The Standard Normal distribution

Save this PDF as:

Size: px
Start display at page:

Transcription

1 The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance on deviations from the mean. For example if we produce piston rings which have a target mean inside diameter of 45mm then realistically we expect the diameter to deviate slightly from this value. The deviations from the mean value are often modelled very well by the Normal distribution. Suppose we decide that diameters in the range 44.95mm to 45.05mm are acceptable, then what proportion of the output is satisfactory? In this Block we shall see how to use the normal distribution to answer questions like this. Prerequisites Before starting this Block you should... Learning Outcomes After completing this Block you should be able to... recognise the shape of the frequency curve for the standard normal distribution calculate probabilities using the standard normal distribution recognise key areas under the frequency curve 1 be familiar with the basic properties of probability 2 be familiar with continuous random variables Learning Style To achieve what is expected of you... allocate sufficient study time briefly revise the prerequisite material attempt every guided exercise and most of the other exercises

2 1. A Key Transformation A normal distribution is, perhaps, the most important example of a continuous random variable. The probability density function of a normal distribution is y = 1 σ 2π e (a µ) 2 2σ 2 This curve is always bell-shaped with the centre of the bell located at the value of µ. The depth of the bell is controlled by the value of σ. As with all normal distribution curves it is symmetrical about the centre and decays exponentially as x ±. As with any probability density function the area under the curve is equal to 1. See Figure 1. y y = 1 σ (x µ) 2 2π e 2σ 2 µ x Figure 1 A normal distribution is completely defined by specifying its mean (the value of µ) and its variance (the value of σ 2 ). The normal distribution with mean µ and variance σ 2 is written N(µ, σ 2 ). Hence the disribution N(20, 25) has a mean of 20 and a standard deviation of 5; remember that the second coordinate is the variance and the variance is the square of the standard deviation. The standard normal distribution curve. At this stage we shall, for simplicity, consider what is known as a standard normal distribution which is obtained by choosing particularly simple values for µ and σ. Key Point The standard normal distribution has a mean of zero and a variance of one. In Figure 2 we show the graph of the standard normal bistribution which has probability density function y = 1 2π e x2 /2 y y = 1 2π e x2 /2 0 x Figure 2. The standard normal distribution curve Engineering Mathematics: Open Learning Unit Level 1 2

3 The result which makes the standard normal distribution so important is as follows: Key Point If the behaviour of a continuous random variable X is described by the distribution N(µ, σ 2 ) then the behaviour of the random variable Z = X µ is described by the standard normal distribution σ N(0, 1). We call Z the standardised normal variable. Example If the random variable X is described by the distribution N(45, ) then what is the transformation to obtain the standardised normal variable? Here, µ = 45and σ 2 = so that σ = Hence Z =(X 45)/0.025is the required transformation. Example When the random variable X takes values between and 45.05, between which values does the random variable Z lie? When X =45.05, Z= = When X =44.95, Z= = Hence Z lies between 2 and 2. Now do this exercise The random variable X follows a normal distribution with mean 1000 and variance 100. When X takes values between 1005and 1010, between which values does the standardised normal variable Z lie? Answer 2. Probabilities and the standard normal distribution Since the standard normal distribution is used so frequently tables have been produced to help us calculate probabilities. This table is located at the end of this block. It is based upon the following diagram: 0 z 1 Figure 3 3 Engineering Mathematics: Open Learning Unit Level 1

4 Since the total area under the curve is equal to 1 it follows from the symmetry in the curve that the area under the curve in the region x>0 is equal to 0.5. In Figure 3 the shaded area is the probability that Z takes values between 0 and z 1. When we look-up a value in the table we obtain the value of the shaded area. Example What is the probability that Z takes values between 0 and 1.9? The second column headed 0 is the one to choose and its entry in the row neginning 1.9 is This is to be read as (we omitted the 0 in each entry for clarity) The interpretation is that the probability that Z takes values between 0 and 1.9 is Example What is the probability that Z takes values between 0 and 1.96? This time we want the column headed 6 and the row beginning 1.9. The entry is 4750 so that the required probability is Example What is the probability that Z takes values between 0 and 1.965? There is no entry corresponding to 1.965so we take the average of the values for 1.96 and (This linear interpolation is not strictly correct but is acceptable). The two values are 4750 and 4756 with an average of Hence the required probability is Now do this exercise What are the probabilities that Z takes values between (i) 0 and 2 (ii) 0 and 2.3 (iii) 0 and 2.33 (iv) 0 and 2.333? Answer Note from the table that as Z increases from 0 the entries increase, rapidly at first and then more slowly, toward 5000 i.e. a probability of 0.5. This is consistent with the shape of the curve. After Z = 3 the increase is quite slow so that we tabulate entries for values of Z rising by 0.1 instead of 0.01 as in the rest of the table. Engineering Mathematics: Open Learning Unit Level 1 4

5 3. Calculating other probabilities In this section we see how to calculate probabilities represented by areas other than those of the type shown in Figure 3. Case 1 Figure 4 illustrates what we do if both Z values are positive. By using the properties of the standard normal distribution we can organise matters so that any required area is always of standard form. Here the shaded region can be represented by the difference between two shaded areas 0 z 1 z 2 0 z 2 0 z 1 Figure 4 Example Find the probability that Z takes values bwteen 1 and 2. Using the table P (Z = z 2 ) i.e. P (Z = 2) is P (Z = z 1 ) i.e. P (Z = 1) is Hence P (1 <Z<2)= = (Remember that with a continuous distribution, P (Z = 1) is meaningless so that P (1 Z 2) is also Engineering Mathematics: Open Learning Unit Level 1

6 Case 2 The following diagram illustrates the procedure to be followed when finding probabilities of the form P (Z >z 1 ). This time the shaded area is the difference between the right-hand half of the total area and a tabulatedflarea. 0 z 1 area z 1 Figure 5 Example What is the probability that Z>2? P (0 <Z<2)= (from the table) Hence the probability is = Case 3 Here we consider the procedure to be followed when calculating probabilities of the form P (Z <z 1 ). Here the shaded area is the sum of the left-hand half of the total area and a standard area. 0 z 1 area z 1 Figure 6 Engineering Mathematics: Open Learning Unit Level 1 6

7 Example What is the probability that Z<2? P (Z >2) = = Case 4 Here we consider what needs to be done when calculating probabilities of the form P ( z 1 <Z<0) where z 1 is positive. This time we make use of the symmetry in the standard normal distribution curve. z 1 0 by symmetry this shaded area is equal in value to the one above. 0 z 1 Figure 7 Example What is the probability that 2 <Z<0? The area is equal to that corresponding to P (0 <Z<2) = Engineering Mathematics: Open Learning Unit Level 1

8 Case 5 Finally we consider probabilities of the form P ( z 2 <Z<z 1 ). Here we use the sum property and the symmetry property. z 1 0 z 2 0 z 1 0 z 2 Figure 8 Example What is the probability that 1 <Z<2? P ( 1 <Z<0) = P (0 <Z<1) = P (0 < Z < 2) = Hence the required probability, P ( 1 < Z < 2) is Other cases can be handed by a combination of the ideas already used. Now do this exercise Find the following probabilities. (i) P (0 <Z<1.5) (ii) P (Z >1.8) (iii) P (1.5 < Z < 1.8) (iv) P (Z < 1.8) (v) P ( 1.5 < Z < 0) (vi) P (Z < 1.5) (vii) P ( 1.8 < Z < 1.5) (viii) P ( 1.5 < Z < 1.8) (A simple sketch of the standard normal curve will help). Answer Engineering Mathematics: Open Learning Unit Level 1 8

9 4. Confidence Intervals We use probability models to make predictions in situations where there is not sufficient data available to make a definite statement. Any statement based on these models carries with it a risk of being proved incorrect by events. Notice that the normal probability curve extends to infinity in both directions. Theoretically any value of the normal random variable is possible, although, of course, values far from the mean position, zero, are very unlikely. Consider the diagram in Figure 9, 95% Figure 9 The shaded area is 95% of the total area. If we look at the entry in Table 1 corresponding to Z =1.96 we see the value This means that the probability of Z taking a value between 0 and 1.96 is By symmetry, the probability that Z takes a value between 1.96 and 0 is also Combining these results we see that P ( 1.96 < Z < 1.96)= 0.95or 95% We say that the 95% confidence interval for Z (about its mean of 0) is ( 1.96, 1.96). It follows that there is a 5% chance that Z lies outside this interval. Now do this exercise We wish to find the 99% confidence interval for Z about its mean, i.e. the value of z 1 in Figure 10 99% z 1 0 z 1 Figure 10 The shaded area is 99% of the total area. First, note that 99% corresponds to a probability of Find z 1 such that P (0 <Z<z 1 )= = Answer Now do this exercise Now quote the 99% confidence interval. Notice that the risk of Z lying outside this wider interval is reduced to 1%. Answer 9 Engineering Mathematics: Open Learning Unit Level 1

10 Now do this exercise Find the value of Z (i) which is exceeded on 5% of occasions (ii) which is exceeded on 99% of occasions. Answer Engineering Mathematics: Open Learning Unit Level 1 10

11 The Normal Probability Integral Z= x µ σ Engineering Mathematics: Open Learning Unit Level 1

12 End of Block 21.2 Engineering Mathematics: Open Learning Unit Level 1 12

13 The transformation is Z = X when X = 1005,Z = 5 10 =0.5 when X = 1010, Z= =1. Hence Z lies between 0.5and 1. Back to the theory 13 Engineering Mathematics: Open Learning Unit Level 1

14 (i) The entry is 4772; the probability is (ii) The entry is 4803; the probability is (iii) The entry is 4901; the probability is (iv) The entry for 2.33 is 4901, that for 2.34 is Linear interpolation gives a value of ( ) i.e. about 4903; the probability is Back to the theory Engineering Mathematics: Open Learning Unit Level 1 14

15 (i) (direct from table) (ii) = (iii) P (0 <Z<1.8) P (0 <Z<1.5 )= = (iv) = (v) P ( 1.5 <Z<0) = P (0 <Z<1.5 )= (vi) P (Z < 1.5 )=P (Z >1.5 )= = (vii) P ( 1.8 <Z< 1.5 )=P (1.5 <Z<1.8) = (viii) P (0 <Z<1.5)+P (0 <Z<1.8)= Back to the theory 15 Engineering Mathematics: Open Learning Unit Level 1

16 We look for a table value of The nearest we get is 4949 and 4951 corresponding to Z =2.57 and Z =2.58 respectively. We choose Z =2.58 Back to the theory Engineering Mathematics: Open Learning Unit Level 1 16

17 ( 2.58, 2.58) or 2.58 <Z<2.58. Back to the theory 17 Engineering Mathematics: Open Learning Unit Level 1

18 (i) The value is z 1, where P (Z >z 1 )=0.05. Hence P (0 <Z<z 1 )= = 0.45This corresponds to a table entry of The nearest values are 4495 (Z =1.64) and 4505 (Z =1.65). (ii) Values less than z 1 occur on 1% of occasions. By symmetry values greater than ( z 1 ) occur on 1% of occasions so that P (0 <z< z 1 )=0.49. The nearest table corresponding to 4900 is 4901 (Z =2.33). Hence the required value is z 1 = Back to the theory Engineering Mathematics: Open Learning Unit Level 1 18

4. 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

The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

AP 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, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

Unit 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

Normal 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

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part 1

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part Note: Assume missing data (if any) and mention the same. Q. Suppose X has a normal

How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

6.4 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

39.2. The Normal Approximation to the Binomial Distribution. Introduction. Prerequisites. Learning Outcomes

The Normal Approximation to the Binomial Distribution 39.2 Introduction We have already seen that the Poisson distribution can be used to approximate the binomial distribution for large values of n and

CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS

CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having

Normal and Binomial. Distributions

Normal and Binomial Distributions Library, Teaching and Learning 14 By now, you know about averages means in particular and are familiar with words like data, standard deviation, variance, probability,

MEASURES 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

Chapter 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

Statistics 641 - EXAM II - 1999 through 2003

Statistics 641 - EXAM II - 1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the

STT315 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

Density 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

Statistical 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

Estimation and Confidence Intervals

Estimation and Confidence Intervals Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Properties of Point Estimates 1 We have already encountered two point estimators: th e

2 ESTIMATION. Objectives. 2.0 Introduction

2 ESTIMATION Chapter 2 Estimation Objectives After studying this chapter you should be able to calculate confidence intervals for the mean of a normal distribution with unknown variance; be able to calculate

Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

8. 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,

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

Lesson 20. Probability and Cumulative Distribution Functions

Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

Continuous Random Variables Random variables whose values can be any number within a specified interval.

Section 10.4 Continuous Random Variables and the Normal Distribution Terms Continuous Random Variables Random variables whose values can be any number within a specified interval. Examples include: fuel

Chi Square Tests. Chapter 10. 10.1 Introduction

Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

CHI-SQUARE: TESTING FOR GOODNESS OF FIT

CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

HISTOGRAMS, 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

Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

Summarizing Data: Measures of Variation

Summarizing Data: Measures of Variation One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance

3.4 The Normal Distribution

3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

Unit 21 Student s t Distribution in Hypotheses Testing

Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between

6.1. The Exponential Function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The Exponential Function 6.1 Introduction In this block we revisit the use of exponents. We consider how the expression a x is defined when a is a positive number and x is irrational. Previously we have

MATHEMATICS 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

6 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

39.2. The Normal Approximation to the Binomial Distribution. Introduction. Prerequisites. Learning Outcomes

The Normal Approximation to the Binomial Distribution 39.2 Introduction We have already seen that the Poisson distribution can be used to approximate the binomial distribution for large values of n and

Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker

HYPOTHESIS TESTING PHILOSOPHY 1 The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker Question: So I'm hypothesis testing. What's the hypothesis I'm testing? Answer: When you're

Normal 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:

MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem

MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

Math 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

the 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

Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

The 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

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1: THE NORMAL CURVE AND "Z" SCORES: The Normal Curve: The "Normal" curve is a mathematical abstraction which conveniently

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

Unit 7: Normal Curves

Unit 7: Normal Curves Summary of Video Histograms of completely unrelated data often exhibit similar shapes. To focus on the overall shape of a distribution and to avoid being distracted by the irregularities

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

Exact Confidence Intervals

Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter

The 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

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

Week 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

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

Curving sketching using calculus. Jackie Nicholas

Mathematics Learning Centre Curving sketching using calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Curve sketching using calculus 1.1 Some General

SOLVING TRIGONOMETRIC EQUATIONS

Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC

6.5. Log-linear Graphs. Introduction. Prerequisites. Learning Outcomes. Learning Style

Log-linear Graphs 6.5 Introduction In this block we employ our knowledge of logarithms to simplify plotting the relation between one variable and another. In particular we consider those situations in

Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

Probability 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

Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

Sampling (cont d) and Confidence Intervals 11.220 Lecture 9 8 March 2006 R. Ryznar Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form

Lecture 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 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables

STAT 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

Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

1. How different is the t distribution from the normal?

Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions.

4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

Statistics 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

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

Lesson 4 Measures of Central Tendency

Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central

Lecture 8: More Continuous Random Variables

Lecture 8: More Continuous Random Variables 26 September 2005 Last time: the eponential. Going from saying the density e λ, to f() λe λ, to the CDF F () e λ. Pictures of the pdf and CDF. Today: the Gaussian

Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

Key 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

The 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

Extended control charts

Extended control charts The control chart types listed below are recommended as alternative and additional tools to the Shewhart control charts. When compared with classical charts, they have some advantages

OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance

Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance (if the Gauss-Markov

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

Basic Statistics. Probability and Confidence Intervals

Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

Continuous Random Variables

Chapter 5 Continuous Random Variables 5.1 Continuous Random Variables 1 5.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand continuous

Exemplar 6 Cumulative Frequency Polygon

Exemplar 6 Cumulative Frequency Polygon Objectives (1) To construct a cumulative frequency polygon from a set of data (2) To interpret a cumulative frequency polygon Learning Unit Construction and Interpretation

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

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.

Introduction to Descriptive Statistics

Mathematics Learning Centre Introduction to Descriptive Statistics Jackie Nicholas c 1999 University of Sydney Acknowledgements Parts of this booklet were previously published in a booklet of the same

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.