CHI-SQUARE: TESTING FOR GOODNESS OF FIT

Size: px
Start display at page: Transcription

1 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 we called Φ in order to determine best estimates for certain function parameters, such as (for a straight line) a slope and an intercept. Φ is proportional to (or in some cases equal to) a statistical measure called χ 2, or chi-square, a quantity commonly used to test whether any given data are well described by some hypothesized function. Such a determination is called a chi-square test for goodness of fit. In the following, we discuss χ 2 and its statistical distribution, and show how it can be used as a test for goodness of fit. 1 The definition of χ 2 If ν independent variables x i are each normally distributed with mean µ i and variance σ 2 i, then the quantity known as chi-square 2 is defined by χ 2 (x 1 µ 1 ) 2 σ (x 2 µ 2 ) 2 σ (x ν µ ν ) 2 σ 2 ν = ν (x i µ i ) 2 σ 2 i=1 i (1) Note that ideally, given the random fluctuations of the values of x i about their mean values µ i, each term in the sum will be of order unity. Hence, if we have chosen the µ i and the σ i correctly, we may expect that a calculated value of χ 2 will be approximately equal to ν. If it is, then we may conclude that the data are well described by the values we have chosen for the µ i, that is, by the hypothesized function. If a calculated value of χ 2 turns out to be much larger than ν, and we have correctly estimated the values for the σ i, we may possibly conclude that our data are not welldescribed by our hypothesized set of the µ i. This is the general idea of the χ 2 test. In what follows we spell out the details of the procedure. 1 The chi-square distribution and some examples of its use as a statistical test are also described in the references listed at the end of this chapter. 2 The notation of χ 2 is traditional and possibly misleading. It is a single statistical variable, and not the square of some quantity χ. It is therefore not chi squared, but chi-square. The notation is merely suggestive of its construction as the sum of squares of terms. Perhaps it would have been better, historically, to have called it ξ or ζ. 4 1

2 4 2 Chi-square: Testing for goodness of fit The χ 2 distribution The quantity χ 2 defined in Eq. 1 has the probability distribution given by f(χ 2 ) = 1 2 ν/2 Γ(ν/2) e χ2 /2 (χ 2 ) (ν/2) 1 (2) This is known as the χ 2 -distribution with ν degrees of freedom. ν is a positive integer. 3 Sometimes we write it as f(χ 2 ν) when we wish to specify the value of ν. f(χ 2 ) d(χ 2 ) is the probability that a particular value of χ 2 falls between χ 2 and χ 2 + d(χ 2 ). Here are graphs of f(χ 2 ) versus χ 2 for three values of ν: Figure 1 The chi-square distribution for ν = 2, 4, and 10. Note that χ 2 ranges only over positive values: 0 < χ 2 <. The mean value of χ 2 ν is equal to ν, and the variance of χ 2 ν is equal to 2ν. The distribution is highly skewed for small values of ν, and becomes more symmetric as ν increases, approaching a Gaussian distribution for large ν, just as predicted by the Central Limit Theorem. 3 Γ(p) is the Gamma function, defined by Γ(p + 1) 0 x p e x dx. It is a generalization of the factorial function to non-integer values of p. If p is an integer, Γ(p+1) = p!. In general, Γ(p+1) = pγ(p), and Γ(1/2) = π.

3 Chi-square: Testing for goodness of fit 4 3 How to use χ 2 to test for goodness of fit Suppose we have a set of N experimentally measured quantities x i. We want to test whether they are well-described by some set of hypothesized values µ i. We form a sum like that shown in Eq. 1. It will contain N terms, constituting a sample value for χ 2. In forming the sum, we must use estimates for the σ i that are independently obtained for each x i. 4 Now imagine, for a moment, that we could repeat our experiment many times. Each time, we would obtain a data sample, and each time, a sample value for χ 2. If our data were well-described by our hypothesis, we would expect our sample values of χ 2 to be distributed according to Eq. 2, and illustrated by example in Fig. 1. However, we must be a little careful. The expected distribution of our samples of χ 2 will not be one of N degrees of freedom, even though there are N terms in the sum, because our sample variables x i will invariably not constitute a set of N independent variables. There will, typically, be at least one, and often as many as three or four, relations connecting the x i. Such relations are needed in order to make estimates of hypothesized parameters such as the µ i, and their presence will reduce the number of degrees of freedom. With r such relations, or constraints, the number of degrees of freedom becomes ν = N r, and the resulting χ 2 sample will be one having ν (rather than N) degrees of freedom. As we repeat our experiment and collect values of χ 2, we expect, if our model is a valid one, that they will be clustered about the median value of χ 2 ν, with about half of these collected values being greater than the median value, and about half being less than the median value. This median value, which we denote by χ 2 ν,0.5, is determined by f(χ 2 ) dχ 2 = 0.5 χ 2 ν,0.5 Note that because of the skewed nature of the distribution function, the median value of χ 2 ν will be somewhat less than the mean (or average) value of χ 2 ν, which as we have noted, is equal to ν. For example, for ν = 10 degrees of freedom, χ 2 10, , a number slightly less than 10. Put another way, we expect that a single measured value of χ 2 will have a probability of 0.5 of being greater than χ 2 ν, In the previous chapter, we showed how a hypothesized function may be fit to a set of data points. There we noted that it may be either impossible or inconvenient to make independent estimates of the σ i, in which case estimates of the σ i can be made only by assuming an ideal fit of the function to the data. That is, we assumed χ 2 to be equal to its mean value, and from that, estimated uncertainties, or confidence intervals, for the values of the determined parameters. Such a procedure precludes the use of the χ 2 test.

4 4 4 Chi-square: Testing for goodness of fit We can generalize from the above discussion, to say that we expect a single measured value of χ 2 will have a probability α ( alpha ) of being greater than χ 2 ν,α, where χ 2 ν,α is defined by χ 2 ν,α f(χ 2 ) dχ 2 = α This definition is illustrated by the inset in Fig. 2 on page 4 9. Here is how the χ 2 test works: (a) We hypothesize that our data are appropriately described by our chosen function, or set of µ i. This is the hypothesis we are going to test. (b) From our data sample we calculate a sample value of χ 2 (chi-square), along with ν (the number of degrees of freedom), and so determine χ 2 /ν (the normalized chi-square, or the chi-square per degree of freedom) for our data sample. (c) We choose a value of the significance level α (a common value is.05, or 5 per cent), and from an appropriate table or graph (e.g., Fig. 2), determine the corresponding value of χ 2 ν,α/ν. We then compare this with our sample value of χ 2 /ν. (d) If we find that χ 2 /ν > χ 2 ν,α/ν, we may conclude that either (i) the model represented by the µ i is a valid one but that a statistically improbable excursion of χ 2 has occurred, or (ii) that our model is so poorly chosen that an unacceptably large value of χ 2 has resulted. (i) will happen with a probability α, so if we are satisfied that (i) and (ii) are the only possibilities, (ii) will happen with a probability 1 α. Thus if we find that χ 2 /ν > χ 2 ν,α/ν, we are 100 (1 α) per cent confident in rejecting our model. Note that this reasoning breaks down if there is a possibility (iii), for example if our data are not normally distributed. The theory of the chi-square test relies on the assumption that chi-square is the sum of the squares of random normal deviates, that is, that each x i is normally distributed about its mean value µ i. However for some experiments, there may be occasional non-normal data points that are too far from the mean to be real. A truck passing by, or a glitch in the electrical power could be the cause. Such points, sometimes called outliers, can unexpectedly increase the sample value of chi-square. It is appropriate to discard data points that are clearly outliers. (e) If we find that χ 2 is too small, that is, if χ 2 /ν < χ 2 ν,1 α /ν, we may conclude only that either (i) our model is valid but that a statistically improbable excursion of χ 2 has occurred, or (ii) we have, too conservatively, over-estimated the values of σ i, or (iii) someone has given us fraudulent data, that is, data too good to be true. A too-small value of χ 2 cannot be indicative of a poor model. A poor model can only increase χ 2.

5 Chi-square: Testing for goodness of fit 4 5 Generally speaking, we should be pleased to find a sample value of χ 2 /ν that is near 1, its mean value for a good fit. In the final analysis, we must be guided by our own intuition and judgment. The chi-square test, being of a statistical nature, serves only as an indicator, and cannot be iron clad. An example The field of particle physics provides numerous situations where the χ 2 test can be applied. A particularly simple example 5 involves measurements of the mass M Z of the Z 0 boson by experimental groups at CERN. The results of measurements of M Z made by four different detectors (L3, OPAL, Aleph and Delphi) are as follows: Detector Mass in GeV/c 2 L ± OPAL ± Aleph ± Delphi ± The listed uncertainties are estimates of the σ i, the standard deviations for each of the measurements. The figure below shows these measurements plotted on a horizontal mass scale (vertically displaced for clarity). Measurements of the Z 0 boson. The question arises: Can these data be well described by a single number, namely an estimate of M Z made by determining the weighted mean of the four measurements? 5 This example is provided by Pat Burchat.

6 4 6 Chi-square: Testing for goodness of fit to find We find the weighted mean M Z, and its standard deviation σ M Z Mi /σi 2 M Z = 1/σ 2 and σ 2 = 1 MZ i 1/σ 2 i like this:6 M Z ± σ M Z = ± Then we form χ 2 : χ 2 = 4 (M i M Z ) 2 σ 2 i=1 i 2.78 We expect this value of χ 2 to be drawn from a chi-square distribution with 3 degrees of freedom. The number is 3 (not 4) because we have used the mean of the four measurements to estimate the value of µ, the true mass of the Z 0 boson, and this uses up one degree of freedom. Hence χ 2 /ν = 2.78/ Now from the graph of α versus χ 2 /ν shown in Fig. 2, we find that for 3 degrees of freedom, α is about 0.42, meaning that if we were to repeat the experiments we would have about a 42 per cent chance of finding a χ 2 for the new measurement set larger than 2.78, assuming our hypothesis is correct. We have therefore no good reason to reject the hypothesis, and conclude that the four measurements of the Z 0 boson mass are consistent with each other. We would have had to have found χ 2 in the vicinity of 8.0 (leading to an α of about 0.05) to have been justified in suspecting the consistency of the measurements. The fact that our sample value of χ 2 /3 is close to 1 is reassuring. Using χ 2 to test hypotheses regarding statistical distributions The χ 2 test is used most commonly to test the nature of a statistical distribution from which some random sample is drawn. It is this kind of application that is described by Evans in his text, and is the kind of application for which the χ 2 test was first formulated. Situations frequently arise where data can be classified into one of k classes, with probabilities p 1, p 2,..., p k of falling into each class. If all the data are accounted for, pi = 1. Now suppose we take data by classifying it: We count the number of observations falling into each of the k classes. We ll have n 1 in the first class, n 2 in the second, and so on, up to n k in the k th class. We suppose there are a total of N observations, so n i = N. 6 These expressions are derived on page 2 12.

7 Chi-square: Testing for goodness of fit 4 7 It can be shown by non-trivial methods that the quantity (n 1 Np 1 ) 2 Np 1 + (n 2 Np 2 ) 2 Np (n k Np k ) 2 Np k = k (n i Np i ) 2 (3) Np i has approximately the χ 2 distribution with k r degrees of freedom, where r is the number of constraints, or relations used to estimate the p i from the data. r will always be at least 1, since it must be that n i = Np i = N p i = N. Since Np i is the mean, or expected value of n i, the form of χ 2 given by Eq. 3 corresponds to summing, over all classes, the squares of the deviations of the observed n i from their mean values divided by their mean values. At first glance, this special form looks different from that shown in Eq. 1, since the variance for each point is replaced by the mean value of n i for each point. Such an estimate for the variance makes sense in situations involving counting, where the counted numbers are distributed according to the Poisson distribution, for which the mean is equal to the variance: µ = σ 2. Equation 3 forms the basis of what is sometimes called Pearson s Chi-square Test. Unfortunately some authors (incorrectly) use this equation to define χ 2. However, this form is not the most general form, in that it applies only to situations involving counting, where the data variables are dimensionless. Another example Here is an example in which the chi-square test is used to test whether a data sample consisting of the heights of 66 women can be assumed to be drawn from a Gaussian distribution. 7 We first arrange the data in the form of a frequency distribution, listing for each height h, the value of n(h), the number of women in the sample whose height is h (h is in inches): h n(h) We make the hypothesis that the heights are distributed according to the Gaussian distribution (see page 2 6 of this manual), namely that the probability p(h) dh that a height falls between h and h + dh is given by p(h) dh = 1 σ 2 /2σ 2π e (h µ) 2 dh i=1 7 This sample was collected by Intermediate Laboratory students in 1991.

8 4 8 Chi-square: Testing for goodness of fit This expression, if multiplied by N, will give, for a sample of N women, the number of women n th (h) dh theoretically expected to have a height between h and h + dh: n th (h) dh = N σ 2 /2σ 2π e (h µ) 2 dh (4) In our example, N = 66. Note that we have, in our table of data above, grouped the data into bins (we ll label them with the index j), each of size 1 inch. A useful approximation to Eq. 4, in which dh is taken to be 1 inch, gives the expected number of women n th (j) having a height h j : n th (j) = N σ 2 2π e (h j µ) /2σ 2 (5) Now the sample mean h and the sample standard deviation s are our best estimates of µ and σ. We find, calculating from the data: h = 64.9 inches, and s = 2.7 inches Using these values we may calculate, from Eq. 5, the number expected in each bin, with the following results: h n th (j) In applying the chi-square test to a situation of this type, it is advisable to re-group the data into new bins (classes) such that the expected number occurring in each bin is greater than 4 or 5; otherwise the theoretical distributions within each bin become too highly skewed for meaningful results. Thus in this situation we shall put all the heights of 61 inches or less into a single bin, and all the heights of 69 inches or more into a single bin. This groups the data into a total of 9 bins (or classes), with actual numbers and expected numbers in each bin being given as follows (note the bin sizes need not be equal): h n(h) n th (j) Now we calculate the value of χ 2 using these data, finding χ 2 = (6 6.5) (6 5.5) (6 5.8)2 5.8 = 6.96 Since we have grouped our data into 9 classes, and since we have used up three degrees of freedom by demanding (a) that the sum of the n j be equal to N, (b) that

9 Chi-square: Testing for goodness of fit 4 9 the mean of the hypothesized distribution be equal to the sample mean, and (c) that the variance of the hypothesized distribution be equal to the sample variance, there are 6 degrees of freedom left. 8 Hence χ 2 /ν = 6.96/6 1.16, leading to an α of about Therefore we have no good reason to reject our hypothesis that our data are drawn from a Gaussian distribution function. Figure 2 α versus the normalized chi-square: χ 2 ν/ν. α is the probability that a sample chi-square will be larger than χ 2 ν, as shown in the inset. Each curve is labeled by ν, the number of degrees of freedom. References 1. Press, William H. et. al., Numerical Recipes in C The Art of Scientific Computing, 2nd Ed. (Cambridge University Press, New York, 1992). Press devotes considerable discussion to the subject of fitting parameters to data, including the use of the chi-square test. Noteworthy are his words of advice, appearing on pages : To be genuinely useful, a fitting procedure should provide (i) parameters, (ii) error estimates on the parameters, and (iii) a statistical measure of goodness-of-fit. When the third item suggests that the model is an unlikely match to the data, then items (i) and (ii) are probably worthless. Unfortunately, many practitioners of parameter estimation never proceed beyond item (i). They deem a fit acceptable 8 Note that if we were hypothesizing a Poisson distribution (as in a counting experiment), there would be 7 degrees of freedom (only 2 less than the number of classes). For a Poisson distribution the variance is equal to the mean, so there is only 1 parameter to be determined, not 2.

10 4 10 Chi-square: Testing for goodness of fit if a graph of data and model looks good. This approach is known as chi-by-eye. Luckily, its practitioners get what they deserve. 2. Bennett, Carl A., and Franklin, Norman L., Statistical Analysis in Chemistry and the Chemical Industry (Wiley, 1954). An excellent discussion of the chi-square distribution function, with good examples illustrating its use, may be found on pages 96 and Evans, Robley D., The Atomic Nucleus (McGraw-Hill, 1969). Chapter 27 of this advanced text contains a description of Pearson s Chi-square Test.

12.5: 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

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

Lecture 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

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

Nuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics Nuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics PART I Geiger Tube: Optimal Operating Voltage and Resolving Time Objective: To become acquainted with the operation and characteristics

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

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

Simple 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

" Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

statistics Chi-square tests and nonparametric Summary sheet from last time: Hypothesis testing Summary sheet from last time: Confidence intervals Summary sheet from last time: Confidence intervals Confidence intervals take on the usual form: parameter = statistic ± t crit SE(statistic) parameter SE a s e sqrt(1/n + m x 2 /ss xx ) b s e /sqrt(ss

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

HYPOTHESIS TESTING: POWER OF THE TEST HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

3.4 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

1 Sufficient statistics 1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives. The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

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

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

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

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

MEASURES OF VARIATION NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

Exercise 1.12 (Pg. 22-23) Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

Calculating 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.

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

Premaster Statistics Tutorial 4 Full solutions Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for

Confidence Intervals for Cp Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

The 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

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

Statistics 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

UNDERSTANDING THE TWO-WAY ANOVA UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

CURVE FITTING LEAST SQUARES APPROXIMATION CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

HYPOTHESIS 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

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1 THE STATISTICAL TREATMET OF EXPERIMETAL DATA Introduction The subject of statistical data analysis is regarded as crucial by most scientists, since error-free measurement is impossible in virtually all

AP Physics 1 and 2 Lab Investigations AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

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

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

EMPIRICAL FREQUENCY DISTRIBUTION INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1 when some empirical distribution approximates

1 Prior Probability and Posterior Probability Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

Comparing Means in Two Populations Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

6.2 Permutations continued 6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

Normality Testing in Excel Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

Using Excel for inferential statistics FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

PROBABILITY AND SAMPLING DISTRIBUTIONS PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 seema@iasri.res.in. Introduction The concept of probability

5.1 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

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so: Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

Study 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

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

Two-sample inference: Continuous data Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

Experimental 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

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

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r), Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

Two-Sample T-Tests Assuming Equal Variance (Enter Means) Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

17. SIMPLE LINEAR REGRESSION II 17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

Hypothesis Testing for Beginners Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

LAB 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.

X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1) CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

HYPOTHESIS 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

Chapter 2. Hypothesis testing in one population Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

Standard Deviation Estimator CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

DATA INTERPRETATION AND STATISTICS PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

Topic 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

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

Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

Confidence Intervals for One Standard Deviation Using Standard Deviation Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

Testing 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

AP Statistics Solutions to Packet 2 AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

Fairfield 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

research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other 1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric

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

How 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

1. 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

Unit 26 Estimation with Confidence Intervals Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance 2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

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

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

Means, standard deviations and. and standard errors CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

CHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13 COMMON DESCRIPTIVE STATISTICS / 13 CHAPTER THREE COMMON DESCRIPTIVE STATISTICS The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance,

Lecture 5 : The Poisson Distribution Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

A 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 Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

Variables Control Charts 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. Variables

Econometrics Simple Linear Regression Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations. 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