Statistical Inference
|
|
|
- Nathaniel Arnold
- 5 years ago
- Views:
Transcription
1 Chapter 14 Confidence Intervals: The Basic Statistical Inference Situation: We are interested in estimating some parameter (population mean, μ) that is unknown. We take a random sample from this population. Goal: Draw inference about the population parameter from the sample data. Example: Beetle cars Confidence Intervals Suppose EPA wants to estimate μ, the average CO 2 emitted by all Beetle cars. 49 Beetle cars are randomly selected at the VW plant in Detroit and tested for CO 2 emissions. The test results show these 49 cars have an average of 1.5 grams CO 2 emission per mile. What is our best guess about the unknown parameter μ? But we need to be careful in making a conclusion about μ because We know any other random sample of 49 cars would give a different value of We can t say how much confident we are with our estimate. So instead of just giving one number (the value of x ) as our estimate of μ, it seems more desirable to give an of values that may contain μ with some degree of. Let s try to find an interval of values within which we can say that average CO 2 emission (μ) of Beetle cars lies with some high degree of confidence, say 95%. For this, let s recall from Chapter 11 how x behaves in repeated sampling. 1
2 In the Beetle cars example, suppose we repeatedly sample 49 cars and for each sample note their mean CO 2 emission. We know that the distribution of sample mean, x is then Assume we know that the standard deviation (σ) of CO 2 emissions of Beetle cars is 0.84 (it is unrealistic to assume σ to be known, we would get rid of this assumption in Chapter 18!). Then, Using the rule, we know that x will fall within 2 standard deviations σ 2 = of the population mean μ with probability n This means that in 95% of all samples, the observed mean score x will be within points of the population mean μ. (Note: μ is unknown and fixed - it doesn t vary from sample to sample). To say that x is within 0.24 points of μ is equivalent to saying that μ is within 0.24 points of observed x. This happens in 95% of all samples. 2
3 Combining these facts we can say, In 95% of all samples (of size n = 49) the true but unknown mean μ lies in the interval. We can rewrite this interval as Recall that mean of our sample was 1.5 grams. So we say that We are 95% confident that This interval we just calculated is a 95% confidence interval for the unknown average CO 2 emission (μ) of all Beetle cars. In general, confidence intervals for any parameter consists of two parts: 1) An interval calculated from the data of the form The margin of error conveys how accurate we believe our guess of the true parameter value is, based on the variability of the estimate. 2) A confidence level, C, gives the probability that the random interval captures the true parameter value in repeated samples. (Note that it is NOT the probability that any one specific interval calculated from a random sample captures the true parameter.) Many types of confidence intervals exist for various kinds of parameters. Ch. 14: Confidence Intervals for mean μ (s.d. σ is known). Ch. 18: Confidence Intervals for mean μ (s.d. σ is unknown). Ch. 19: Confidence Intervals for difference between two population means. Ch. 20: Confidence Intervals for population proportion. Ch. 21: Confidence Intervals for comparing two proportions. We will concentrate on confidence intervals for the mean μ of a population. 3
4 Confidence Intervals for a Population Mean (standard deviation is known) Confidence Intervals for a Population Mean Choose an SRS of size n from a population having unknown mean μ and known standard deviation σ. A level C confidence interval for μ is Here, z* is the value on the standard normal curve with area C between -z* and z*. The interval is exact when the original population distribution is normal and is approximately correct for large n otherwise. The most commonly used confidence levels are C 90% 95% 99% z* z* for other confidence levels can be found similarly from Table A or more conveniently from Table C. 4
5 Ex: Beetle Cars Recall that the CO 2 emissions of Beetle cars have a mean μ and standard deviation σ = Our sample mean was x = 1.5 and n = 49. We had calculated the 95% confidence interval for the mean CO 2 emission as a. Find a 90% confidence interval for the mean CO 2 emission. b. Find a 99% confidence interval for the mean CO 2 emission. c. Find an 80% confidence interval for the mean CO 2 emission. Note: We don t know if any of the above confidence intervals contain μ or not! Then what do we mean by confidence? 5
6 The meaning of Confidence : When we say 95% confident, we mean that Our confidence is in the procedure, not in any one specific interval. So it is completely wrong to say: (because any one interval either contains the parameter or not, there is no randomness in it!) Remember that probability (chance) is associated only with a random phenomenon. After you have constructed a CI from a random sample, there is no randomness left it. Hence it doesn t make sense to attach any probability statement to a specific (numerical) CI. 6
7 Behavior of Confidence Intervals What happens to the margin of error if we increase the confidence level C? Does it increase, decrease or stay the same? (Hint: what happens to the value of z*?) How does this affect the width of the resulting confidence interval? Note the tradeoff: We would like to have a smaller margin of error (narrower interval) as well as high confidence but Q: What could we do to get a narrower interval (smaller margin of error) without lowering confidence? E.g. in the Beetle cars example the 99% confidence interval for μ was found to be with a sample of size 49. If instead we had taken a sample of size 100 cars and suppose their mean CO 2 emission was 1.5 grams, then How does the size of σ affect the margin of error? Thus we have 3 ways of reducing the width of the confidence interval: 1) 2) 3) 7
8 Choosing the Sample Size We saw that we can have high degree of confidence as well as small margin of error by Usually researchers will have a desired confidence level and margin of error they want to attain. So one aspect of designing any study is to decide the number of observations needed. Let m represent the desired margin of error. Recall the formula of margin of error: Solving for n we get: *****Always round your answer up!!***** Ex: Suppose PGSA (Poor Graduate Students Association) at the Texas state wants to estimate the mean monthly income of SMU graduate students within $100 with 95% confidence. How many students should PGSA sample? Assume that the standard deviation of incomes of SMU graduate students is $421. 8
Notes 3: Statistical Inference: Sampling, Sampling Distributions Confidence Intervals, and Hypothesis Testing
Notes 3: Statistical Inference: Sampling, Sampling Distributions Confidence Intervals, and Hypothesis Testing 1. Purpose of statistical inference Statistical inference provides a means of generalizing
1 Binomial Probability [15 points]
![1 Binomial Probability [15 points]](/thumbs/80/81467318.jpg "1 Binomial Probability [15 points]")
Economics 250 Assignment 2 (Due November 13, 2017, in class) i) You should do the assignment on your own, Not group work! ii) Submit the completed work in class on the due date. iii) Remember to include
STA Module 10 Comparing Two Proportions
STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare
Estimating a Population Mean
Estimating a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives At the end of this lesson we will be able to: obtain a point estimate for
Confidence intervals
Confidence intervals We now want to take what we ve learned about sampling distributions and standard errors and construct confidence intervals. What are confidence intervals? Simply an interval for which
Statistical inference provides methods for drawing conclusions about a population from sample data.
Introduction to inference Confidence Intervals Statistical inference provides methods for drawing conclusions about a population from sample data. 10.1 Estimating with confidence SAT σ = 100 n = 500 µ
CHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.1 Comparing Two Proportions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Proportions
Confidence Intervals for Population Mean
Confidence Intervals for Population Mean Reading: Sections 7.1, 7.2, 7.3 Learning Objectives: Students should be able to: Understand the meaning and purpose of confidence intervals Calculate a confidence
Chapter 22. Comparing Two Proportions 1 /29
Chapter 22 Comparing Two Proportions 1 /29 Homework p519 2, 4, 12, 13, 15, 17, 18, 19, 24 2 /29 Objective Students test null and alternate hypothesis about two population proportions. 3 /29 Comparing Two
Harvard University. Rigorous Research in Engineering Education
Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected
Interval estimation. October 3, Basic ideas CLT and CI CI for a population mean CI for a population proportion CI for a Normal mean
Interval estimation October 3, 2018 STAT 151 Class 7 Slide 1 Pandemic data Treatment outcome, X, from n = 100 patients in a pandemic: 1 = recovered and 0 = not recovered 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0
Statistic: a that can be from a sample without making use of any unknown. In practice we will use to establish unknown parameters.
Chapter 9: Sampling Distributions 9.1: Sampling Distributions IDEA: How often would a given method of sampling give a correct answer if it was repeated many times? That is, if you took repeated samples
MAT Mathematics in Today's World
MAT 1000 Mathematics in Today's World Last Time We discussed the four rules that govern probabilities: 1. Probabilities are numbers between 0 and 1 2. The probability an event does not occur is 1 minus
One-sample categorical data: approximate inference
One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
Comparing Means from Two-Sample
Comparing Means from Two-Sample Kwonsang Lee University of Pennsylvania kwonlee@wharton.upenn.edu April 3, 2015 Kwonsang Lee STAT111 April 3, 2015 1 / 22 Inference from One-Sample We have two options to
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
Central Limit Theorem Confidence Intervals Worked example #6. July 24, 2017
Central Limit Theorem Confidence Intervals Worked example #6 July 24, 2017 10 8 Raw scores 6 4 Mean=71.4% 2 0 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90+ Scaling is to add 3.6% to bring mean
Note that we are looking at the true mean, μ, not y. The problem for us is that we need to find the endpoints of our interval (a, b).
Confidence Intervals 1) What are confidence intervals? Simply, an interval for which we have a certain confidence. For example, we are 90% certain that an interval contains the true value of something
CHAPTER 7 THE SAMPLING DISTRIBUTION OF THE MEAN. 7.1 Sampling Error; The need for Sampling Distributions
CHAPTER 7 THE SAMPLING DISTRIBUTION OF THE MEAN 7.1 Sampling Error; The need for Sampling Distributions Sampling Error the error resulting from using a sample characteristic (statistic) to estimate a population
Probability and Statistics
Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT
8.2 Margin of Error, Sample Size (old 8.3)
8.2 Margin of Error, Sample Size (old 8.3) GOALS: 1. Understand the Margin of Error is a measure of sampling error, and is directly proportional to the standard error. 2. Understand how the Margin of Error
Business Statistics. Lecture 5: Confidence Intervals
Business Statistics Lecture 5: Confidence Intervals Goals for this Lecture Confidence intervals The t distribution 2 Welcome to Interval Estimation! Moments Mean 815.0340 Std Dev 0.8923 Std Error Mean
Do students sleep the recommended 8 hours a night on average?
BIEB100. Professor Rifkin. Notes on Section 2.2, lecture of 27 January 2014. Do students sleep the recommended 8 hours a night on average? We first set up our null and alternative hypotheses: H0: μ= 8
CSE 103 Homework 8: Solutions November 30, var(x) = np(1 p) = P r( X ) 0.95 P r( X ) 0.
() () a. X is a binomial distribution with n = 000, p = /6 b. The expected value, variance, and standard deviation of X is: E(X) = np = 000 = 000 6 var(x) = np( p) = 000 5 6 666 stdev(x) = np( p) = 000
Statistical Intervals (One sample) (Chs )
7 Statistical Intervals (One sample) (Chs 8.1-8.3) Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to normally distributed with expected value µ and
Sampling Distributions
Sampling Distributions Sampling Distribution of the Mean & Hypothesis Testing Remember sampling? Sampling Part 1 of definition Selecting a subset of the population to create a sample Generally random sampling
Chapter 22. Comparing Two Proportions 1 /30
Chapter 22 Comparing Two Proportions 1 /30 Homework p519 2, 4, 12, 13, 15, 17, 18, 19, 24 2 /30 3 /30 Objective Students test null and alternate hypothesis about two population proportions. 4 /30 Comparing
Last few slides from last time
Last few slides from last time Example 3: What is the probability that p will fall in a certain range, given p? Flip a coin 50 times. If the coin is fair (p=0.5), what is the probability of getting an
Chapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 8 Estimating with Confidence n 8.1 Confidence Intervals: The Basics n 8.2
Chapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
Objectives Simple linear regression. Statistical model for linear regression. Estimating the regression parameters
Objectives 10.1 Simple linear regression Statistical model for linear regression Estimating the regression parameters Confidence interval for regression parameters Significance test for the slope Confidence
Two-sample inference: Continuous data
Two-sample inference: Continuous data Patrick Breheny November 11 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data
Chapter 18. Sampling Distribution Models /51
Chapter 18 Sampling Distribution Models 1 /51 Homework p432 2, 4, 6, 8, 10, 16, 17, 20, 30, 36, 41 2 /51 3 /51 Objective Students calculate values of central 4 /51 The Central Limit Theorem for Sample
Intro to Confidence Intervals: A estimate is a single statistic based on sample data to estimate a population parameter Simplest approach But not always very precise due to variation in the sampling distribution
CENTRAL LIMIT THEOREM (CLT)
CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic
Ch. 7 Statistical Intervals Based on a Single Sample
Ch. 7 Statistical Intervals Based on a Single Sample Before discussing the topics in Ch. 7, we need to cover one important concept from Ch. 6. Standard error The standard error is the standard deviation
Statistics for IT Managers
Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample
Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals
Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals Lecture 9 Justin Kern April 9, 2018 Measuring Effect Size: Cohen s d Simply finding whether a
Matrices, Row Reduction of Matrices
Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only
Lecture 22/Chapter 19 Part 4. Statistical Inference Ch. 19 Diversity of Sample Proportions
Lecture 22/Chapter 19 Part 4. Statistical Inference Ch. 19 Diversity of Sample Proportions Probability versus Inference Behavior of Sample Proportions: Example Behavior of Sample Proportions: Conditions
The Mean Value Theorem Rolle s Theorem
The Mean Value Theorem In this section, we will look at two more theorems that tell us about the way that derivatives affect the shapes of graphs: Rolle s Theorem and the Mean Value Theorem. Rolle s Theorem
Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.
Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,
Inferential Statistics. Chapter 5
Inferential Statistics Chapter 5 Keep in Mind! 1) Statistics are useful for figuring out random noise from real effects. 2) Numbers are not absolute, and they can be easily manipulated. 3) Always scrutinize
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
Confidence intervals CE 311S
CE 311S PREVIEW OF STATISTICS The first part of the class was about probability. P(H) = 0.5 P(T) = 0.5 HTTHHTTTTHHTHTHH If we know how a random process works, what will we see in the field? Preview of
Inferential Statistics
Inferential Statistics Part 1 Sampling Distributions, Point Estimates & Confidence Intervals Inferential statistics are used to draw inferences (make conclusions/judgements) about a population from a sample.
Confidence Intervals. - simply, an interval for which we have a certain confidence.
Confidence Intervals I. What are confidence intervals? - simply, an interval for which we have a certain confidence. - for example, we are 90% certain that an interval contains the true value of something
Statistics and Quantitative Analysis U4320. Segment 5: Sampling and inference Prof. Sharyn O Halloran
Statistics and Quantitative Analysis U4320 Segment 5: Sampling and inference Prof. Sharyn O Halloran Sampling A. Basics 1. Ways to Describe Data Histograms Frequency Tables, etc. 2. Ways to Characterize
SECTION 3.5: DIFFERENTIALS and LINEARIZATION OF FUNCTIONS
(Section 3.5: Differentials and Linearization of Functions) 3.5.1 SECTION 3.5: DIFFERENTIALS and LINEARIZATION OF FUNCTIONS LEARNING OBJECTIVES Use differential notation to express the approximate change
5/15/2017. Sample Size and Power. Henry Glick May 12, Goals of Sample Size and Power Analysis
Sample Size and Power Henry Glick www.uphs.upenn.edu/dgimhsr May 1, 017 Goals of Sample Size and Power Analysis Sample size calculation Given a desired alpha (α) and beta (β), proactively manages probability
AP Statistics Ch 12 Inference for Proportions
Ch 12.1 Inference for a Population Proportion Conditions for Inference The statistic that estimates the parameter p (population proportion) is the sample proportion p ˆ. p ˆ = Count of successes in the
Ch Inference for Linear Regression
Ch. 12-1 Inference for Linear Regression ACT = 6.71 + 5.17(GPA) For every increase of 1 in GPA, we predict the ACT score to increase by 5.17. population regression line β (true slope) μ y = α + βx mean
Chapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.
Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:
Sampling Distributions
Sampling Error As you may remember from the first lecture, samples provide incomplete information about the population In particular, a statistic (e.g., M, s) computed on any particular sample drawn from
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
Two-sample inference: Continuous data
Two-sample inference: Continuous data Patrick Breheny April 6 Patrick Breheny University of Iowa to Biostatistics (BIOS 4120) 1 / 36 Our next several lectures will deal with two-sample inference for continuous
M(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
*Karle Laska s Sections: There is no class tomorrow and Friday! Have a good weekend! Scores will be posted in Compass early Friday morning
STATISTICS 100 EXAM 3 Spring 2016 PRINT NAME (Last name) (First name) *NETID CIRCLE SECTION: Laska MWF L1 Laska Tues/Thurs L2 Robin Tu Write answers in appropriate blanks. When no blanks are provided CIRCLE
Boyle s Law and Charles Law Activity
Boyle s Law and Charles Law Activity Introduction: This simulation helps you to help you fully understand 2 Gas Laws: Boyle s Law and Charles Law. These laws are very simple to understand, but are also
F = ma W = mg v = D t
Forces and Gravity Car Lab Name: F = ma W = mg v = D t p = mv Part A) Unit Review at D = f v = t v v Please write the UNITS for each item below For example, write kg next to mass. Name: Abbreviation: Units:
2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
Chapter 12: Inference about One Population
Chapter 1: Inference about One Population 1.1 Introduction In this chapter, we presented the statistical inference methods used when the problem objective is to describe a single population. Sections 1.
INTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
211 Real Analysis. f (x) = x2 1. x 1. x 2 1
Part. Limits of functions. Introduction 2 Real Analysis Eample. What happens to f : R \ {} R, given by f () = 2,, as gets close to? If we substitute = we get f () = 0 which is undefined. Instead we 0 might
Introduction to Design of Experiments
Introduction to Design of Experiments Jean-Marc Vincent and Arnaud Legrand Laboratory ID-IMAG MESCAL Project Universities of Grenoble {Jean-Marc.Vincent,Arnaud.Legrand}@imag.fr November 20, 2011 J.-M.
Chapter 16. Simple Linear Regression and Correlation
Chapter 16 Simple Linear Regression and Correlation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
Business Statistics. Lecture 10: Course Review
Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,
10.1. Comparing Two Proportions. Section 10.1
/6/04 0. Comparing Two Proportions Sectio0. Comparing Two Proportions After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET
CH 14 MORE DIVISION, SIGNED NUMBERS, & EQUATIONS
1 CH 14 MORE DIVISION, SIGNED NUMBERS, & EQUATIONS Division and Those Pesky Zeros O ne of the most important facts in all of mathematics is that the denominator (bottom) of a fraction can NEVER be zero.
1 Matched pair comparison(p430-)
[1] ST301(AKI) LEC 25 2010/11/30 ST 301 (AKI) LECTURE #25 1 Matched pair comparison(p430-) This has a quite different assumption (matched pair) from the other three methods. Remember LEC 32 page 1 example:
Confidence Intervals with σ unknown
STAT 141 Confidence Intervals and Hypothesis Testing 10/26/04 Today (Chapter 7): CI with σ unknown, t-distribution CI for proportions Two sample CI with σ known or unknown Hypothesis Testing, z-test Confidence
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:
7 Estimation. 7.1 Population and Sample (P.91-92)
7 Estimation MATH1015 Biostatistics Week 7 7.1 Population and Sample (P.91-92) Suppose that we wish to study a particular health problem in Australia, for example, the average serum cholesterol level for
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
Chapter 24. Comparing Means
Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing
Northwood High School Algebra 2/Honors Algebra 2 Summer Review Packet
Northwood High School Algebra 2/Honors Algebra 2 Summer Review Packet This assignment should serve as a review of the Algebra 1 skills necessary for success. Our hope is that this review will keep your
NOTES: EXPONENT RULES
NOTES: EXPONENT RULES DAY 2 Topic Definition/Rule Example(s) Multiplication (add exponents) x a x b = x a+b x 4 x 8 x 5 y 2 x 2 y Power to a Power (multiply exponents) x a ( ) b = x ab ( x ) 7 ( x ) 2
3.3 Estimator quality, confidence sets and bootstrapping
Estimator quality, confidence sets and bootstrapping 109 3.3 Estimator quality, confidence sets and bootstrapping A comparison of two estimators is always a matter of comparing their respective distributions.
Assessment Schedule 2011 Statistics and Modelling: Calculate confidence intervals for population parameters (90642)
NCEA Level 3 Statistics & Modelling (90642) 2011 page 1 of 5 Assessment Schedule 2011 Statistics and Modelling: Calculate confidence intervals for population parameters (90642) Evidence Statement Q Achievement
Chapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
Lecture 8 Hypothesis Testing
Lecture 8 Hypothesis Testing Taylor Ch. 6 and 10.6 Introduction l The goal of hypothesis testing is to set up a procedure(s) to allow us to decide if a mathematical model ("theory") is acceptable in light
Experiment 2 Random Error and Basic Statistics
PHY191 Experiment 2: Random Error and Basic Statistics 7/12/2011 Page 1 Experiment 2 Random Error and Basic Statistics Homework 2: turn in the second week of the experiment. This is a difficult homework
Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing
Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology
Week 11 Sample Means, CLT, Correlation
Week 11 Sample Means, CLT, Correlation Slides by Suraj Rampure Fall 2017 Administrative Notes Complete the mid semester survey on Piazza by Nov. 8! If 85% of the class fills it out, everyone will get a
Linear Regression with one Regressor
1 Linear Regression with one Regressor Covering Chapters 4.1 and 4.2. We ve seen the California test score data before. Now we will try to estimate the marginal effect of STR on SCORE. To motivate these
Ch 7: Dummy (binary, indicator) variables
Ch 7: Dummy (binary, indicator) variables :Examples Dummy variable are used to indicate the presence or absence of a characteristic. For example, define female i 1 if obs i is female 0 otherwise or male
Examine characteristics of a sample and make inferences about the population
Chapter 11 Introduction to Inferential Analysis Learning Objectives Understand inferential statistics Explain the difference between a population and a sample Explain the difference between parameter and
Chapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
Linear Classifiers and the Perceptron
Linear Classifiers and the Perceptron William Cohen February 4, 2008 1 Linear classifiers Let s assume that every instance is an n-dimensional vector of real numbers x R n, and there are only two possible
Chapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY
EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY HIGHER CERTIFICATE IN STATISTICS, 2013 MODULE 5 : Further probability and inference Time allowed: One and a half hours Candidates should answer THREE questions.
Ch18 links / ch18 pdf links Ch18 image t-dist table
Ch18 links / ch18 pdf links Ch18 image t-dist table ch18 (inference about population mean) exercises: 18.3, 18.5, 18.7, 18.9, 18.15, 18.17, 18.19, 18.27 CHAPTER 18: Inference about a Population Mean The
Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Probability and statistics: Module 25. Inference for means
1 Supporting Australian Mathematics Project 2 3 4 6 7 8 9 1 11 12 A guide for teachers Years 11 and 12 Probability and statistics: Module 2 Inference for means Inference for means A guide for teachers
Physics 225 Relativity and Math Applications. Fall Unit 7 The 4-vectors of Dynamics
Physics 225 Relativity and Math Applications Fall 2011 Unit 7 The 4-vectors of Dynamics N.C.R. Makins University of Illinois at Urbana-Champaign 2010 Physics 225 7.2 7.2 Physics 225 7.3 Unit 7: The 4-vectors
Lecture 8 Sampling Theory
Lecture 8 Sampling Theory Thais Paiva STA 111 - Summer 2013 Term II July 11, 2013 1 / 25 Thais Paiva STA 111 - Summer 2013 Term II Lecture 8, 07/11/2013 Lecture Plan 1 Sampling Distributions 2 Law of Large
SECTION 3.5: DIFFERENTIALS and LINEARIZATION OF FUNCTIONS
(Section 3.5: Differentials and Linearization of Functions) 3.5.1 SECTION 3.5: DIFFERENTIALS and LINEARIZATION OF FUNCTIONS LEARNING OBJECTIVES Use differential notation to express the approximate change
Basic Math Problems Unit 1
Basic Math Problems Unit 1 Name Period Using fractions: When you are using fractions in science, we need to convert them into decimals. You can do this by dividing the top number by the bottom number.
QUADRATICS 3.2 Breaking Symmetry: Factoring
QUADRATICS 3. Breaking Symmetry: Factoring James Tanton SETTING THE SCENE Recall that we started our story of symmetry with a rectangle of area 36. Then you would say that the picture is wrong and that
9. Linear Regression and Correlation
9. Linear Regression and Correlation Data: y a quantitative response variable x a quantitative explanatory variable (Chap. 8: Recall that both variables were categorical) For example, y = annual income,
Chapter 16. Simple Linear Regression and dcorrelation
Chapter 16 Simple Linear Regression and dcorrelation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will