Statistical Analysis of Chemical Data Chapter 4
|
|
- Arabella Brittany Ward
- 5 years ago
- Views:
Transcription
1 Statistical Analysis of Chemical Data Chapter 4
2 Random errors arise from limitations on our ability to make physical measurements and on natural fluctuations
3 Random errors arise from limitations on our ability to make physical measurements and on natural fluctuations
4 Histogram: (Bar Graph) Normal Curve: (Line Graph)
5 Data that vary because of random errors only will be normally distributed around a mean value. The distribution of random data around the mean is characterized by a Gaussian Distribution. Characteristics: Bell-shaped Center: Mean = Median = Mode Standard Deviation width of the distribution
6 SAMPLE vs. POPULATION Population is a set of entities concerning which statistical inferences are to be drawn. Sample is the subset of a manageable size of population. Statistics calculated from the sample are used to infer or extrapolate about the population. Population Sample
7 SAMPLE vs. POPULATION Population Mean ( ) - mean of entire population Sample Mean (x) mean of a given sample When N (usually 20-30) is big x When N is small there is a bigger deviation between x and
8 SAMPLE vs. POPULATION Population Standard Deviation ( ) measures the width of distribution of a population Sample Standard Deviation (s) applicable to finite samples When N (usually 20-30) is big s When N is small there is a bigger deviation between s and
9 SAMPLE vs. POPULATION
10 Standard Deviation and Probability
11 Confidence Interval Confidence Interval (CI) is a range of values within which there is a specified probability of finding the true mean If you only take a single measurement in a population then that single measurement will have a confidence interval of: If you take a lot of measurements, the mean of all the measurements will have a confidence interval of: µ = x ± zσ µ = x ± zσ n **NOTE: This is for cases where there is a good estimate of the population standard deviation (s ) or it is known.
12 Confidence Intervals
13 Confidence Intervals There is an incorrect notion of confidence interval: Given the true value and a specified confidence interval, the measurements will fall within in this interval at a certain probability The correct notion is that, Given the sample/population mean and a specified confidence interval, the true mean will fall in this confidence interval at a certain probability
14 Confidence Intervals Student s t is a statistical tool used to express confidence intervals We use t when we don t know the population standard deviation, the confidence interval can be estimated as:
15 Confidence Intervals Student s t is a statistical tool used to express confidence intervals We use t when we don t know the population standard deviation, the confidence interval can be estimated as: µ = x ± ts n
16 Hypothesis testing employs Student s t statistics Student s t can be used to compare two sets of measurements to decided whether they are the same or different CASE 1: Comparing measured value to theoretical value CASE 2: Comparing replicate sets of measurements (with different means and standard deviations) CASE 3: Comparing paired data
17 Hypothesis testing employs Student s t statistics CASE 1: Comparing measured value to theoretical value We measure a quantity several times, obtaining an average value and a standard deviation. We need to compare our answer with a known, accepted answer. The average does not agree exactly with the accepted answer. Does our measured answer agree or disagree with the known value within experimental error? Null Hypothesis (H 0 ): x = 0 Use the t-statistic: Alternative Hypothesis (H a ): x 0 (two-tailed) if t calc = x µ s n t calc t table or t calc t table x < 0 (one-tailed) if x > 0 (one-tailed) if t calc t table t calc t table
18 Hypothesis testing employs Student s t statistics CASE 1: Comparing measured value to theoretical value EXAMPLE 1. A new procedure for the rapid determination of the percentage of sulfur in kerosene was tested on a sample known from its method of preparation to contain 0.123% Sulfur. The results were % S= 0.112, 0.118, and Do the data indicate that there is a bias in the method at the 95% confidence interval?
19 Hypothesis testing employs Student s t statistics CASE 1: Comparing measured value to theoretical value EXAMPLE 2. Sewage and industrial pollutants dumped into a body of water can reduce the dissolved oxygen concentration and adversely affect aquatic species. In one study, weekly readings are taken from the same location in a river over a 2-month period (see table). Some scientists think that 5.0 ppm is a dissolved O 2 level that is marginal for fish to live. Conduct a statistical test to determine whether the mean dissolved O 2 concentration is less than 5.0 ppm at 95% confidence level. Week Dissolved O 2, ppm
20 Hypothesis testing employs Student s t statistics CASE 2: Comparing replicate measurements We measure a quantity multiple times by two different Methods that give two different answers, each with its own standard deviation. Do the two results agree with each other within experimental error, or do they disagree? Null Hypothesis (H 0 ): x 1 = x 2 Use the t-statistic: t calc = x 1 x 2 s pooled n 1 n 2 n 1 + n 2 Alternative Hypothesis (H a ): 0 if t calc s pooled = s 2 1 n 1 1 t table ( ) + s 2 ( 2 n 2 1) n 1 + n 2 2
21 Hypothesis testing employs Student s t statistics CASE 2: Comparing replicate measurements We measure a quantity multiple times by two different Methods that give two different answers, each with its own standard deviation. Do the two results agree with each other within experimental error, or do they disagree?
22 Hypothesis testing employs Student s t statistics CASE 2: Comparing replicate measurement EXAMPLE 3. Lord Rayleigh measured the mass of dry air (O 2 -free) and chemically generated N 2 of the same volume. Is dry air the same as chemically generated N 2? From air (g) Average From chemical composition (g) Standard Deviation
23 Hypothesis testing employs Student s t statistics CASE 2: Comparing replicate measurement EXAMPLE 3. Lord Rayleigh measured the mass of dry air (O 2 -free) and chemically generated N 2 of the same volume. Is dry air the same as chemically generated N 2?
24 Hypothesis testing employs Student s t statistics CASE 2: Comparing replicate measurement EXAMPLE 4. A reliable assay of ATP in a certain type of cell gives a value of mol/100 ml, with a standard deviation of 2.8 in four replicate measurements. You have developed a new assay which gave the following values in replicate analyses: 117, , 115, 120 mol/100 ml a). Find the mean and standard deviation of your new analysis b). Can you be 95% confident that your method produces a result different from the reliable value?
25 Hypothesis testing employs Student s t statistics CASE 3: Comparing paired data Sample 1 is measured once by Method A and once by Method B, which do not give exactly the same result. Then a different sample, designated as sample 2, is also measured once by Method A and once by Method B; and again the results are not exactly equal. The procedure is repeated for n different samples. Do the two methods agree with each other within experimental error, or is one systematically different from the other? Null Hypothesis (H 0 ): d = 0 often times 0 = 0 Use the t-statistic: Alternative Hypothesis (H a ): 0 (two-tailed) if t calc = d 0 s d n t calc t table or t calc t table
26 Hypothesis testing employs Student s t statistics CASE 3: Comparing paired data EXAMPLE 5. A new automated procedure for determining glucose in serum (Method A) is to be compared with an established method (Method B). Both methods are performed on the serum from six patients to eliminate patient-to-patient variability. Do the following results confirm a difference in the two methods at 95% confidence level? Method A, mg/l Method B, mg/l Difference, mg/l
27 Hypothesis testing employs Student s t statistics CASE 3: Comparing paired data EXAMPLE 6. Two different analytical methods were used to determine residual chlorine in sewage effluents. Both methods were used on the same samples, but each sample came from various locations, with differing amounts of contact time. The concentration of Cl in mg/l are given in the Table. Do the two methods give different results for 90%, 95% and 99% confidence levels? Sample Method A Method B
28 Dealing with BAD DATA Bad data are due to GROSS ERRORS, and result in outliers. We use Q test to determine whether we can reject or we need to retain an outlier. Q = x (questionable data ) x nearest neighbor spread Q calc > Q table Discard data EXAMPLE 7. The analysis of calcite sample yielded % CaO of 55.95, 56.00, 56.04, and The last value appears anomalous; should it be retained or discarded at 95% confidence level?
Statistics: Error (Chpt. 5)
Statistics: Error (Chpt. 5) Always some amount of error in every analysis (How much can you tolerate?) We examine error in our measurements to know reliably that a given amount of analyte is in the sample
More informationBasic Statistics. 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation).
Basic Statistics There are three types of error: 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation). 2. Systematic error - always too high or too low
More informationTopic 2 Measurement and Calculations in Chemistry
Topic Measurement and Calculations in Chemistry Nature of Measurement Quantitative observation consisting of two parts. number scale (unit) Examples 0 grams 6.63 10 34 joule seconds The Fundamental SI
More informationCh18 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
More informationBusiness 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,
More informationLecture 3. - all digits that are certain plus one which contains some uncertainty are said to be significant figures
Lecture 3 SIGNIFICANT FIGURES e.g. - all digits that are certain plus one which contains some uncertainty are said to be significant figures 10.07 ml 0.1007 L 4 significant figures 0.10070 L 5 significant
More information4.1 Hypothesis Testing
4.1 Hypothesis Testing z-test for a single value double-sided and single-sided z-test for one average z-test for two averages double-sided and single-sided t-test for one average the F-parameter and F-table
More informationBasic Statistics. 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation).
Basic Statistics There are three types of error: 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation). 2. Systematic error - always too high or too low
More informationHow to Describe Accuracy
OK, so what s s the speed of dark? When everything is coming your way, you're obviously in the wrong lane MARS 450 Thursday, Feb 14 2008 A) Standard deviation B) Student s t-test - Test of a mean C) Q-test
More information-However, this definition can be expanded to include: biology (biometrics), environmental science (environmetrics), economics (econometrics).
Chemometrics Application of mathematical, statistical, graphical or symbolic methods to maximize chemical information. -However, this definition can be expanded to include: biology (biometrics), environmental
More informationChapter 7. Inference for Distributions. Introduction to the Practice of STATISTICS SEVENTH. Moore / McCabe / Craig. Lecture Presentation Slides
Chapter 7 Inference for Distributions Introduction to the Practice of STATISTICS SEVENTH EDITION Moore / McCabe / Craig Lecture Presentation Slides Chapter 7 Inference for Distributions 7.1 Inference for
More informationChemometrics. Matti Hotokka Physical chemistry Åbo Akademi University
Chemometrics Matti Hotokka Physical chemistry Åbo Akademi University Hypothesis testing Inference method Confidence levels Descriptive statistics Hypotesis testing Predictive statistics Hypothesis testing
More informationAIM HIGH SCHOOL. Curriculum Map W. 12 Mile Road Farmington Hills, MI (248)
AIM HIGH SCHOOL Curriculum Map 2923 W. 12 Mile Road Farmington Hills, MI 48334 (248) 702-6922 www.aimhighschool.com COURSE TITLE: Statistics DESCRIPTION OF COURSE: PREREQUISITES: Algebra 2 Students will
More informationPSY 305. Module 3. Page Title. Introduction to Hypothesis Testing Z-tests. Five steps in hypothesis testing
Page Title PSY 305 Module 3 Introduction to Hypothesis Testing Z-tests Five steps in hypothesis testing State the research and null hypothesis Determine characteristics of comparison distribution Five
More informationChapter 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,
More informationChapter 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
More informationSingle Sample Means. SOCY601 Alan Neustadtl
Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size
More informationSTA 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
More informationSection 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples
Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means
More informationCBA4 is live in practice mode this week exam mode from Saturday!
Announcements CBA4 is live in practice mode this week exam mode from Saturday! Material covered: Confidence intervals (both cases) 1 sample hypothesis tests (both cases) Hypothesis tests for 2 means as
More informationLecture 26: Chapter 10, Section 2 Inference for Quantitative Variable Confidence Interval with t
Lecture 26: Chapter 10, Section 2 Inference for Quantitative Variable Confidence Interval with t t Confidence Interval for Population Mean Comparing z and t Confidence Intervals When neither z nor t Applies
More informationThe Normal Distribution. Chapter 6
+ The Normal Distribution Chapter 6 + Applications of the Normal Distribution Section 6-2 + The Standard Normal Distribution and Practical Applications! We can convert any variable that in normally distributed
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationBackground to Statistics
FACT SHEET Background to Statistics Introduction Statistics include a broad range of methods for manipulating, presenting and interpreting data. Professional scientists of all kinds need to be proficient
More informationHow do we compare the relative performance among competing models?
How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More informationANALYTICAL CHEMISTRY - CLUTCH 1E CH STATISTICS, QUALITY ASSURANCE AND CALIBRATION METHODS
!! www.clutchprep.com CONCEPT: MEAN EVALUATION The measures how close data results are in relation to the mean or average value. s = i (x i x) n 1 = Individual Measurement = Average or Mean = variance
More informationSign test. Josemari Sarasola - Gizapedia. Statistics for Business. Josemari Sarasola - Gizapedia Sign test 1 / 13
Josemari Sarasola - Gizapedia Statistics for Business Josemari Sarasola - Gizapedia 1 / 13 Definition is a non-parametric test, a special case for the binomial test with p = 1/2, with these applications:
More information11: Comparing Group Variances. Review of Variance
11: Comparing Group Variances Review of Variance Parametric measures of variability are often based on sum of squares (SS) around e mean: (1) For e data set {3, 4, 5, 8}, = 5 and SS = (3 5) + (4 5) + (5
More informationMedian Statistics Analysis of Non- Gaussian Astrophysical and Cosmological Data Compilations
Median Statistics Analysis of Non- Gaussian Astrophysical and Cosmological Data Compilations Amber Thompson Mentor: Dr. Bharat Ratra Graduate Student: Tia Camarillo Background Motivation Scientific integrity
More informationProbability 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
More informationy n 1 ( x i x )( y y i n 1 i y 2
STP3 Brief Class Notes Instructor: Ela Jackiewicz Chapter Regression and Correlation In this chapter we will explore the relationship between two quantitative variables, X an Y. We will consider n ordered
More informationGlossary. The ISI glossary of statistical terms provides definitions in a number of different languages:
Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the
More informationTables Table A Table B Table C Table D Table E 675
BMTables.indd Page 675 11/15/11 4:25:16 PM user-s163 Tables Table A Standard Normal Probabilities Table B Random Digits Table C t Distribution Critical Values Table D Chi-square Distribution Critical Values
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationMICROPIPETTE CALIBRATIONS
Physics 433/833, 214 MICROPIPETTE CALIBRATIONS I. ABSTRACT The micropipette set is a basic tool in a molecular biology-related lab. It is very important to ensure that the micropipettes are properly calibrated,
More informationPrentice Hall Stats: Modeling the World 2004 (Bock) Correlated to: National Advanced Placement (AP) Statistics Course Outline (Grades 9-12)
National Advanced Placement (AP) Statistics Course Outline (Grades 9-12) Following is an outline of the major topics covered by the AP Statistics Examination. The ordering here is intended to define the
More informationPhysics 509: Bootstrap and Robust Parameter Estimation
Physics 509: Bootstrap and Robust Parameter Estimation Scott Oser Lecture #20 Physics 509 1 Nonparametric parameter estimation Question: what error estimate should you assign to the slope and intercept
More informationInference for Regression Inference about the Regression Model and Using the Regression Line, with Details. Section 10.1, 2, 3
Inference for Regression Inference about the Regression Model and Using the Regression Line, with Details Section 10.1, 2, 3 Basic components of regression setup Target of inference: linear dependency
More informationStatistical 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
More informationMath Review Sheet, Fall 2008
1 Descriptive Statistics Math 3070-5 Review Sheet, Fall 2008 First we need to know about the relationship among Population Samples Objects The distribution of the population can be given in one of the
More informationPurposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions
Part 1: Probability Distributions Purposes of Data Analysis True Distributions or Relationships in the Earths System Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution
More informationIntroduction 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.
More informationSTAT Chapter 8: Hypothesis Tests
STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More information4.12 Sampling Distributions 183
4.12 Sampling Distributions 183 FIGURE 4.19 Sampling distribution for y Example 4.22 illustrates for a very small population that we could in fact enumerate every possible sample of size 2 selected from
More informationHYPOTHESIS TESTING. Hypothesis Testing
MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.
More informationChapter 7 Comparison of two independent samples
Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N
More informationHarris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR
Harris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR Chapter 3. Experimental Error -There is error associated with every measurement. -There is no way to measure the true
More informationChapter 1 Statistical Inference
Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations
More informationReview of Statistics 101
Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods
More informationChapter 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
More informationM(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)
More informationHarris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR
Harris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR Chapter 3. Experimental Error -There is error associated with every measurement. -There is no way to measure the true
More informationBusiness Statistics MEDIAN: NON- PARAMETRIC TESTS
Business Statistics MEDIAN: NON- PARAMETRIC TESTS CONTENTS Hypotheses on the median The sign test The Wilcoxon signed ranks test Old exam question HYPOTHESES ON THE MEDIAN The median is a central value
More informationGAISE Framework 3. Formulate Question Collect Data Analyze Data Interpret Results
Project-SET Variability Final Learning Trajectory 1,2 Loop 1 4 Concept of a Distribution GAISE Framework 3 Formulate Question Collect Data Analyze Data Interpret Results a. How can we discover a. Describe
More informationElementary Statistics Triola, Elementary Statistics 11/e Unit 17 The Basics of Hypotheses Testing
(Section 8-2) Hypotheses testing is not all that different from confidence intervals, so let s do a quick review of the theory behind the latter. If it s our goal to estimate the mean of a population,
More informationThe Purpose of Hypothesis Testing
Section 8 1A:! An Introduction to Hypothesis Testing The Purpose of Hypothesis Testing See s Candy states that a box of it s candy weighs 16 oz. They do not mean that every single box weights exactly 16
More informationDover- Sherborn High School Mathematics Curriculum Probability and Statistics
Mathematics Curriculum A. DESCRIPTION This is a full year courses designed to introduce students to the basic elements of statistics and probability. Emphasis is placed on understanding terminology and
More informationInferential 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.
More informationGROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION
FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89
More informationStatistics 4. Experimental measurements always contain some variability, so no conclusion can be. Is My Red Blood Cell Count High Today?
Statistics 4 Is My Red Blood Cell Count High Today? Red blood cells (erythrocytes, Er) tangled in fibrin threads (Fi) in a blood clot. Stacks of erythrocytes in a clot are called a rouleaux formation (Ro).
More information9/2/2010. Wildlife Management is a very quantitative field of study. throughout this course and throughout your career.
Introduction to Data and Analysis Wildlife Management is a very quantitative field of study Results from studies will be used throughout this course and throughout your career. Sampling design influences
More informationNull Hypothesis Significance Testing p-values, significance level, power, t-tests Spring 2017
Null Hypothesis Significance Testing p-values, significance level, power, t-tests 18.05 Spring 2017 Understand this figure f(x H 0 ) x reject H 0 don t reject H 0 reject H 0 x = test statistic f (x H 0
More informationSampling, Confidence Interval and Hypothesis Testing
Sampling, Confidence Interval and Hypothesis Testing Christopher Grigoriou Executive MBA HEC Lausanne 2007-2008 1 Sampling : Careful with convenience samples! World War II: A statistical study to decide
More informationOriginality in the Arts and Sciences: Lecture 2: Probability and Statistics
Originality in the Arts and Sciences: Lecture 2: Probability and Statistics Let s face it. Statistics has a really bad reputation. Why? 1. It is boring. 2. It doesn t make a lot of sense. Actually, the
More informationVocabulary: Samples and Populations
Vocabulary: Samples and Populations Concept Different types of data Categorical data results when the question asked in a survey or sample can be answered with a nonnumerical answer. For example if we
More informationDensity Temp vs Ratio. temp
Temp Ratio Density 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Density 0.0 0.2 0.4 0.6 0.8 1.0 1. (a) 170 175 180 185 temp 1.0 1.5 2.0 2.5 3.0 ratio The histogram shows that the temperature measures have two peaks,
More informationInference for the Regression Coefficient
Inference for the Regression Coefficient Recall, b 0 and b 1 are the estimates of the slope β 1 and intercept β 0 of population regression line. We can shows that b 0 and b 1 are the unbiased estimates
More informationHypothesis testing: Steps
Review for Exam 2 Hypothesis testing: Steps Repeated-Measures ANOVA 1. Determine appropriate test and hypotheses 2. Use distribution table to find critical statistic value(s) representing rejection region
More informationPolitical Science 236 Hypothesis Testing: Review and Bootstrapping
Political Science 236 Hypothesis Testing: Review and Bootstrapping Rocío Titiunik Fall 2007 1 Hypothesis Testing Definition 1.1 Hypothesis. A hypothesis is a statement about a population parameter The
More informationFinal Exam - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis March 19, 2010 Instructor: John Parman Final Exam - Solutions You have until 5:30pm to complete this exam. Please remember to put your
More informationEstimating the accuracy of a hypothesis Setting. Assume a binary classification setting
Estimating the accuracy of a hypothesis Setting Assume a binary classification setting Assume input/output pairs (x, y) are sampled from an unknown probability distribution D = p(x, y) Train a binary classifier
More informationSurvey on Population Mean
MATH 203 Survey on Population Mean Dr. Neal, Spring 2009 The first part of this project is on the analysis of a population mean. You will obtain data on a specific measurement X by performing a random
More informationQuestions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.
Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized
More informationUCLA STAT 10 Statistical Reasoning - Midterm Review Solutions Observational Studies, Designed Experiments & Surveys
UCLA STAT 10 Statistical Reasoning - Midterm Review Solutions Observational Studies, Designed Experiments & Surveys.. 1. (i) The treatment being compared is: (ii). (5) 3. (3) 4. (4) Study 1: the number
More informationError Analysis, Statistics and Graphing Workshop
Error Analysis, Statistics and Graphing Workshop Percent error: The error of a measurement is defined as the difference between the experimental and the true value. This is often expressed as percent (%)
More informationDescriptive Statistics
Descriptive Statistics Once an experiment is carried out and the results are measured, the researcher has to decide whether the results of the treatments are different. This would be easy if the results
More informationStat 427/527: Advanced Data Analysis I
Stat 427/527: Advanced Data Analysis I Review of Chapters 1-4 Sep, 2017 1 / 18 Concepts you need to know/interpret Numerical summaries: measures of center (mean, median, mode) measures of spread (sample
More informationChapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
More informationTwo Sample Hypothesis Tests
Note Packet #21 Two Sample Hypothesis Tests CEE 3710 November 13, 2017 Review Possible states of nature: H o and H a (Null vs. Alternative Hypothesis) Possible decisions: accept or reject Ho (rejecting
More informationAdvanced Experimental Design
Advanced Experimental Design Topic Four Hypothesis testing (z and t tests) & Power Agenda Hypothesis testing Sampling distributions/central limit theorem z test (σ known) One sample z & Confidence intervals
More information(Re)introduction to statistics: dusting off the cobwebs
(Re)introduction to statistics: dusting off the cobwebs Vicki Barwick LGC Aoife Morrin Insight Centre for Data Analysis DCU Data Quality, analysis and integrity workshop Dublin Castle 14-15 May 018 Overview
More informationLecture 26. December 19, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University.
s Sign s Lecture 26 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University December 19, 2007 s Sign s 1 2 3 s 4 Sign 5 6 7 8 9 10 s s Sign 1 Distribution-free
More informationCENTRAL 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
More informationSTAB57: Quiz-1 Tutorial 1 (Show your work clearly) 1. random variable X has a continuous distribution for which the p.d.f.
STAB57: Quiz-1 Tutorial 1 1. random variable X has a continuous distribution for which the p.d.f. is as follows: { kx 2.5 0 < x < 1 f(x) = 0 otherwise where k > 0 is a constant. (a) (4 points) Determine
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More informationHomework Assignment - Chapter 4 - Fall 2011
59-320 Homework - Chapter 4 - Fall 2011 On differences between the 7th and 8th edition: In 4-A in the 7th edition, you are asked to perform a Q test instead of a Grubbs test; the former is no longer discussed
More informationCS 5014: Research Methods in Computer Science. Bernoulli Distribution. Binomial Distribution. Poisson Distribution. Clifford A. Shaffer.
Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2015 by Clifford A. Shaffer Computer Science Title page Computer Science Clifford A. Shaffer Fall 2015 Clifford A. Shaffer
More information2.0 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
More informationWarm-up Using the given data Create a scatterplot Find the regression line
Time at the lunch table Caloric intake 21.4 472 30.8 498 37.7 335 32.8 423 39.5 437 22.8 508 34.1 431 33.9 479 43.8 454 42.4 450 43.1 410 29.2 504 31.3 437 28.6 489 32.9 436 30.6 480 35.1 439 33.0 444
More informationThe Difference in Proportions Test
Overview The Difference in Proportions Test Dr Tom Ilvento Department of Food and Resource Economics A Difference of Proportions test is based on large sample only Same strategy as for the mean We calculate
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors
More informationChapter 7 Sampling Distributions
Statistical inference looks at how often would this method give a correct answer if it was used many many times. Statistical inference works best when we produce data by random sampling or randomized comparative
More informationChapter 23. Inference About Means
Chapter 23 Inference About Means 1 /57 Homework p554 2, 4, 9, 10, 13, 15, 17, 33, 34 2 /57 Objective Students test null and alternate hypotheses about a population mean. 3 /57 Here We Go Again Now that
More informationIntroduction to Statistics and Data Analysis
Introduction to Statistics and Data Analysis RSI 2005 Staff July 15, 2005 Variation and Statistics Good experimental technique often requires repeated measurements of the same quantity These repeatedly
More informationMidterm 2 - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis February 24, 2010 Instructor: John Parman Midterm 2 - Solutions You have until 10:20am to complete this exam. Please remember to put
More informationExperimental design. Matti Hotokka Department of Physical Chemistry Åbo Akademi University
Experimental design Matti Hotokka Department of Physical Chemistry Åbo Akademi University Contents Elementary concepts Regression Validation Hypotesis testing ANOVA PCA, PCR, PLS Clusters, SIMCA Design
More informationSociology 6Z03 Review I
Sociology 6Z03 Review I John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review I Fall 2016 1 / 19 Outline: Review I Introduction Displaying Distributions Describing
More informationEC2001 Econometrics 1 Dr. Jose Olmo Room D309
EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population:
More informationOutline. PubH 5450 Biostatistics I Prof. Carlin. Confidence Interval for the Mean. Part I. Reviews
Outline Outline PubH 5450 Biostatistics I Prof. Carlin Lecture 11 Confidence Interval for the Mean Known σ (population standard deviation): Part I Reviews σ x ± z 1 α/2 n Small n, normal population. Large
More information