Chapter 12: Chi-Square Procedures
|
|
- Rosaline Hancock
- 7 years ago
- Views:
Transcription
1 January 6, 2010
2 Chapter Outline 12.1 The Chi-Square Distribution 12.2 Chi-Square Goodness-of-Fit Test 12.3 Contingency Tables: Association 12.4 Chi-Square Test of Independence
3 General Objective: In Chapter 11 in Section 11.2 we learned to test hypothesis regarding a single population proportion. For example, we looked at the problem of Playing Hooky From Work. In that example each person either plays Hooky or does not plays Hooky. For each person there are only two possible responses. These are known as binary response. In many other situations outcomes can be classified in more than two groups. These are known as multinomial response. In Section 12.2, we will extend the hypothesis testing methods of Section 11.2 to accommodate multinomial responses. This is known as Test of Goodness-of-Fit. In previous chapters, we performed statistical inference for only one variable. In Section 12.3 we will learn the technique of assessing the Association between two variables. Finally in Section 12.4 we will learn to test for the independence of two variables. For example, suppose a violent crime has been committed. It can
4 12.1 The Chi-Square Distribution So far we have used standard normal distribution, N(0,1) and the t distribution for confidence interval and hypothesis testing. In this chapter we will need to use another probability distribution which is known as Chi-square distribution. The Chi-square random variable is always positive. Like t distribution, Chi-square distribution also depends on degrees of freedom. Selected critical values for various degreees of freedom are listed in Table V. Figure 12.1 below shows how the curve changes when the degrees of freedom changes.
5 As the degrees of freedom increases the curve looks more and more symmetric.
6 The grapgh shows the critical value for α = Table V lists the critical values for selected values of α StatCrunch can be used to get these Follow: Stat > Calculator > Chisquare.
7 Change the entries in the graph on the left. Enter 9 in DF box Select => in the left box on the second row Enter the value of t o in the middle box. Say Press Compute. You should see the graph on the right. The area of the shaded region will appear in the last box.
8 12.2 Chi-Square Gooness-of-Fit Test Controlling Road Rage: Road rage is defined as.. an incident in which an angry motorist tries to harm another motorist. Suppose we want to find out whether any particular day of the week is more succeptible to road rage compared to other days of the week. This can be formulated as hypothesis test as follows: H 0 : It is equally likely to happen on any day of the week. H a : It is NOT equally likely to happen on any day of the week.
9 Suppose we define p m = chance that it will happen on Monday p t = chance that it will happen on Tuesday p w = chance that it will happen on Wednesday p th = chance that it will happen on Thursday p f = chance that it will happen on Friday p sa = chance that it will happen on Saturday p su = chance that it will happen on Sunday We can state the null hypothesis as: H 0 : p m = p t = p w = p th = p f = p sa = p su = 1/7. Since p m + p t + p w + p th + p f + p sa + p su = 1, and all are equal, each one must be = 1/7. This is similar to testing H 0 : p = p 0 in one proportion case as we have seen in Chapter 11.
10 Road Rage Sample data: A random sample of 69 cases were taken from all police records. The reults are summarized below. Day Observed Count Null Hypothesis Expected Count M 5 p m = 1/ Tu 11 p t = 1/ W 12 p w = 1/ Th 11 p th = 1/ F 18 p f = 1/ Sa 7 p sa = 1/ Su 5 p su = 1/ Total If the null hypothesis is true then we should expect equal number of cases on each day which = 69/7 =
11 To verify the null hypothesis we need to compare the observed counts with the expected counts. Observed Expected Square of Chi-Square Day Freq. Freq. Diff. Diff. sub-total O E O - E (O E) 2 (O E) 2 /E M T W Th F Sa Su If the Null Hypothesis is true then in we should expect O E in each category.
12 To measure the discrepency between observed and expected counts the difference, (O E), is calculated. Since the differences could be positive or negative, they are squared. Finally the squraed difference (O E) 2 is standardized by diving by E. Sum of these gives a overall measure of discrepency between observed and expected counts. This denoted by Chi-Square = χ 2 = (O E) 2 /E = If the hypothesis is true the we expect O E. Hence each term in the last column should be small. Hence the total should be small. If the hypothesis is NOT true then at least one (O E) 2 /E will be large. Hence the total will be a large positive quantity. This suggest the rejection rule: Reject H 0 : if χ 2 = (O E) 2 /E χ 2 α, where χ 2 α is obtained from the Chi-Square distribution.
13 The df in this case is (7-1) =6 and χ 2 05 = Since the observed value > , the null hyppothesis is rejected. Conclusion: There is enough evidence to conclude that some days are more succesptible to road rage than others. Use StatCrunch to compute the P value: Open StatCrunch Follow: Stat > Calculator > Chi-Square Enter DF = 6, Select and Enter in the middle box. Press Calculate In this case P value = P(χ ) = Sonce P value < 0.05 reject H 0. Also see the instruction in Section 12.1 for calculating P values.
14 Example 12.2: Violent Crimes: Table 12.1 below shows the relative frequency of four different crimes in the year Table 12.2 below shows the frequency distribution of 500 randomly selected violent crime cases in We want to test whether the year 2000 proportions are still valid in the year Table 12.1 (Yr 2000) Table 12.2 (Yr 2008) Type of Relative Type of Violent Crime Frequency Violent Crime Frequency Murder Murder 3 Forcible Rape Forcible Rape 37 Robbery Robbery 154 Agg Assault Agg. Assault
15 Set up the Null Hypothesis: Define p M = Probability of committing a Murder p F = Probability of committing a Forcible Rape p R = Probability of committing a Robbery p A = Probability of committing an Agg. Assault Expected Counts H 0 : p m = E 1 = = 5.5 p F = E 2 = = 31.5 p R = E 3 = = p A = E 4 = = The first column specifies the null hypothesis. The second shows the expected counts in each category out of 500 cases if the null hypothesis is true. The expected counts will be compared to observed counts to test H 0.
16 Type of Observed Expected Square of Chi-Square Crime Freq. Freq. Diff. Diff. sub-total O E = np O - E (O E) 2 (O E) 2 /E Murder Rape Robbery Assault If the Null Hypothesis is true then in we should expect O E in each category. To measure the discrepency between observed and expected counts the difference, (O E), is calculated. Since the differences could be positive or negative, they are squared. Finally the squraed difference (O E) 2 is standardized by diving by E.
17 Sum of these gives a overall measure of discrepency between observed and expected counts. This denoted by Chi-Square = χ 2 = (O E) 2 /E = If the hypothesis is true the we expect O E. Hence each term in the last column should be small. Hence the total should be small. If the hypothesis is NOT true then at least one (O E) 2 /E will be large. Hence the total will be a large positive quantity. This suggest the rejection rule: Reject H 0 : if χ 2 = (O E) 2 /E χ 2 α, where χ 2 α is obtained from the Chi-Square distribution. The degrees of freedom = (k-1), k= number of groups. P value = P(χ ) = 0.314) (using StatCrunch). Since P value > 0.05, do not reject H 0. Conclusion: The crime rates remained same in 2008.
18 Note: The χ 2 procedures are approximate procedure. For the approximation to be valid all expected counts should be at least 1. At most 20% expected frequencies could be less than 5 Two rules above are adhoc rules. For Goodness-of-Fit test the expected frequencies are calculated as E = np for each category. The degrees of freedom = (k-1), k= number of groups.
19 Steps in Computing χ 2 Observed Specified Expected Square of Chi-Square Freq. Prop Freq. Diff. Diff. sub-total O p E = np O - E (O E) 2 (O E) 2 /E n 1 n 0 χ 2 = State the null hypothesis with specified proportions. Complete the entries in the table above. Decide the degrees of freedom = (k-1)=(# of groups) -1 Compute χ 2 α or P value Make a decision.
20 12.3 Association in Contingency Table:
21 The previous page shows the information collected from a randomly selected sample of 40 students. For each student two pieces of information were collected, political party affiliation and class level. Goal is to find out if the these two characteristics are associated or dependent. The information is summarized in a two-way table, also called contingency table.
22 The two-way frequency table shows the joint distribution of the two characteristics. To assess the association we can compare the any two columns to see if the frequency distribution are similar. However, due to unequal column total direct comparision of frequencies in the columns is NOT appropriate. To remedy this, we make total frquency in column to 1. This will put the column comparisions on equal footing.
23 If class level is NOT associated with political party then proportions in column Freshman should be similar to proportion in Sophomore and so on. If these proportions differ substantially from column to column then they are associated. These information can also be presented in a graphical form. Segmented bar graph is used here. One bar for each column. In this case bar graphs for Junior and Senior look different from those of Freshman and Sophomore.
24 Conclusion: Political Party and Class level are associated.
25 Output 12.2 Using StatCrunch Open StatCrunch and load the data from Table Follow: Stat > Tables > Contingency > with data Select a column for row variable and for column variable and press next. Select appropriate items from the list. For this problem Select column percentage.
26 12.3 Chi-Square Independent Test Graphical method presented in last section does NOT provide a quantitative method for evaluating the association between two characteristics. In this section we describe a way to calculate χ 2 statistic that will be used evaluate the dependence between any two characteristics. To test whether Drinking habit is related to Marital status, 1772 individulas were interviewed. The information is summarized in next table.
27 Example: 12.9 Marital Status and Drinking. The following two-way frequency table was based on a sample of 1772 individuals. The counts in each cell is the observed frequency count. We need to calculate the expected count in each cell when the two characteristics are independent. The mathematical formulation is beyond the scope of this class. We will simply present the formula to calculate the expected frequency in each cell. E = (R C)/n, where, R = Row total, C = column total, n = Grand total. Next table shows both observed and expected counts.
28 The observed count in the first cell is 67. To calculate the corresponding expected count: Row total corresponding to 67 is R = 354 Column total corresponding to 67 is C = 590 Grand total n = 1772 Hence E = (R C)/n = ( )/1772 = This formula is applied to all cells to get the expected counts.
29 To compare the observed counts with the expected counts we follow the same method as in Goodness-of-Fit Test. This means calculate (O E) 2 /E for each cell in the table. Then add these quantities. χ 2 = [ (O E) 2 E ] The degrees of freedom in this case: df = (number of rows -1 )(number of columns -1 )=(r-1)(c-1) Reject H 0 if χ 2 χ 2 α. The critical value is obtained from the table with df = (r-1)(c-1).
30 Example Chi-square calculations are shown at the bottom of the table. For first cell: O = 67, E = Hence (O E) 2 /E = ( ) 2 / = Similarly for other cells.
31 χ 2 value is df = (r-1)(c-1) = (4-1)(3-1) = 6. Hence P value = P(χ ) = The null hypothesis is: H 0 : Drinking habit and marital status are independent. Reject null hypothesis since P value < 0.05.
32 Output 12.4 Using StatCrunch Open StatCrunch and load the data from Table Follow: Stat > Tables > Contingency > with summary Select: columns for data in the table Select: column for Row labels and press Next Select: expected counts from the list. Press calculate Note: StatCrunch will not give individual (O E) 2 /E. It will give the calculated value of χ 2 = [ (O E) 2 /E ]
Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationChapter 23. Two Categorical Variables: The Chi-Square Test
Chapter 23. Two Categorical Variables: The Chi-Square Test 1 Chapter 23. Two Categorical Variables: The Chi-Square Test Two-Way Tables Note. We quickly review two-way tables with an example. Example. Exercise
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationOdds ratio, Odds ratio test for independence, chi-squared statistic.
Odds ratio, Odds ratio test for independence, chi-squared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationRecommend Continued CPS Monitoring. 63 (a) 17 (b) 10 (c) 90. 35 (d) 20 (e) 25 (f) 80. Totals/Marginal 98 37 35 170
Work Sheet 2: Calculating a Chi Square Table 1: Substance Abuse Level by ation Total/Marginal 63 (a) 17 (b) 10 (c) 90 35 (d) 20 (e) 25 (f) 80 Totals/Marginal 98 37 35 170 Step 1: Label Your Table. Label
More informationTest Positive True Positive False Positive. Test Negative False Negative True Negative. Figure 5-1: 2 x 2 Contingency Table
ANALYSIS OF DISCRT VARIABLS / 5 CHAPTR FIV ANALYSIS OF DISCRT VARIABLS Discrete variables are those which can only assume certain fixed values. xamples include outcome variables with results such as live
More informationTesting Research and Statistical Hypotheses
Testing Research and Statistical Hypotheses Introduction In the last lab we analyzed metric artifact attributes such as thickness or width/thickness ratio. Those were continuous variables, which as you
More informationComparing Multiple Proportions, Test of Independence and Goodness of Fit
Comparing Multiple Proportions, Test of Independence and Goodness of Fit Content Testing the Equality of Population Proportions for Three or More Populations Test of Independence Goodness of Fit Test 2
More informationAssociation Between Variables
Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi
More informationThe Chi-Square Test. STAT E-50 Introduction to Statistics
STAT -50 Introduction to Statistics The Chi-Square Test The Chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed
More informationChi-square test Fisher s Exact test
Lesson 1 Chi-square test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two- Means
Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More informationThis chapter discusses some of the basic concepts in inferential statistics.
Research Skills for Psychology Majors: Everything You Need to Know to Get Started Inferential Statistics: Basic Concepts This chapter discusses some of the basic concepts in inferential statistics. Details
More informationCrosstabulation & Chi Square
Crosstabulation & Chi Square Robert S Michael Chi-square as an Index of Association After examining the distribution of each of the variables, the researcher s next task is to look for relationships among
More informationCalculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation
Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.
More informationFirst-year Statistics for Psychology Students Through Worked Examples
First-year Statistics for Psychology Students Through Worked Examples 1. THE CHI-SQUARE TEST A test of association between categorical variables by Charles McCreery, D.Phil Formerly Lecturer in Experimental
More informationProbability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationCHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS
CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS CHI-SQUARE TESTS OF INDEPENDENCE (SECTION 11.1 OF UNDERSTANDABLE STATISTICS) In chi-square tests of independence we use the hypotheses. H0: The variables are independent
More informationGoodness of Fit. Proportional Model. Probability Models & Frequency Data
Probability Models & Frequency Data Goodness of Fit Proportional Model Chi-square Statistic Example R Distribution Assumptions Example R 1 Goodness of Fit Goodness of fit tests are used to compare any
More informationBivariate Statistics Session 2: Measuring Associations Chi-Square Test
Bivariate Statistics Session 2: Measuring Associations Chi-Square Test Features Of The Chi-Square Statistic The chi-square test is non-parametric. That is, it makes no assumptions about the distribution
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationStatistics 2014 Scoring Guidelines
AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home
More information12: Analysis of Variance. Introduction
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
More information5.1 Identifying the Target Parameter
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying
More informationChapter 13. Chi-Square. Crosstabs and Nonparametric Tests. Specifically, we demonstrate procedures for running two separate
1 Chapter 13 Chi-Square This section covers the steps for running and interpreting chi-square analyses using the SPSS Crosstabs and Nonparametric Tests. Specifically, we demonstrate procedures for running
More informationCharts, Tables, and Graphs
Charts, Tables, and Graphs The Mathematics sections of the SAT also include some questions about charts, tables, and graphs. You should know how to (1) read and understand information that is given; (2)
More informationSAS Software to Fit the Generalized Linear Model
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
More informationHypothesis Testing: Two Means, Paired Data, Two Proportions
Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this
More informationCHAPTER 15 NOMINAL MEASURES OF CORRELATION: PHI, THE CONTINGENCY COEFFICIENT, AND CRAMER'S V
CHAPTER 15 NOMINAL MEASURES OF CORRELATION: PHI, THE CONTINGENCY COEFFICIENT, AND CRAMER'S V Chapters 13 and 14 introduced and explained the use of a set of statistical tools that researchers use to measure
More informationPart 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217
Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing
More informationIs it statistically significant? The chi-square test
UAS Conference Series 2013/14 Is it statistically significant? The chi-square test Dr Gosia Turner Student Data Management and Analysis 14 September 2010 Page 1 Why chi-square? Tests whether two categorical
More information8 6 X 2 Test for a Variance or Standard Deviation
Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion
More informationData Analysis Tools. Tools for Summarizing Data
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationMath 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2
Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable
More informationChapter 19 The Chi-Square Test
Tutorial for the integration of the software R with introductory statistics Copyright c Grethe Hystad Chapter 19 The Chi-Square Test In this chapter, we will discuss the following topics: We will plot
More informationHaving a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.
Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal
More informationCharacteristics of Binomial Distributions
Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation
More informationSection 12 Part 2. Chi-square test
Section 12 Part 2 Chi-square test McNemar s Test Section 12 Part 2 Overview Section 12, Part 1 covered two inference methods for categorical data from 2 groups Confidence Intervals for the difference of
More informationOnce saved, if the file was zipped you will need to unzip it. For the files that I will be posting you need to change the preferences.
1 Commands in JMP and Statcrunch Below are a set of commands in JMP and Statcrunch which facilitate a basic statistical analysis. The first part concerns commands in JMP, the second part is for analysis
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationLesson 3: Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables
Calculating Conditional Probabilities and Evaluating Independence Using Two-Way Tables Classwork Example 1 Students at Rufus King High School were discussing some of the challenges of finding space for
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationChi Square Tests. Chapter 10. 10.1 Introduction
Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square
More informationChi Square Distribution
17. Chi Square A. Chi Square Distribution B. One-Way Tables C. Contingency Tables D. Exercises Chi Square is a distribution that has proven to be particularly useful in statistics. The first section describes
More informationIntroduction to Analysis of Variance (ANOVA) Limitations of the t-test
Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only
More informationStatistical tests for SPSS
Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly
More informationA Study to Predict No Show Probability for a Scheduled Appointment at Free Health Clinic
A Study to Predict No Show Probability for a Scheduled Appointment at Free Health Clinic Report prepared for Brandon Slama Department of Health Management and Informatics University of Missouri, Columbia
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationCHI-SQUARE: TESTING FOR GOODNESS OF FIT
CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity
More informationGraphs and charts - quiz
Level A 1. In a tally chart, what number does this represent? A) 2 B) 4 C) 8 D) 10 2. In a pictogram if represents 2 people, then how many people do these symbols represent? A) 3 people B) 5 people C)
More informationSAMPLING & INFERENTIAL STATISTICS. Sampling is necessary to make inferences about a population.
SAMPLING & INFERENTIAL STATISTICS Sampling is necessary to make inferences about a population. SAMPLING The group that you observe or collect data from is the sample. The group that you make generalizations
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationFixture List 2018 FIFA World Cup Preliminary Competition
Fixture List 2018 FIFA World Cup Preliminary Competition MATCHDAY 1 4-6 September 2016 4 September Sunday 18:00 Group C 4 September Sunday 20:45 Group C 4 September Sunday 20:45 Group C 4 September Sunday
More informationTEXAS CRIME ANALYSIS 2
2011 CRIME IN TEXAS TEXAS CRIME ANALYSIS 2 CRIME MEASUREMENTS Crime affects every Texan in some fashion. To gain a measurement of crime trends, Texas participates in the Uniform Crime Reporting (UCR) program.
More information11. Analysis of Case-control Studies Logistic Regression
Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More informationSTA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance
Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis
More informationABSORBENCY OF PAPER TOWELS
ABSORBENCY OF PAPER TOWELS 15. Brief Version of the Case Study 15.1 Problem Formulation 15.2 Selection of Factors 15.3 Obtaining Random Samples of Paper Towels 15.4 How will the Absorbency be measured?
More informationContingency Tables and the Chi Square Statistic. Interpreting Computer Printouts and Constructing Tables
Contingency Tables and the Chi Square Statistic Interpreting Computer Printouts and Constructing Tables Contingency Tables/Chi Square Statistics What are they? A contingency table is a table that shows
More informationTesting differences in proportions
Testing differences in proportions Murray J Fisher RN, ITU Cert., DipAppSc, BHSc, MHPEd, PhD Senior Lecturer and Director Preregistration Programs Sydney Nursing School (MO2) University of Sydney NSW 2006
More informationDescriptive Analysis
Research Methods William G. Zikmund Basic Data Analysis: Descriptive Statistics Descriptive Analysis The transformation of raw data into a form that will make them easy to understand and interpret; rearranging,
More information12.5: CHI-SQUARE GOODNESS OF FIT TESTS
125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationHow To Check For Differences In The One Way Anova
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way
More informationUnit 9 Describing Relationships in Scatter Plots and Line Graphs
Unit 9 Describing Relationships in Scatter Plots and Line Graphs Objectives: To construct and interpret a scatter plot or line graph for two quantitative variables To recognize linear relationships, non-linear
More informationExamining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish
Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish Statistics Statistics are quantitative methods of describing, analysing, and drawing inferences (conclusions)
More informationSolutions to Homework 10 Statistics 302 Professor Larget
s to Homework 10 Statistics 302 Professor Larget Textbook Exercises 7.14 Rock-Paper-Scissors (Graded for Accurateness) In Data 6.1 on page 367 we see a table, reproduced in the table below that shows the
More informationThe Big Picture. Describing Data: Categorical and Quantitative Variables Population. Descriptive Statistics. Community Coalitions (n = 175)
Describing Data: Categorical and Quantitative Variables Population The Big Picture Sampling Statistical Inference Sample Exploratory Data Analysis Descriptive Statistics In order to make sense of data,
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationCruise Line Agencies of Alaska. Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL
Cruise Line Agencies of Alaska Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL Page 1 of 5 Sunday, May 1 07:0-18:0 Monday, May 2 Tuesday, May 3 Wednesday, May 4 Thursday,
More informationHypothesis testing - Steps
Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationOne-Way Analysis of Variance (ANOVA) Example Problem
One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means
More informationAP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics
Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,
More informationEMPLOYMENT APPLICATION {PLEASE Print Clearly}
Date Received: Next Step: EMPLOYMENT APPLICATION {PLEASE Print Clearly} Date: Position applied for: Personal Information Legal Name: First Last Middle Initial Address: Street City State Zip code How long
More informationProjects Involving Statistics (& SPSS)
Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,
More informationTopic 8. Chi Square Tests
BE540W Chi Square Tests Page 1 of 5 Topic 8 Chi Square Tests Topics 1. Introduction to Contingency Tables. Introduction to the Contingency Table Hypothesis Test of No Association.. 3. The Chi Square Test
More informationMath 108 Exam 3 Solutions Spring 00
Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8
More informationCourse Syllabus MATH 110 Introduction to Statistics 3 credits
Course Syllabus MATH 110 Introduction to Statistics 3 credits Prerequisites: Algebra proficiency is required, as demonstrated by successful completion of high school algebra, by completion of a college
More informationSimulating Chi-Square Test Using Excel
Simulating Chi-Square Test Using Excel Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 524 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu
More informationSCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES
SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR
More informationCruise Line Agencies of Alaska. Cruise Ship Calendar for 2016 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL
Cruise Line Agencies of Alaska Cruise Ship Calendar for 06 FOR PORT(S) = KTN AND SHIP(S) = ALL AND VOYAGES = ALL 6: Friday, April 5, 06 Cruise Line Agencies of Alaska, Cruise Ship Calendar for 06 Page
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationChi Squared and Fisher's Exact Tests. Observed vs Expected Distributions
BMS 617 Statistical Techniques for the Biomedical Sciences Lecture 11: Chi-Squared and Fisher's Exact Tests Chi Squared and Fisher's Exact Tests This lecture presents two similarly structured tests, Chi-squared
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationMind on Statistics. Chapter 15
Mind on Statistics Chapter 15 Section 15.1 1. A student survey was done to study the relationship between class standing (freshman, sophomore, junior, or senior) and major subject (English, Biology, French,
More informationElementary Statistics
lementary Statistics Chap10 Dr. Ghamsary Page 1 lementary Statistics M. Ghamsary, Ph.D. Chapter 10 Chi-square Test for Goodness of fit and Contingency tables lementary Statistics Chap10 Dr. Ghamsary Page
More informationHYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationUnit 13 Handling data. Year 4. Five daily lessons. Autumn term. Unit Objectives. Link Objectives
Unit 13 Handling data Five daily lessons Year 4 Autumn term (Key objectives in bold) Unit Objectives Year 4 Solve a problem by collecting quickly, organising, Pages 114-117 representing and interpreting
More informationThe Standard Normal distribution
The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance
More informationHYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More information