normal distribution. Lemma: Is x normal distributed with population mean µ and population standard deviation then the standardized random variable Z = X µ is normal distributed with µ = 0 and = 1. 11

12 Example: Let X be normal distributed with µ = 100 and = Calculate the area under the curve between 98 and 107 for the distribution chosen above. P (98 < X < 107) = P ( < X 100 < ) = P ( 2 5 < Z < 7 5 ) = P ( 0.4 < Z < 1.4) This can be calculated using Table IV. P ( 0.4 < z < 1.4) = P (z < 1.4) P (z < 0.4) = = The first step you have to take is to standardize the rv, so that the result is standard normal distributed. The Lemma above tells you how it is done, subtract the mean µ and divide by the standard deviation. In a second step you use the table for the standard normal distribution to find the probability. 2. To find the 0.3 th percentile for this distribution, that is x 0.3, use 0.3 = P (X x 0.3 ) = P ( X 100 = P (Z x x ) 5 5 ) 5 But then x equals the 0.3 th percentile from a standard normal distribution, which 5 we can find in Table A. x = This is equivalent to x 0.3 = = = So that the lower 30% of a normal distributed random variable with mean µ = 100 and = 5 fall into the interval (, 90.6]. Again, in a first step standardize the rv, so that the result is standard normal distributed. This is done by subtracting the mean µ and dividing by the standard deviation. In a second step use the table for the standard normal distribution to find the percentile. 12

13 Examples for Calculating Probabilities of a Standard Normal distribution Assumption: The random variable Z is standard normal distributed, that is population mean µ = 0 and standard deviation = Calculate P (Z < 0.53): In order to find the probability use Table A: (a) Find 0.5 in the left hand side column, this determines the row (b) then find 0.03 in the top row, this determines the column (c) now check for the value where the row and the column intersect In this example Result: P (Z < 0.53) = Calculate P (Z > 0.79): Rule for Compliments: P (z > 0.79) = 1 P (z 0.79). Now use Table A again: (a) Find -.7 in the left hand side column, this determines the row (b) then find.09 in the top row, this determines the column (c) now check for the value where the row and the column intersect In this example Result: P (Z > 0.79) = 1 P (Z 0.79) = = Calculate P (2.1 < Z < 4.79): It is P (2.1 < Z < 4.79) = P (Z 4.79) P (Z < 2.1). Use Table A: (a) You find that 4.79 is larger than the largest value in the Table. That means that P (z < 4.79) = 1. (b) Find 2.1 in the left hand side column, this determines the row (c) then find.00 in the top row, this determines the column (d) now check for the value where the row and the column intersect In this example Result: P (2.1 < Z < 4.79) = P (Z 4.79) P (Z < 2.1) = = Examples for Calculating Probabilities of a Normal distribution (not necessarily standard) Assumption: The random variable x is normal distributed with mean µ = 80 and standard deviation =

14 1. Calculate P (X < 100): First standardize: P (X < 100) = P ( X µ < 100 µ ) = P (Z < ) = P (Z < 2) 10 Now find this from Table A applying the method from above and find P (Z < 100) = Calculate P (X > 79): Rule for Compliments: P (X > 79) = 1 P (X 79). First standardize: P (X < 79) = P ( X µ Now use Table A again: < 79 µ ) = P (Z < ) = P (Z < 0.1) 10 (a) Find -.1 in the left hand side column, this determines the row (b) then find.090 in the top row, this determines the column (c) now check for the value where the row and the column intersect In this example Result: P (X > 79) = 1 P (X 79) = = Calculate P (70 < X < 80): It is P (70 < X < 80) = P (X 80) P (X < 70). First standardize: P (X 80) P (X < 70) = P ( X µ < 80 µ ) P ( X µ P (Z < 0) P (Z < 1) = = < 70 µ ) = Examples for Calculating Percentiles of a Standard Normal distribution Assumption: The random variable z is standard normal distributed, that is population mean µ = 0 and standard deviation = Calculate the 0.9th percentile z.9 : P (Z < z.9 ) = 0.9: In order to find the percentile use Table A: (a) Find 0.9 in the body of the table, the closest you find is (b) go to the left and find in the left column 1.2 (c) got to the top and find in the top row 0.08 Result: The 0.9th percentile equals z.9 =

15 2. Find the interval which contains the middle 50%: The middle 50% of the standard normal distribution can be found between the 0.25th percentile and 0.75 percentile. Now use Table A again: (a) Find 0.25 in the body of the table, the closest you find is (b) go to the left and find in the left column -0.6 (c) got to the top and find in the top row 0.07 (a) Find 0.75 in the body of the table, the closest you find is (b) go to the left and find in the left column 0.6 (c) got to the top and find in the top row 0.07 Result: The middle 50% of a standard normal distribution can be found between and Examples for Calculating Percentiles of ANY Normal distribution Assumption: The random variable X is normal distributed, with mean µ = 50 and =. 1. Calculate the 0.1th percentile x.9 : P (X < x.1 ) = 0.1: First standardize: P (X < x.1 ) = P ( X µ < x 0.1 µ ) = P (Z < x ) = 0.1 For this equation to hold x has to be the 0.1th percentile of the standard normal distribution. So z 0.1 = x x 0.1 = 50 + z 0.1 In order to find the percentile z 0.1 use Table A: (a) Find 0.1 in the body of the table, the closest you find is (b) go to the left and find in the left column -1.2 (c) got to the top and find in the top row 0.08 Result: The 0.1th percentile equals z.1 = 1.28 so that x 0.1 = 50 + ( 1.28) = Find the interval which contains the middle 50% of this distribution: The middle 50% of the normal distribution can be found between the 0.25th percentile and 0.75 percentile. First standardize: and P (X < x.25 ) = P ( X µ P (X < x.75 ) = P ( X µ < x 0.25 µ < x 0.75 µ 15 ) = P (Z < x ) = 0.25 ) = P (Z < x ) = 0.75

16 For these equations to hold x has to be the 0.25th percentile of the standard normal distribution and x has to be the 0.75th percentile of the standard normal distribution. So and z 0.25 = x z 0.75 = x x 0.25 = 50 + z 0.25 = 50 + ( 0.67) = 36.6 x 0.75 = 50 + z 0.75 = 50 + (0.67) = 63.4 Result: The middle 50% of this normal distribution can be found between 36.6 and

What Is a Sampling Distribution? DISTINGUISH between a parameter and a statistic

What Is a Sampling Distribution? DISTINGUISH between a parameter and a statistic Section 8.1A What Is a Sampling Distribution? Learning Objectives After this section, you should be able to DISTINGUISH between a parameter and a statistic DEFINE sampling distribution DISTINGUISH between

More information

Chapter 5. Understanding and Comparing. Distributions

Chapter 5. Understanding and Comparing. Distributions STAT 141 Introduction to Statistics Chapter 5 Understanding and Comparing Distributions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 27 Boxplots How to create a boxplot? Assume

More information

Normal Distribution: Calculations of Probabilities

Normal Distribution: Calculations of Probabilities OpenStax-CNX module: m46212 1 Normal Distribution: Calculations of Probabilities Irene Mary Duranczyk Suzanne Loch Janet Stottlemyer Based on Normal Distribution: Calculations of Probabilities by Susan

More information

Example A. Define X = number of heads in ten tosses of a coin. What are the values that X may assume?

Example A. Define X = number of heads in ten tosses of a coin. What are the values that X may assume? Stat 400, section.1-.2 Random Variables & Probability Distributions notes by Tim Pilachowski For a given situation, or experiment, observations are made and data is recorded. A sample space S must contain

More information

Lecture 10: The Normal Distribution. So far all the random variables have been discrete.

Lecture 10: The Normal Distribution. So far all the random variables have been discrete. Lecture 10: The Normal Distribution 1. Continuous Random Variables So far all the random variables have been discrete. We need a different type of model (called a probability density function) for continuous

More information

Business Statistics: A Decision-Making Approach, 6e. Chapter Goals

Business Statistics: A Decision-Making Approach, 6e. Chapter Goals Chapter 4 Student Lecture Notes 4-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 4 Using Probability and Probability Distributions Fundamentals of Business Statistics Murali Shanker

More information

Random variables (section 6.1)

Random variables (section 6.1) Random variables (section 6.1) random variable: a number attached to the outcome of a random process (r.v is usually denoted with upper case letter, such as \) discrete random variables discrete random

More information

To find the median, find the 40 th quartile and the 70 th quartile (which are easily found at y=1 and y=2, respectively). Then we interpolate:

To find the median, find the 40 th quartile and the 70 th quartile (which are easily found at y=1 and y=2, respectively). Then we interpolate: Joel Anderson ST 37-002 Lecture Summary for 2/5/20 Homework 0 First, the definition of a probability mass function p(x) and a cumulative distribution function F(x) is reviewed: Graphically, the drawings

More information

STAT 200 Chapter 1 Looking at Data - Distributions

STAT 200 Chapter 1 Looking at Data - Distributions STAT 200 Chapter 1 Looking at Data - Distributions What is Statistics? Statistics is a science that involves the design of studies, data collection, summarizing and analyzing the data, interpreting the

More information

Discrete and continuous

Discrete and continuous Discrete and continuous A curve, or a function, or a range of values of a variable, is discrete if it has gaps in it - it jumps from one value to another. In practice in S2 discrete variables are variables

More information

Algebra 2 Practice Midterm

Algebra 2 Practice Midterm Name: Algebra 2 Practice Midterm Circle the letter for the correct answer. 1. A study comparing patients who received a new medicine with those who did not is. A. an observational study C. an experiment

More information

Section 5.4. Ken Ueda

Section 5.4. Ken Ueda Section 5.4 Ken Ueda Students seem to think that being graded on a curve is a positive thing. I took lasers 101 at Cornell and got a 92 on the exam. The average was a 93. I ended up with a C on the test.

More information

Section 7.1 Properties of the Normal Distribution

Section 7.1 Properties of the Normal Distribution Section 7.1 Properties of the Normal Distribution In Chapter 6, talked about probability distributions. Coin flip problem: Difference of two spinners: The random variable x can only take on certain discrete

More information

Random variables, Expectation, Mean and Variance. Slides are adapted from STAT414 course at PennState

Random variables, Expectation, Mean and Variance. Slides are adapted from STAT414 course at PennState Random variables, Expectation, Mean and Variance Slides are adapted from STAT414 course at PennState https://onlinecourses.science.psu.edu/stat414/ Random variable Definition. Given a random experiment

More information

Unit 4 Probability. Dr Mahmoud Alhussami

Unit 4 Probability. Dr Mahmoud Alhussami Unit 4 Probability Dr Mahmoud Alhussami Probability Probability theory developed from the study of games of chance like dice and cards. A process like flipping a coin, rolling a die or drawing a card from

More information

AP Final Review II Exploring Data (20% 30%)

AP Final Review II Exploring Data (20% 30%) AP Final Review II Exploring Data (20% 30%) Quantitative vs Categorical Variables Quantitative variables are numerical values for which arithmetic operations such as means make sense. It is usually a measure

More information

Probability Distributions

Probability Distributions CONDENSED LESSON 13.1 Probability Distributions In this lesson, you Sketch the graph of the probability distribution for a continuous random variable Find probabilities by finding or approximating areas

More information

MATH 1150 Chapter 2 Notation and Terminology

MATH 1150 Chapter 2 Notation and Terminology MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the

More information

Chapter 18 Sampling Distribution Models

Chapter 18 Sampling Distribution Models Chapter 18 Sampling Distribution Models The histogram above is a simulation of what we'd get if we could see all the proportions from all possible samples. The distribution has a special name. It's called

More information

CHAPTER 5: EXPLORING DATA DISTRIBUTIONS. Individuals are the objects described by a set of data. These individuals may be people, animals or things.

CHAPTER 5: EXPLORING DATA DISTRIBUTIONS. Individuals are the objects described by a set of data. These individuals may be people, animals or things. (c) Epstein 2013 Chapter 5: Exploring Data Distributions Page 1 CHAPTER 5: EXPLORING DATA DISTRIBUTIONS 5.1 Creating Histograms Individuals are the objects described by a set of data. These individuals

More information

Biostatistics in Dentistry

Biostatistics in Dentistry Biostatistics in Dentistry Continuous probability distributions Continuous probability distributions Continuous data are data that can take on an infinite number of values between any two points. Examples

More information

1 Normal Distribution.

1 Normal Distribution. Normal Distribution.. Introduction A Bernoulli trial is simple random experiment that ends in success or failure. A Bernoulli trial can be used to make a new random experiment by repeating the Bernoulli

More information

STAT/SOC/CSSS 221 Statistical Concepts and Methods for the Social Sciences. Random Variables

STAT/SOC/CSSS 221 Statistical Concepts and Methods for the Social Sciences. Random Variables STAT/SOC/CSSS 221 Statistical Concepts and Methods for the Social Sciences Random Variables Christopher Adolph Department of Political Science and Center for Statistics and the Social Sciences University

More information

4/1/2012. Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20. Skipping Topics 11 and 15. Topic 12. Normal Distribution

4/1/2012. Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20. Skipping Topics 11 and 15. Topic 12. Normal Distribution Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20 Skipping Topics 11 and 15 Topic 12 Normal Distribution 1 Normal Distribution If Density Curve is symmetric, single peaked, bell-shaped then it is

More information

Econ 371 Problem Set #1 Answer Sheet

Econ 371 Problem Set #1 Answer Sheet Econ 371 Problem Set #1 Answer Sheet 2.1 In this question, you are asked to consider the random variable Y, which denotes the number of heads that occur when two coins are tossed. a. The first part of

More information

Math Review Sheet, Fall 2008

Math Review Sheet, Fall 2008 1 Descriptive Statistics Math 3070-5 Review Sheet, Fall 2008 First we need to know about the relationship among Population Samples Objects The distribution of the population can be given in one of the

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES VARIABLE Studying the behavior of random variables, and more importantly functions of random variables is essential for both the

More information

Chapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67

Chapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67 Chapter 6 The Standard Deviation as a Ruler and the Normal Model 1 /67 Homework Read Chpt 6 Complete Reading Notes Do P129 1, 3, 5, 7, 15, 17, 23, 27, 29, 31, 37, 39, 43 2 /67 Objective Students calculate

More information

OPIM 303, Managerial Statistics H Guy Williams, 2006

OPIM 303, Managerial Statistics H Guy Williams, 2006 OPIM 303 Lecture 6 Page 1 The height of the uniform distribution is given by 1 b a Being a Continuous distribution the probability of an exact event is zero: 2 0 There is an infinite number of points in

More information

are the objects described by a set of data. They may be people, animals or things.

are the objects described by a set of data. They may be people, animals or things. ( c ) E p s t e i n, C a r t e r a n d B o l l i n g e r 2016 C h a p t e r 5 : E x p l o r i n g D a t a : D i s t r i b u t i o n s P a g e 1 CHAPTER 5: EXPLORING DATA DISTRIBUTIONS 5.1 Creating Histograms

More information

Lecture 1: Description of Data. Readings: Sections 1.2,

Lecture 1: Description of Data. Readings: Sections 1.2, Lecture 1: Description of Data Readings: Sections 1.,.1-.3 1 Variable Example 1 a. Write two complete and grammatically correct sentences, explaining your primary reason for taking this course and then

More information

Essential Statistics Chapter 6

Essential Statistics Chapter 6 1 Essential Statistics Chapter 6 By Navidi and Monk Copyright 2016 Mark A. Thomas. All rights reserved. 2 Continuous Probability Distributions chapter 5 focused upon discrete probability distributions,

More information

Chapter 2 Solutions Page 15 of 28

Chapter 2 Solutions Page 15 of 28 Chapter Solutions Page 15 of 8.50 a. The median is 55. The mean is about 105. b. The median is a more representative average" than the median here. Notice in the stem-and-leaf plot on p.3 of the text that

More information

Chapter 5: Exploring Data: Distributions Lesson Plan

Chapter 5: Exploring Data: Distributions Lesson Plan Lesson Plan Exploring Data Displaying Distributions: Histograms Interpreting Histograms Displaying Distributions: Stemplots Describing Center: Mean and Median Describing Variability: The Quartiles The

More information

Discrete Probability distribution Discrete Probability distribution

Discrete Probability distribution Discrete Probability distribution 438//9.4.. Discrete Probability distribution.4.. Binomial P.D. The outcomes belong to either of two relevant categories. A binomial experiment requirements: o There is a fixed number of trials (n). o On

More information

Statistics 1. Edexcel Notes S1. Mathematical Model. A mathematical model is a simplification of a real world problem.

Statistics 1. Edexcel Notes S1. Mathematical Model. A mathematical model is a simplification of a real world problem. Statistics 1 Mathematical Model A mathematical model is a simplification of a real world problem. 1. A real world problem is observed. 2. A mathematical model is thought up. 3. The model is used to make

More information

Describing distributions with numbers

Describing distributions with numbers Describing distributions with numbers A large number or numerical methods are available for describing quantitative data sets. Most of these methods measure one of two data characteristics: The central

More information

Density Curves & Normal Distributions

Density Curves & Normal Distributions Density Curves & Normal Distributions Sections 4.1 & 4.2 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu

More information

Sampling Distribution Models. Chapter 17

Sampling Distribution Models. Chapter 17 Sampling Distribution Models Chapter 17 Objectives: 1. Sampling Distribution Model 2. Sampling Variability (sampling error) 3. Sampling Distribution Model for a Proportion 4. Central Limit Theorem 5. Sampling

More information

STAT 430/510 Probability Lecture 7: Random Variable and Expectation

STAT 430/510 Probability Lecture 7: Random Variable and Expectation STAT 430/510 Probability Lecture 7: Random Variable and Expectation Pengyuan (Penelope) Wang June 2, 2011 Review Properties of Probability Conditional Probability The Law of Total Probability Bayes Formula

More information

Probability Distribution for a normal random variable x:

Probability Distribution for a normal random variable x: Chapter5 Continuous Random Variables 5.3 The Normal Distribution Probability Distribution for a normal random variable x: 1. It is and about its mean µ. 2. (the that x falls in the interval a < x < b is

More information

Introduction to Probability and Statistics Twelfth Edition

Introduction to Probability and Statistics Twelfth Edition Introduction to Probability and Statistics Twelfth Edition Robert J. Beaver Barbara M. Beaver William Mendenhall Presentation designed and written by: Barbara M. Beaver Introduction to Probability and

More information

LC OL - Statistics. Types of Data

LC OL - Statistics. Types of Data LC OL - Statistics Types of Data Question 1 Characterise each of the following variables as numerical or categorical. In each case, list any three possible values for the variable. (i) Eye colours in a

More information

University of Jordan Fall 2009/2010 Department of Mathematics

University of Jordan Fall 2009/2010 Department of Mathematics handouts Part 1 (Chapter 1 - Chapter 5) University of Jordan Fall 009/010 Department of Mathematics Chapter 1 Introduction to Introduction; Some Basic Concepts Statistics is a science related to making

More information

Elementary Statistics

Elementary Statistics Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q:

More information

MA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.

MA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table. MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at

More information

Chapter 7: Sampling Distributions

Chapter 7: Sampling Distributions Chapter 7: Sampling Distributions Section 7.1 What is a Sampling Distribution? The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 7 Sampling Distributions 7.1 What is a Sampling

More information

FREQUENCY DISTRIBUTIONS AND PERCENTILES

FREQUENCY DISTRIBUTIONS AND PERCENTILES FREQUENCY DISTRIBUTIONS AND PERCENTILES New Statistical Notation Frequency (f): the number of times a score occurs N: sample size Simple Frequency Distributions Raw Scores The scores that we have directly

More information

3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability

3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values

More information

a table or a graph or an equation.

a table or a graph or an equation. Topic (8) POPULATION DISTRIBUTIONS 8-1 So far: Topic (8) POPULATION DISTRIBUTIONS We ve seen some ways to summarize a set of data, including numerical summaries. We ve heard a little about how to sample

More information

Chapter 4. Displaying and Summarizing. Quantitative Data

Chapter 4. Displaying and Summarizing. Quantitative Data STAT 141 Introduction to Statistics Chapter 4 Displaying and Summarizing Quantitative Data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 31 4.1 Histograms 1 We divide the range

More information

6 THE NORMAL DISTRIBUTION

6 THE NORMAL DISTRIBUTION CHAPTER 6 THE NORMAL DISTRIBUTION 341 6 THE NORMAL DISTRIBUTION Figure 6.1 If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described

More information

Counting principles, including permutations and combinations.

Counting principles, including permutations and combinations. 1 Counting principles, including permutations and combinations. The binomial theorem: expansion of a + b n, n ε N. THE PRODUCT RULE If there are m different ways of performing an operation and for each

More information

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation

More information

STAT Section 2.1: Basic Inference. Basic Definitions

STAT Section 2.1: Basic Inference. Basic Definitions STAT 518 --- Section 2.1: Basic Inference Basic Definitions Population: The collection of all the individuals of interest. This collection may be or even. Sample: A collection of elements of the population.

More information

Sampling, Frequency Distributions, and Graphs (12.1)

Sampling, Frequency Distributions, and Graphs (12.1) 1 Sampling, Frequency Distributions, and Graphs (1.1) Design: Plan how to obtain the data. What are typical Statistical Methods? Collect the data, which is then subjected to statistical analysis, which

More information

MATH 2560 C F03 Elementary Statistics I Lecture 1: Displaying Distributions with Graphs. Outline.

MATH 2560 C F03 Elementary Statistics I Lecture 1: Displaying Distributions with Graphs. Outline. MATH 2560 C F03 Elementary Statistics I Lecture 1: Displaying Distributions with Graphs. Outline. data; variables: categorical & quantitative; distributions; bar graphs & pie charts: What Is Statistics?

More information

Properties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area

Properties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area Properties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. The curve is called the probability

More information

Business Statistics. Lecture 3: Random Variables and the Normal Distribution

Business Statistics. Lecture 3: Random Variables and the Normal Distribution Business Statistics Lecture 3: Random Variables and the Normal Distribution 1 Goals for this Lecture A little bit of probability Random variables The normal distribution 2 Probability vs. Statistics Probability:

More information

EQ: What is a normal distribution?

EQ: What is a normal distribution? Unit 5 - Statistics What is the purpose EQ: What tools do we have to assess data? this unit? What vocab will I need? Vocabulary: normal distribution, standard, nonstandard, interquartile range, population

More information

AP Statistics Cumulative AP Exam Study Guide

AP Statistics Cumulative AP Exam Study Guide AP Statistics Cumulative AP Eam Study Guide Chapters & 3 - Graphs Statistics the science of collecting, analyzing, and drawing conclusions from data. Descriptive methods of organizing and summarizing statistics

More information

Chapter 1: Introduction. Material from Devore s book (Ed 8), and Cengagebrain.com

Chapter 1: Introduction. Material from Devore s book (Ed 8), and Cengagebrain.com 1 Chapter 1: Introduction Material from Devore s book (Ed 8), and Cengagebrain.com Populations and Samples An investigation of some characteristic of a population of interest. Example: Say you want to

More information

Algebra 2. Outliers. Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)

Algebra 2. Outliers. Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves) Algebra 2 Outliers Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves) Algebra 2 Notes #1 Chp 12 Outliers In a set of numbers, sometimes there will be

More information

What is a parameter? What is a statistic? How is one related to the other?

What is a parameter? What is a statistic? How is one related to the other? Chapter Seven: SAMPLING DISTRIBUTIONS 7.1 Sampling Distributions Read 424 425 What is a parameter? What is a statistic? How is one related to the other? Example: Identify the population, the parameter,

More information

9. DISCRETE PROBABILITY DISTRIBUTIONS

9. DISCRETE PROBABILITY DISTRIBUTIONS 9. DISCRETE PROBABILITY DISTRIBUTIONS Random Variable: A quantity that takes on different values depending on chance. Eg: Next quarter s sales of Coca Cola. The proportion of Super Bowl viewers surveyed

More information

Chapter 18. Sampling Distribution Models. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 18. Sampling Distribution Models. Copyright 2010, 2007, 2004 Pearson Education, Inc. Chapter 18 Sampling Distribution Models Copyright 2010, 2007, 2004 Pearson Education, Inc. Normal Model When we talk about one data value and the Normal model we used the notation: N(μ, σ) Copyright 2010,

More information

Continuous Probability Distributions

Continuous Probability Distributions 1 Chapter 5 Continuous Probability Distributions 5.1 Probability density function Example 5.1.1. Revisit Example 3.1.1. 11 12 13 14 15 16 21 22 23 24 25 26 S = 31 32 33 34 35 36 41 42 43 44 45 46 (5.1.1)

More information

STT 315 Problem Set #3

STT 315 Problem Set #3 1. A student is asked to calculate the probability that x = 3.5 when x is chosen from a normal distribution with the following parameters: mean=3, sd=5. To calculate the answer, he uses this command: >

More information

Math Bootcamp 2012 Miscellaneous

Math Bootcamp 2012 Miscellaneous Math Bootcamp 202 Miscellaneous Factorial, combination and permutation The factorial of a positive integer n denoted by n!, is the product of all positive integers less than or equal to n. Define 0! =.

More information

Density curves and the normal distribution

Density curves and the normal distribution Density curves and the normal distribution - Imagine what would happen if we measured a variable X repeatedly and make a histogram of the values. What shape would emerge? A mathematical model of this shape

More information

Essentials of Statistics and Probability

Essentials of Statistics and Probability May 22, 2007 Department of Statistics, NC State University dbsharma@ncsu.edu SAMSI Undergrad Workshop Overview Practical Statistical Thinking Introduction Data and Distributions Variables and Distributions

More information

You try: What is the equation of the line on the graph below? What is the equation of the line on the graph below?

You try: What is the equation of the line on the graph below? What is the equation of the line on the graph below? 1 What is the equation of the line on the graph below? 2 3 1a What is the equation of the line on the graph below? y-intercept Solution: To write an equation in slope-intercept form, identify the slope

More information

Sets and Set notation. Algebra 2 Unit 8 Notes

Sets and Set notation. Algebra 2 Unit 8 Notes Sets and Set notation Section 11-2 Probability Experimental Probability experimental probability of an event: Theoretical Probability number of time the event occurs P(event) = number of trials Sample

More information

(It's not always good, but we can always make it.) (4) Convert the normal distribution N to the standard normal distribution Z. Specically.

(It's not always good, but we can always make it.) (4) Convert the normal distribution N to the standard normal distribution Z. Specically. . Introduction The quick summary, going forwards: Start with random variable X. 2 Compute the mean EX and variance 2 = varx. 3 Approximate X by the normal distribution N with mean µ = EX and standard deviation.

More information

Lecture 1: Descriptive Statistics

Lecture 1: Descriptive Statistics Lecture 1: Descriptive Statistics MSU-STT-351-Sum 15 (P. Vellaisamy: MSU-STT-351-Sum 15) Probability & Statistics for Engineers 1 / 56 Contents 1 Introduction 2 Branches of Statistics Descriptive Statistics

More information

A constant is a value that is always the same. (This means that the value is constant / unchanging). o

A constant is a value that is always the same. (This means that the value is constant / unchanging). o Math 8 Unit 7 Algebra and Graphing Relations Solving Equations Using Models We will be using algebra tiles to help us solve equations. We will practice showing work appropriately symbolically and pictorially

More information

The Normal Distribution (Pt. 2)

The Normal Distribution (Pt. 2) Chapter 5 The Normal Distribution (Pt 2) 51 Finding Normal Percentiles Recall that the Nth percentile of a distribution is the value that marks off the bottom N% of the distribution For review, remember

More information

Final Exam STAT On a Pareto chart, the frequency should be represented on the A) X-axis B) regression C) Y-axis D) none of the above

Final Exam STAT On a Pareto chart, the frequency should be represented on the A) X-axis B) regression C) Y-axis D) none of the above King Abdul Aziz University Faculty of Sciences Statistics Department Final Exam STAT 0 First Term 49-430 A 40 Name No ID: Section: You have 40 questions in 9 pages. You have 90 minutes to solve the exam.

More information

Index I-1. in one variable, solution set of, 474 solving by factoring, 473 cubic function definition, 394 graphs of, 394 x-intercepts on, 474

Index I-1. in one variable, solution set of, 474 solving by factoring, 473 cubic function definition, 394 graphs of, 394 x-intercepts on, 474 Index A Absolute value explanation of, 40, 81 82 of slope of lines, 453 addition applications involving, 43 associative law for, 506 508, 570 commutative law for, 238, 505 509, 570 English phrases for,

More information

Continuous Probability Distributions

Continuous Probability Distributions 1 Chapter 5 Continuous Probability Distributions 5.1 Probability density function Example 5.1.1. Revisit Example 3.1.1. 11 12 13 14 15 16 21 22 23 24 25 26 S = 31 32 33 34 35 36 41 42 43 44 45 46 (5.1.1)

More information

1/18/2011. Chapter 6: Probability. Introduction to Probability. Probability Definition

1/18/2011. Chapter 6: Probability. Introduction to Probability. Probability Definition Chapter 6: Probability Introduction to Probability The role of inferential statistics is to use the sample data as the basis for answering questions about the population. To accomplish this goal, inferential

More information

IV. The Normal Distribution

IV. The Normal Distribution IV. The Normal Distribution The normal distribution (a.k.a., the Gaussian distribution or bell curve ) is the by far the best known random distribution. It s discovery has had such a far-reaching impact

More information

AP STATISTICS. 7.3 Probability Distributions for Continuous Random Variables

AP STATISTICS. 7.3 Probability Distributions for Continuous Random Variables AP STATISTICS 7.3 Probability Distributions for Continuous Random Variables 7.3 Objectives: Ø Understand the definition and properties of continuous random variables Ø Be able to represent the probability

More information

Ch. 7: Estimates and Sample Sizes

Ch. 7: Estimates and Sample Sizes Ch. 7: Estimates and Sample Sizes Section Title Notes Pages Introduction to the Chapter 2 2 Estimating p in the Binomial Distribution 2 5 3 Estimating a Population Mean: Sigma Known 6 9 4 Estimating a

More information

Raquel Prado. Name: Department of Applied Mathematics and Statistics AMS-131. Spring 2010

Raquel Prado. Name: Department of Applied Mathematics and Statistics AMS-131. Spring 2010 Raquel Prado Name: Department of Applied Mathematics and Statistics AMS-131. Spring 2010 Final Exam (Type B) The midterm is closed-book, you are only allowed to use one page of notes and a calculator.

More information

Chapter 1. Looking at Data

Chapter 1. Looking at Data Chapter 1 Looking at Data Types of variables Looking at Data Be sure that each variable really does measure what you want it to. A poor choice of variables can lead to misleading conclusions!! For example,

More information

Math 140 Introductory Statistics

Math 140 Introductory Statistics Math 140 Introductory Statistics Professor Silvia Fernández Chapter 2 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Visualizing Distributions Recall the definition: The

More information

Math 140 Introductory Statistics

Math 140 Introductory Statistics Visualizing Distributions Math 140 Introductory Statistics Professor Silvia Fernández Chapter Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Recall the definition: The

More information

8.1 Graphing Data. Series1. Consumer Guide Dealership Word of Mouth Internet. Consumer Guide Dealership Word of Mouth Internet

8.1 Graphing Data. Series1. Consumer Guide Dealership Word of Mouth Internet. Consumer Guide Dealership Word of Mouth Internet 8.1 Graphing Data In this chapter, we will study techniques for graphing data. We will see the importance of visually displaying large sets of data so that meaningful interpretations of the data can be

More information

4.2 The Normal Distribution. that is, a graph of the measurement looks like the familiar symmetrical, bell-shaped

4.2 The Normal Distribution. that is, a graph of the measurement looks like the familiar symmetrical, bell-shaped 4.2 The Normal Distribution Many physiological and psychological measurements are normality distributed; that is, a graph of the measurement looks like the familiar symmetrical, bell-shaped distribution

More information

Chapter 3: The Normal Distributions

Chapter 3: The Normal Distributions Chapter 3: The Normal Distributions http://www.yorku.ca/nuri/econ2500/econ2500-online-course-materials.pdf graphs-normal.doc / histogram-density.txt / normal dist table / ch3-image Ch3 exercises: 3.2,

More information

Binomial and Poisson Probability Distributions

Binomial and Poisson Probability Distributions Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What

More information

Chapter 2 Linear Equations and Inequalities in One Variable

Chapter 2 Linear Equations and Inequalities in One Variable Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound

More information

There are two basic kinds of random variables continuous and discrete.

There are two basic kinds of random variables continuous and discrete. Summary of Lectures 5 and 6 Random Variables The random variable is usually represented by an upper case letter, say X. A measured value of the random variable is denoted by the corresponding lower case

More information

Math 2311 Sections 4.1, 4.2 and 4.3

Math 2311 Sections 4.1, 4.2 and 4.3 Math 2311 Sections 4.1, 4.2 and 4.3 4.1 - Density Curves What do we know about density curves? Example: Suppose we have a density curve defined for defined by the line y = x. Sketch: What percent of observations

More information

Bemidji Area Schools Outcomes in Mathematics Algebra 2 Applications. Based on Minnesota Academic Standards in Mathematics (2007) Page 1 of 7

Bemidji Area Schools Outcomes in Mathematics Algebra 2 Applications. Based on Minnesota Academic Standards in Mathematics (2007) Page 1 of 7 9.2.1.1 Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain. For example: If f x 1, find f(-4). x2 3 Understand the concept of function,

More information

3.2 Probability Rules

3.2 Probability Rules 3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need

More information

ACMS Statistics for Life Sciences. Chapter 13: Sampling Distributions

ACMS Statistics for Life Sciences. Chapter 13: Sampling Distributions ACMS 20340 Statistics for Life Sciences Chapter 13: Sampling Distributions Sampling We use information from a sample to infer something about a population. When using random samples and randomized experiments,

More information

Using the z-table: Given an Area, Find z ID1050 Quantitative & Qualitative Reasoning

Using the z-table: Given an Area, Find z ID1050 Quantitative & Qualitative Reasoning Using the -Table: Given an, Find ID1050 Quantitative & Qualitative Reasoning between mean and beyond 0.0 0.000 0.500 0.1 0.040 0.460 0.2 0.079 0.421 0.3 0.118 0.382 0.4 0.155 0.345 0.5 0.192 0.309 0.6

More information

STATISTICS 1 REVISION NOTES

STATISTICS 1 REVISION NOTES STATISTICS 1 REVISION NOTES Statistical Model Representing and summarising Sample Data Key words: Quantitative Data This is data in NUMERICAL FORM such as shoe size, height etc. Qualitative Data This is

More information