Key Concept. Properties. February 23, S6.4_3 Sampling Distributions and Estimators
|
|
|
- Delphia Weaver
- 6 years ago
- Views:
Transcription
1 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 Distributions 6 4 Sampling Distributions and Estimators 6 5 The Central Limit Theorem 6 6 Normal as Approximation to Binomial 6 7 Assessing Normality Check out the Sample Means (xls) Excel program in the Technology section, at of the Important Links webpage or go directly to Key Concept The main objective of this section is to understand the concept of a sampling distribution of a statistic, which is the distribution of all values of that statistic when all possible samples of the same size are taken from the same population. We will also see that some statistics are better than others for estimating population parameters. Definitions The sampling distribution of a statistic (such as the sample mean or sample proportion) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. (The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) The sampling distribution of the mean is the distribution of sample means, with all samples having the same sample size n taken from the same population. (The sampling distribution of the mean is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) Properties Sample means target the value of the population mean. (That is, the mean of the sample means is the population mean. The expected value of the sample mean is equal to the population mean.) The distribution of the sample means tends to be a normal distribution. 1
2 Definition The sampling distribution of the variance is the distribution of sample variances, with all samples having the same sample size n taken from the same population. (The sampling distribution of the variance is typically represented as a probability distribution in the format of a table, probability histogram, or formula.) Properties Sample variances target the value of the population variance. (That is, the mean of the sample variances is the population variance. The expected value of the sample variance is equal to the population variance.) The distribution of the sample variances tends to be a distribution skewed to the right. Definition The sampling distribution of the proportion is the distribution of sample proportions, with all samples having the same sample size n taken from the same population. Definition We need to distinguish between a population proportion p and some sample proportion: p = population proportion = sample proportion Properties Sample proportions target the value of the population proportion. (That is, the mean of the sample proportions is the population proportion. The expected value of the sample proportion is equal to the population proportion.) The distribution of the sample proportion tends to be a normal distribution. Unbiased Sample means, variances, and proportions are unbiased estimators. That is they target the population parameter. These statistics are better in estimating the population parameter. Biased Sample medians, ranges, and standard deviations are biased estimators. That is they do NOT target the population parameter. Note: the bias with the standard deviation is relatively small in large samples so s is often used to estimate σ. 2
3 Specific results from 10,000 trials Consider repeating this process: Roll a die 5 times, find the mean, variance s 2, and the proportion of odd numbers of the results. What do we know about the behavior of all sample means that are generated as this process continues indefinitely? All outcomes are equally likely so the population mean is 3.5; the mean of the 10,000 trials is If continued indefinitely, the sample mean will be 3.5. Also, notice the distribution is normal. Specific results from 10,000 trials Specific results from 10,000 trials All outcomes are equally likely so the population variance is 2.9; the mean of the 10,000 trials is If continued indefinitely, the sample variance will be 2.9. Also, notice the distribution is skewed to the right. All outcomes are equally likely so the population proportion of odd numbers is 0.50; the proportion of the 10,000 trials is If continued indefinitely, the mean of sample proportions will be Also, notice the distribution is approximately normal. 3
4 Why Sample with Replacement? Sampling without replacement would have the very practical advantage of avoiding wasteful duplication whenever the same item is selected more than once. However, we are interested in sampling with replacement for these two reasons: 1. When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement. 2. Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and result in simpler calculations and formulas. Caution Many methods of statistics require a simple random sample. Some samples, such as voluntary response samples or convenience samples, could easily result in very wrong results. Recap In this section we have discussed: Sampling distribution of a statistic. Sampling distribution of the mean. Sampling distribution of the variance. Sampling distribution of the proportion. Estimators. 292/10. In Exercises 9 12, refer to the population and list of samples in Example 4. Sampling Distribution of the Standard Deviation. From Example 4: Sampling Distribution of the Range Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10 (based on Data Set 22 in Appendix B). Consider the values of 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and
5 292/12. In Exercises 9 12, refer to the population and list of samples in Example 4. Sampling Distribution of the Mean. From Example 4: Sampling Distribution of the Mean Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10 (based on Data Set 22 in Appendix B). Consider the values of 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and /14. Assassinated Presidents: Sampling Distribution of the Median The ages (years) of the four U. S. presidents when they were assassinated in office are 56 (Lincoln), 49 (Garfield), 58 (McKinley), and 46 (Kennedy). a. Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. b. Find the median of each of the 16 samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution. (Use a format similar to Table 6 5 on page 289). c. Compare the population median to the mean of the sample medians. d. Do the sample medians target the value of the population median? In general, do sample medians make good estimators of population median? Why or why not? 292/
6 292/16. Assassinated Presidents: Sampling Distribution of the Variance The ages (years) of the four U. S. presidents when they were assassinated in office are 56 (Lincoln), 49 (Garfield), 58 (McKinley), and 46 (Kennedy). a. Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. b. Find the variance of each of the 16 samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution. (Use a format similar to Table 6 5 on page 289). c. Compare the population variance to the mean of the sample variances. d. Do the sample variances target the value of the population variance? In general, do sample variances make good estimators of population variance? Why or why not? 292/ /18. Births: Sampling Distribution of Proportion When 3 births are randomly selected, the sample space is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those 8 outcomes are equally likely. Describe the sampling distribution of the proportion of girls from 3 births as a probability distribution table. Does the mean of the sample proportions equal the proportion of girls in 3 births? (Hint: See Example 5.) 292/20. Quality Control: Sampling Distribution of Proportion After constructing a new manufacturing machine, 5 prototype integrated circuit chips are produced and it is found that 2 are defective (D) and 3 are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population. a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects. b. Find the mean of the sampling distribution. c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion? 6
7 292/21. Using a Formula to Describe a Sampling Distribution Example 5 includes a table and graph to describe the sampling distribution of the proportions of girls from 2 births. Consider the formula shown below, and evaluate that formula using sample proportions x of 0, 0.5, and 1. Based on the results, does the formula describe the sampling distribution? Why or why not? 292/21. Using a Formula to Describe a Sampling Distribution Example 5 includes a table and graph to describe the sampling distribution of the proportions of girls from 2 births. Consider the formula shown below, and evaluate that formula using sample proportions x of 0, 0.5, and 1. Based on the results, does the formula describe the sampling distribution? Why or why not? Use TI calculator and let Y1 = P(x). 2ND Window, TblStart=0, ΔTbl=0.5 Use TI calculator and let Y1 = P(x). 2ND Window, TblStart=0, ΔTbl=0.5 7
{ } all possible outcomes of the procedure. There are 8 ways this procedure can happen.
Probability with the 3-Kids Procedure Statistics Procedures and Events Definition A procedure is something that produces an outcome. When a procedure produces an outcome, it s called a trial or a run of
Producing data Toward statistical inference. Section 3.3
Producing data Toward statistical inference Section 3.3 Toward statistical inference Idea: Use sampling to understand statistical inference Statistical inference is when a conclusion about a population
ACMS Statistics for Life Sciences. Chapter 9: Introducing Probability
ACMS 20340 Statistics for Life Sciences Chapter 9: Introducing Probability Why Consider Probability? We re doing statistics here. Why should we bother with probability? As we will see, probability plays
MAT 155. Key Concept. Density Curve
MAT 155 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 Distributions 6 4
Chapter 4 Probability
4-1 Review and Preview Chapter 4 Probability 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting
13.1 The Basics of Probability Theory
13.1 The Basics of Probability Theory An experiment is a controlled operation that yields a set of results. The possible results of an experiment are called its outcomes. The set of outcomes are the sample
Chapter Summary. 7.1 Discrete Probability 7.2 Probability Theory 7.3 Bayes Theorem 7.4 Expected value and Variance
Chapter 7 Chapter Summary 7.1 Discrete Probability 7.2 Probability Theory 7.3 Bayes Theorem 7.4 Expected value and Variance Section 7.1 Introduction Probability theory dates back to 1526 when the Italian
Section 13.3 Probability
288 Section 13.3 Probability Probability is a measure of how likely an event will occur. When the weather forecaster says that there will be a 50% chance of rain this afternoon, the probability that it
CHAPTER 3: Quadratic Functions and Equations; Inequalities
171S MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros,
Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last
Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating
CHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
CS 441 Discrete Mathematics for CS Lecture 20. Probabilities. CS 441 Discrete mathematics for CS. Probabilities
CS 441 Discrete Mathematics for CS Lecture 20 Probabilities Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square CS 441 Discrete mathematics for CS Probabilities Three axioms of the probability theory:
CHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
Chapter5 Probability.
Chapter5 Probability. Introduction. We will consider random experiments with chance outcomes. Events are outcomes that may or may not occur. Notation: Capital letters like E will denote events Probability
171S2.2q The Algebra of Functions. February 13, MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
2.3 Estimating PDFs and PDF Parameters
.3 Estimating PDFs and PDF Parameters estimating means - discrete and continuous estimating variance using a known mean estimating variance with an estimated mean estimating a discrete pdf estimating a
Introduction to Probability. Experiments. Sample Space. Event. Basic Requirements for Assigning Probabilities. Experiments
Introduction to Probability Experiments These are processes that generate welldefined outcomes Experiments Counting Rules Combinations Permutations Assigning Probabilities Experiment Experimental Outcomes
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 2: Random Experiments. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 2: Random Experiments Prof. Vince Calhoun Reading This class: Section 2.1-2.2 Next class: Section 2.3-2.4 Homework: Assignment 1: From the
Chapter 13, Probability from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and is
Chapter 13, Probability from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and is available on the Connexions website. It is used under a
Chapter 3 Probability Distributions and Statistics Section 3.1 Random Variables and Histograms
Math 166 (c)2013 Epstein Chapter 3 Page 1 Chapter 3 Probability Distributions and Statistics Section 3.1 Random Variables and Histograms The value of the result of the probability experiment is called
CHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
171S3.2 Quadratic Equations, Functions, Zeros, and Models September 30, Quadratic Equations, Functions, Zeros, and Models
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
CHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3
MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 1. A four engine plane can fly if at least two engines work. a) If the engines operate independently and each malfunctions with probability q, what is the
Solution: Solution: Solution:
Chapter 5: Exponential and Logarithmic Functions a. The exponential growth function is y = f(t) = ab t, where a = 2000 because the initial population is 2000 squirrels The annual growth rate is 3% per
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
171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
CS 361: Probability & Statistics
September 12, 2017 CS 361: Probability & Statistics Correlation Summary of what we proved We wanted a way of predicting y from x We chose to think in standard coordinates and to use a linear predictor
October 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
171S4.6q Polynomial and Rational Inequalities. April 02, Polynomial and Rational Inequalities. Polynomial Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University
Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University Introduction Luca Paccioli(1445-1514), Studies of chances of events Niccolo Tartaglia(1499-1557) Girolamo
MTH302 Long Solved Questions By
MTH30 Long Solved uestions By www.vuattach.ning.com If you toss a die and observe the number of dots that appears on top face then write the events that the even number occurs. Number of Possible outcomes
CHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
The Normal Distribution. The Gaussian Curve. Advantages of using Z-score. Importance of normal or Gaussian distribution (ND)
Importance of normal or Gaussian distribution (ND) The Normal It is the most used distribution Most method are based on the assumption of ND Sum of many independent, random contributions variables (grain
The Central Limit Theorem
Introductory Statistics Lectures The Central Limit Theorem Sampling distributions Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission
Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University
Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University What is probability? A branch of mathematics that deals with calculating the likelihood of a given event
14.2 THREE IMPORTANT DISCRETE PROBABILITY MODELS
14.2 THREE IMPORTANT DISCRETE PROBABILITY MODELS In Section 14.1 the idea of a discrete probability model was introduced. In the examples of that section the probability of each basic outcome of the experiment
green, green, green, green, green The favorable outcomes of the event are blue and red.
0 Chapter Review Review Key Vocabulary experiment, p. 0 outcomes, p. 0 event, p. 0 favorable outcomes, p. 0 probability, p. 08 relative frequency, p. Review Examples and Exercises experimental probability,
Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 4.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola 4.1-1 4-1 Review and Preview Chapter 4 Probability 4-2 Basic Concepts of Probability 4-3 Addition
Chapter 3. Discrete Random Variables and Their Probability Distributions
Chapter 3. Discrete Random Variables and Their Probability Distributions 2.11 Definition of random variable 3.1 Definition of a discrete random variable 3.2 Probability distribution of a discrete random
Part 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a
LOWELL WEEKLY JOURNAL
Y G Bk b $ 6 G Y 7 B B B B - BB -BY- B Bk B Qk Q k Q k B g (- -- k Bk G Bk k q B - - - - - $ gb q g bg g g b b q )( 6 B 7 B B k 6 g k 6 B b Y k b - b b k b b b g ( \ bg Y b b k b /% /% b k b b g Y Y k
Estadística I Exercises Chapter 4 Academic year 2015/16
Estadística I Exercises Chapter 4 Academic year 2015/16 1. An urn contains 15 balls numbered from 2 to 16. One ball is drawn at random and its number is reported. (a) Define the following events by listing
Regression Equation. November 28, S10.3_3 Regression. Key Concept. Chapter 10 Correlation and Regression. Definitions
MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 10 Correlation and Regression 10 1 Review and Preview 10 2 Correlation 10 3 Regression 10 4 Variation and Prediction Intervals
Regression Equation. April 25, S10.3_3 Regression. Key Concept. Chapter 10 Correlation and Regression. Definitions
MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 10 Correlation and Regression 10 1 Review and Preview 10 2 Correlation 10 3 Regression 10 4 Variation and Prediction Intervals
CHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
Answers Part A. P(x = 67) = 0, because x is a continuous random variable. 2. Find the following probabilities:
Answers Part A 1. Woman s heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the probability that a single randomly selected woman will be 67 inches
Statistics 2. Revision Notes
Statistics 2 Revision Notes June 2016 2 S2 JUNE 2016 SDB Statistics 2 1 The Binomial distribution 5 Factorials... 5 Combinations... 5 Properties of n C r... 5 Binomial Theorem... 6 Binomial coefficients...
Probability Distributions.
Probability Distributions http://www.pelagicos.net/classes_biometry_fa18.htm Probability Measuring Discrete Outcomes Plotting probabilities for discrete outcomes: 0.6 0.5 0.4 0.3 0.2 0.1 NOTE: Area within
Discrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan Introduction The markets can be thought of as a complex interaction of a large number of random processes,
Hypothesis Testing. ) the hypothesis that suggests no change from previous experience
Hypothesis Testing Definitions Hypothesis a claim about something Null hypothesis ( H 0 ) the hypothesis that suggests no change from previous experience Alternative hypothesis ( H 1 ) the hypothesis that
Scientific Measurement
Scientific Measurement SPA-4103 Dr Alston J Misquitta Lecture 5 - The Binomial Probability Distribution Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson
Unit 22: Sampling Distributions
Unit 22: Sampling Distributions Summary of Video If we know an entire population, then we can compute population parameters such as the population mean or standard deviation. However, we generally don
1.0 Continuous Distributions. 5.0 Shapes of Distributions. 6.0 The Normal Curve. 7.0 Discrete Distributions. 8.0 Tolerances. 11.
Chapter 4 Statistics 45 CHAPTER 4 BASIC QUALITY CONCEPTS 1.0 Continuous Distributions.0 Measures of Central Tendency 3.0 Measures of Spread or Dispersion 4.0 Histograms and Frequency Distributions 5.0
Statistics for Economists. Lectures 3 & 4
Statistics for Economists Lectures 3 & 4 Asrat Temesgen Stockholm University 1 CHAPTER 2- Discrete Distributions 2.1. Random variables of the Discrete Type Definition 2.1.1: Given a random experiment with
Objectives. 171S5.6_p Applications of Exponential and Logarithmic Functions. April 21, 2011
MAT 171 Precalculus Algebra Trigsted Pilot Test Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential
STA 291 Lecture 16. Normal distributions: ( mean and SD ) use table or web page. The sampling distribution of and are both (approximately) normal
STA 291 Lecture 16 Normal distributions: ( mean and SD ) use table or web page. The sampling distribution of and are both (approximately) normal X STA 291 - Lecture 16 1 Sampling Distributions Sampling
Statistics for Economists Lectures 6 & 7. Asrat Temesgen Stockholm University
Statistics for Economists Lectures 6 & 7 Asrat Temesgen Stockholm University 1 Chapter 4- Bivariate Distributions 41 Distributions of two random variables Definition 41-1: Let X and Y be two random variables
INDEX. Numerical Methods. Finite Differences and Interpolation. Solution of a System of Linear Equations
INDEX GTU Syllabus Tutorial No. Name Of Topic Statistics Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Basic Probability Probability Descriptive Statistics Correlation and Regression Curve Fitting Numerical Methods
Part 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park August 19, 2008 1 Introduction There are three main objectives to this section: 1. To introduce the concepts of probability and random variables.
Math 10 - Compilation of Sample Exam Questions + Answers
Math 10 - Compilation of Sample Exam Questions + Sample Exam Question 1 We have a population of size N. Let p be the independent probability of a person in the population developing a disease. Answer the
6.867 Machine Learning
6.867 Machine Learning Problem set 1 Due Thursday, September 19, in class What and how to turn in? Turn in short written answers to the questions explicitly stated, and when requested to explain or prove.
CSS 211: Statistical Methods I
CSS 211: Statistical Methods I Zhaoxian Zhou School of Computing University of Southern Mississippi Zhaoxian.Zhou@usm.edu January 11, 2018 Zhaoxian Zhou (USM) CSS 211 January 11, 2018 1 / 227 Overview
Essential Question: How are the mean and the standard deviation determined from a discrete probability distribution?
Probability and Statistics The Binomial Probability Distribution and Related Topics Chapter 5 Section 1 Introduction to Random Variables and Probability Distributions Essential Question: How are the mean
Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (NEW)(CSE/IT)/SEM-4/M-401/ MATHEMATICS - III
Name :. Roll No. :..... Invigilator s Signature :.. 202 MATHEMATICS - III Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers
Chapter 1 - Lecture 3 Measures of Location
Chapter 1 - Lecture 3 of Location August 31st, 2009 Chapter 1 - Lecture 3 of Location General Types of measures Median Skewness Chapter 1 - Lecture 3 of Location Outline General Types of measures What
DISCRETE VARIABLE PROBLEMS ONLY
DISCRETE VARIABLE PROBLEMS ONLY. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each
AP Statistics Review Ch. 7
AP Statistics Review Ch. 7 Name 1. Which of the following best describes what is meant by the term sampling variability? A. There are many different methods for selecting a sample. B. Two different samples
(a) Find the value of x. (4) Write down the standard deviation. (2) (Total 6 marks)
1. The following frequency distribution of marks has mean 4.5. Mark 1 2 3 4 5 6 7 Frequency 2 4 6 9 x 9 4 Find the value of x. (4) Write down the standard deviation. (Total 6 marks) 2. The following table
Lecture 10: Probability distributions TUESDAY, FEBRUARY 19, 2019
Lecture 10: Probability distributions DANIEL WELLER TUESDAY, FEBRUARY 19, 2019 Agenda What is probability? (again) Describing probabilities (distributions) Understanding probabilities (expectation) Partial
Chapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic
BSTT523: Pagano & Gavreau, Chapter 7 1 Chapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic Random Variable (R.V.) X Assumes values (x) by chance Discrete R.V.
Sections OPIM 303, Managerial Statistics H Guy Williams, 2006
Sections 3.1 3.5 The three major properties which describe a set of data: Central Tendency Variation Shape OPIM 303 Lecture 3 Page 1 Most sets of data show a distinct tendency to group or cluster around
Chapter (4) Discrete Probability Distributions Examples
Chapter (4) Discrete Probability Distributions Examples Example () Two balanced dice are rolled. Let X be the sum of the two dice. Obtain the probability distribution of X. Solution When the two balanced
Sec$on Summary. Assigning Probabilities Probabilities of Complements and Unions of Events Conditional Probability
Section 7.2 Sec$on Summary Assigning Probabilities Probabilities of Complements and Unions of Events Conditional Probability Independence Bernoulli Trials and the Binomial Distribution Random Variables
the amount of the data corresponding to the subinterval the width of the subinterval e x2 to the left by 5 units results in another PDF g(x) = 1 π
Math 10A with Professor Stankova Worksheet, Discussion #42; Friday, 12/8/2017 GSI name: Roy Zhao Problems 1. For each of the following distributions, derive/find all of the following: PMF/PDF, CDF, median,
Applied Statistics I
Applied Statistics I (IMT224β/AMT224β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna Applied Statistics I(IMT224β/AMT224β) 1/158 Chapter
********************************************************************************************************
QUESTION # 1 1. Let the random variable X represent the number of telephone lines in use by the technical support center of a software manufacturer at noon each day. The probability distribution of X is
IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice
IB Mathematics HL Year 2 Unit 7 (Core Topic 6: Probability and Statistics) Valuable Practice 1. We have seen that the TI-83 calculator random number generator X = rand defines a uniformly-distributed random
Chapter 15 Sampling Distribution Models
Chapter 15 Sampling Distribution Models 1 15.1 Sampling Distribution of a Proportion 2 Sampling About Evolution According to a Gallup poll, 43% believe in evolution. Assume this is true of all Americans.
What is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
MAT 271E Probability and Statistics
MAT 71E Probability and Statistics Spring 013 Instructor : Class Meets : Office Hours : Textbook : Supp. Text : İlker Bayram EEB 1103 ibayram@itu.edu.tr 13.30 1.30, Wednesday EEB 5303 10.00 1.00, Wednesday
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
Chapter. Numerically Summarizing Data Pearson Prentice Hall. All rights reserved
Chapter 3 Numerically Summarizing Data Section 3.1 Measures of Central Tendency Objectives 1. Determine the arithmetic mean of a variable from raw data 2. Determine the median of a variable from raw data
CHAPTER TOPICS. Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population
Distribusi Sampel CHAPTER TOPICS Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population 2 3 WHY STUDY SAMPLING DISTRIBUTIONS
EXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS
EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker
At least 8/9 of th measurements lie in the interval
Tchebysheff s Theorem Given a number k 1 and a population with n measurements, at least [1 (1/k 2 )] of the measurements will lie within k standard deviations of their mean. A Simplified Tchebysheff s
Announcements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias
Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to
Probability Distributions
EXAMPLE: Consider rolling a fair die twice. Probability Distributions Random Variables S = {(i, j : i, j {,...,6}} Suppose we are interested in computing the sum, i.e. we have placed a bet at a craps table.
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
Interpret Standard Deviation. Outlier Rule. Describe the Distribution OR Compare the Distributions. Linear Transformations SOCS. Interpret a z score
Interpret Standard Deviation Outlier Rule Linear Transformations Describe the Distribution OR Compare the Distributions SOCS Using Normalcdf and Invnorm (Calculator Tips) Interpret a z score What is an
Chapter 4: Probability and Probability Distributions
Chapter 4: Probability and Probability Distributions 4.1 How Probability Can Be Used in Making Inferences 4.1 a. Subjective probability b. Relative frequency c. Classical d. Relative frequency e. Subjective
6.4 THE HYPERGEOMETRIC PROBABILITY DISTRIBUTION
Section 6.4 The Hypergeometric Probability Distribution 6 1 6.4 THE HYPERGEOMETRIC PROBABILITY DISTRIBUTION Preparing for This Section Before getting started, review the following: Classical Method (Section
Recall that the standard deviation σ of a numerical data set is given by
11.1 Using Normal Distributions Essential Question In a normal distribution, about what percent of the data lies within one, two, and three standard deviations of the mean? Recall that the standard deviation
Randomized Algorithms
Randomized Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A new 4 credit unit course Part of Theoretical Computer Science courses at the Department of Mathematics There will be 4 hours
Question Bank In Mathematics Class IX (Term II)
Question Bank In Mathematics Class IX (Term II) PROBABILITY A. SUMMATIVE ASSESSMENT. PROBABILITY AN EXPERIMENTAL APPROACH. The science which measures the degree of uncertainty is called probability.. In
Unit 1: Statistics. Mrs. Valentine Math III
Unit 1: Statistics Mrs. Valentine Math III 1.1 Analyzing Data Statistics Study, analysis, and interpretation of data Find measure of central tendency Mean average of the data Median Odd # data pts: middle
Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 3.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola 3.1-1 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview
September 26, S2.5 Variation and Applications
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks
Random Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the