DOCSLIB.ORG

Chapter 9 Sampling Distributions Parameter – Number That Describes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 9 Sampling Distributions Parameter – Number That Describes Chapter 9 Sampling distributions Parameter – number that describes the population A parameter is a fixed number, but in reality we do not know its value because we can not examine the entire population. Statistic – number that describes a sample Use statistic to estimate an unknown parameter. μ = mean of population x = mean of the sample Sampling variability – the differences in each sample mean P(p-hat) - the proportion of the sample 160 out of 515 people believe in ghosts P(hat) = = .31 .31 is a statistic Proportion of US adults – parameter Sampling variability Take a large number of samples from the same population Calculate x or p(hat) for each sample Make a histogram of x(bar) or p(hat) Examine the distribution displayed in the histogram for shape, center, and spread as well as outliers and other deviations Use simulations for multiple samples – much cheaper than using actual samples Sampling distribution of a statistic is the distribution of values taken by the statistics in all possible samples of the same size from the same population Describing sample distributions Ex 9.5 page 494-495, 496 Bias of a statistic Unbiased statistic- a statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. Ex 9.6 page 498 Variability of a statistic is described by the spread of its sampling distribution. The spread is determined by the sampling design and the size of the sample. Larger samples give smaller spreads!! Homework read pages 488 – 503 do problems 1-4, 12- 15 9.2 Sampling proportions P(hat) = = Mean of sampling distributions μp(hat) = p standard deviation of sampling distribution σp(hat) = Must always check these two items!!!!!!!!!!!! Rule of thumb 1- population must be at least 10x greater than sample size (used to see if you can use standard deviation formula) Rule of thumb 2 – we will use normal approximation as long as np 10 n(1-p) 10 example 9.7 Given μp(hat) = .35 n = 1500 What percent fall with 2% of the true value? ______________________________ .33 .35 .37 (check for rule 1) σp(hat) = σp(hat) = = .0123 check to see if we can use normal distribution np 10 n(1-p) 10 1500(.35) 10 1500(.65) 10 Yes yes Check is population > 10 times 1500? Yes Need to find z-scores .33-.35 .37 - .35 .0123 .0123 = -1.63 = 1.63 .9484 .0516 ____________________________ .33 .37 .9484 - .0516 = .8968 89.68% fall within 2% of the true value P(.33 p(hat) .37) Homework read pages 504 – 512 problems 19 – 23 9.3 Sampling μ = mean of population x = mean of sample μx = μ σx = (when taking a sample and finding the standard deviation) N(μ, ) normal distribution of a sample mean Z = (when sample size is 1) Z = (when sample size is greater than 1) Ex. X = height of randomly selected woman μ = 64.5 inches σ = 2.5 inches take a sample of size 10 find σx = = = .79 p(x > 66.5) z = = 2.53 .9943 .0057 __________________ 0.57 % chance that an SRS of 10 women would have an average height of 66.5 inches or taller .7881 .2119 Individual person Z = = .8 ______________________ 21.19 % chance of a woman being taller than 66.5 inches. Do not forget sample of size n z = Homework problems 32, 33, 39 Central limit theorem Draw an SRS of size n from any population what so ever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x is close to normal N(μ, ). Page 522 shows examples of graphs Homework problems 45, 47 – 53 Take home quiz .
Load more

Recommended publications
  • A Recursive Formula for Moments of a Binomial Distribution Arp´ Ad´ Benyi´ ([email protected]), University of Massachusetts, Amherst, MA 01003 and Saverio M
    A Recursive Formula for Moments of a Binomial Distribution Arp´ ad´ Benyi´ ([email protected]), University of Massachusetts, Amherst, MA 01003 and Saverio M. Manago ([email protected]) Naval Postgraduate School, Monterey, CA 93943 While teaching a course in probability and statistics, one of the authors came across an apparently simple question about the computation of higher order moments of a ran- dom variable. The topic of moments of higher order is rarely emphasized when teach- ing a statistics course. The textbooks we came across in our classes, for example, treat this subject rather scarcely; see [3, pp. 265–267], [4, pp. 184–187], also [2, p. 206]. Most of the examples given in these books stop at the second moment, which of course suffices if one is only interested in finding, say, the dispersion (or variance) of a ran- 2 2 dom variable X, D (X) = M2(X) − M(X) . Nevertheless, moments of order higher than 2 are relevant in many classical statistical tests when one assumes conditions of normality. These assumptions may be checked by examining the skewness or kurto- sis of a probability distribution function. The skewness, or the first shape parameter, corresponds to the the third moment about the mean. It describes the symmetry of the tails of a probability distribution. The kurtosis, also known as the second shape pa- rameter, corresponds to the fourth moment about the mean and measures the relative peakedness or flatness of a distribution. Significant skewness or kurtosis indicates that the data is not normal. However, we arrived at higher order moments unintentionally.
    [Show full text]
  • Section 7 Testing Hypotheses About Parameters of Normal Distribution. T-Tests and F-Tests
    Section 7 Testing hypotheses about parameters of normal distribution. T-tests and F-tests. We will postpone a more systematic approach to hypotheses testing until the following lectures and in this lecture we will describe in an ad hoc way T-tests and F-tests about the parameters of normal distribution, since they are based on a very similar ideas to confidence intervals for parameters of normal distribution - the topic we have just covered. Suppose that we are given an i.i.d. sample from normal distribution N(µ, ν2) with some unknown parameters µ and ν2 : We will need to decide between two hypotheses about these unknown parameters - null hypothesis H0 and alternative hypothesis H1: Hypotheses H0 and H1 will be one of the following: H : µ = µ ; H : µ = µ ; 0 0 1 6 0 H : µ µ ; H : µ < µ ; 0 ∼ 0 1 0 H : µ µ ; H : µ > µ ; 0 ≈ 0 1 0 where µ0 is a given ’hypothesized’ parameter. We will also consider similar hypotheses about parameter ν2 : We want to construct a decision rule α : n H ; H X ! f 0 1g n that given an i.i.d. sample (X1; : : : ; Xn) either accepts H0 or rejects H0 (accepts H1). Null hypothesis is usually a ’main’ hypothesis2 X in a sense that it is expected or presumed to be true and we need a lot of evidence to the contrary to reject it. To quantify this, we pick a parameter � [0; 1]; called level of significance, and make sure that a decision rule α rejects H when it is2 actually true with probability �; i.e.
    [Show full text]
  • 11. Parameter Estimation
    11. Parameter Estimation Chris Piech and Mehran Sahami May 2017 We have learned many different distributions for random variables and all of those distributions had parame- ters: the numbers that you provide as input when you define a random variable. So far when we were working with random variables, we either were explicitly told the values of the parameters, or, we could divine the values by understanding the process that was generating the random variables. What if we don’t know the values of the parameters and we can’t estimate them from our own expert knowl- edge? What if instead of knowing the random variables, we have a lot of examples of data generated with the same underlying distribution? In this chapter we are going to learn formal ways of estimating parameters from data. These ideas are critical for artificial intelligence. Almost all modern machine learning algorithms work like this: (1) specify a probabilistic model that has parameters. (2) Learn the value of those parameters from data. Parameters Before we dive into parameter estimation, first let’s revisit the concept of parameters. Given a model, the parameters are the numbers that yield the actual distribution. In the case of a Bernoulli random variable, the single parameter was the value p. In the case of a Uniform random variable, the parameters are the a and b values that define the min and max value. Here is a list of random variables and the corresponding parameters. From now on, we are going to use the notation q to be a vector of all the parameters: Distribution Parameters Bernoulli(p) q = p Poisson(l) q = l Uniform(a,b) q = (a;b) Normal(m;s 2) q = (m;s 2) Y = mX + b q = (m;b) In the real world often you don’t know the “true” parameters, but you get to observe data.
    [Show full text]
  • A Widely Applicable Bayesian Information Criterion
    JournalofMachineLearningResearch14(2013)867-897 Submitted 8/12; Revised 2/13; Published 3/13 A Widely Applicable Bayesian Information Criterion Sumio Watanabe [email protected] Department of Computational Intelligence and Systems Science Tokyo Institute of Technology Mailbox G5-19, 4259 Nagatsuta, Midori-ku Yokohama, Japan 226-8502 Editor: Manfred Opper Abstract A statistical model or a learning machine is called regular if the map taking a parameter to a prob- ability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature 1/logn, where n is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for or unrealizable by a statistical model.
    [Show full text]
  • Statistic: a Quantity That We Can Calculate from Sample Data That Summarizes a Characteristic of That Sample
    STAT 509 – Section 4.1 – Estimation Parameter: A numerical characteristic of a population. Examples: Statistic: A quantity that we can calculate from sample data that summarizes a characteristic of that sample. Examples: Point Estimator: A statistic which is a single number meant to estimate a parameter. It would be nice if the average value of the estimator (over repeated sampling) equaled the target parameter. An estimator is called unbiased if the mean of its sampling distribution is equal to the parameter being estimated. Examples: Another nice property of an estimator: we want it to be as precise as possible. The standard deviation of a statistic’s sampling distribution is called the standard error of the statistic. The standard error of the sample mean Y is / n . Note: As the sample size gets larger, the spread of the sampling distribution gets smaller. When the sample size is large, the sample mean varies less across samples. Evaluating an estimator: (1) Is it unbiased? (2) Does it have a small standard error? Interval Estimates • With a point estimate, we used a single number to estimate a parameter. • We can also use a set of numbers to serve as “reasonable” estimates for the parameter. Example: Assume we have a sample of size n from a normally distributed population. Y T We know: s / n has a t-distribution with n – 1 degrees of freedom. (Exactly true when data are normal, approximately true when data non-normal but n is large.) Y P(t t ) So: 1 – = n1, / 2 s / n n1, / 2 = where tn–1, /2 = the t-value with /2 area to the right (can be found from Table 2) This formula is called a “confidence interval” for .
    [Show full text]
  • THE ONE-SAMPLE Z TEST
    10 THE ONE-SAMPLE z TEST Only the Lonely Difficulty Scale ☺ ☺ ☺ (not too hard—this is the first chapter of this kind, but youdistribute know more than enough to master it) or WHAT YOU WILL LEARN IN THIS CHAPTERpost, • Deciding when the z test for one sample is appropriate to use • Computing the observed z value • Interpreting the z value • Understandingcopy, what the z value means • Understanding what effect size is and how to interpret it not INTRODUCTION TO THE Do ONE-SAMPLE z TEST Lack of sleep can cause all kinds of problems, from grouchiness to fatigue and, in rare cases, even death. So, you can imagine health care professionals’ interest in seeing that their patients get enough sleep. This is especially the case for patients 186 Copyright ©2020 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. Chapter 10 ■ The One-Sample z Test 187 who are ill and have a real need for the healing and rejuvenating qualities that sleep brings. Dr. Joseph Cappelleri and his colleagues looked at the sleep difficul- ties of patients with a particular illness, fibromyalgia, to evaluate the usefulness of the Medical Outcomes Study (MOS) Sleep Scale as a measure of sleep problems. Although other analyses were completed, including one that compared a treat- ment group and a control group with one another, the important analysis (for our discussion) was the comparison of participants’ MOS scores with national MOS norms. Such a comparison between a sample’s mean score (the MOS score for par- ticipants in this study) and a population’s mean score (the norms) necessitates the use of a one-sample z test.
    [Show full text]
  • Skewness Vs. Kurtosis: Implications for Pricing and Hedging Options
    Skewness vs. Kurtosis: Implications for Pricing and Hedging Options Sol Kim College of Business Hankuk University of Foreign Studies 270, Imun-dong, Dongdaemun-Gu, Seoul, Korea Tel: +82-2-2173-3124 Fax: +82-2-959-4645 E-mail: [email protected] Abstract For S&P 500 options, we examine the relative influence of the skewness and kurtosis of the risk- neutral distribution on pricing and hedging performances. Both the nonparametric method suggested by Bakshi, Kapadia and Madan (2003) and the parametric method suggested by Corrado and Su (1996) are used to estimate the risk-neutral skewness and kurtosis. We find that skewness exerts a greater impact on pricing and hedging errors than kurtosis does. The option pricing model that considers skewness shows better performance for pricing and hedging the options than does the model that considers kurtosis. All the results are statistically significant and robust to all sub-periods, which confirms that the risk-neutral skewness is a more important factor than the risk- neutral kurtosis for pricing and hedging stock index options. JEL classification: G13 Keywords: Volatility Smiles; Options Pricing; Risk-neutral Distribution; Skewness; Kurtosis Skewness vs. Kurtosis: Implications for Pricing and Hedging Options Abstract For S&P 500 options, we examine the relative influence of the skewness and kurtosis of the risk- neutral distribution on pricing and hedging performances. Both the nonparametric method suggested by Bakshi, Kapadia and Madan (2003) and the parametric method suggested by Corrado and Su (1996) are used to estimate the risk-neutral skewness and kurtosis. We find that skewness exerts a greater impact on pricing and hedging errors than kurtosis does.
    [Show full text]
  • Chapter 4 Parameter Estimation
    Chapter 4 Parameter Estimation Thus far we have concerned ourselves primarily with probability theory: what events may occur with what probabilities, given a model family and choices for the parameters. This is useful only in the case where we know the precise model family and parameter values for the situation of interest. But this is the exception, not the rule, for both scientific inquiry and human learning & inference. Most of the time, we are in the situation of processing data whose generative source we are uncertain about. In Chapter 2 we briefly covered elemen- tary density estimation, using relative-frequency estimation, histograms and kernel density estimation. In this chapter we delve more deeply into the theory of probability density es- timation, focusing on inference within parametric families of probability distributions (see discussion in Section 2.11.2). We start with some important properties of estimators, then turn to basic frequentist parameter estimation (maximum-likelihood estimation and correc- tions for bias), and finally basic Bayesian parameter estimation. 4.1 Introduction Consider the situation of the first exposure of a native speaker of American English to an English variety with which she has no experience (e.g., Singaporean English), and the problem of inferring the probability of use of active versus passive voice in this variety with a simple transitive verb such as hit: (1) The ball hit the window. (Active) (2) The window was hit by the ball. (Passive) There is ample evidence that this probability is contingent
    [Show full text]
  • Ch. 2 Estimators
    Chapter Two Estimators 2.1 Introduction Properties of estimators are divided into two categories; small sample and large (or infinite) sample. These properties are defined below, along with comments and criticisms. Four estimators are presented as examples to compare and determine if there is a "best" estimator. 2.2 Finite Sample Properties The first property deals with the mean location of the distribution of the estimator. P.1 Biasedness - The bias of on estimator is defined as: Bias(!ˆ ) = E(!ˆ ) - θ, where !ˆ is an estimator of θ, an unknown population parameter. If E(!ˆ ) = θ, then the estimator is unbiased. If E(!ˆ ) ! θ then the estimator has either a positive or negative bias. That is, on average the estimator tends to over (or under) estimate the population parameter. A second property deals with the variance of the distribution of the estimator. Efficiency is a property usually reserved for unbiased estimators. ˆ ˆ 1 ˆ P.2 Efficiency - Let ! 1 and ! 2 be unbiased estimators of θ with equal sample sizes . Then, ! 1 ˆ ˆ ˆ is a more efficient estimator than ! 2 if var(! 1) < var(! 2 ). Restricting the definition of efficiency to unbiased estimators, excludes biased estimators with smaller variances. For example, an estimator that always equals a single number (or a constant) has a variance equal to zero. This type of estimator could have a very large bias, but will always have the smallest variance possible. Similarly an estimator that multiplies the sample mean by [n/(n+1)] will underestimate the population mean but have a smaller variance.
    [Show full text]
  • Estimators, Bias and Variance
    Deep Learning Srihari Machine Learning Basics: Estimators, Bias and Variance Sargur N. Srihari [email protected] This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/CSE676 1 Deep Learning Topics in Basics of ML Srihari 1. Learning Algorithms 2. Capacity, Overfitting and Underfitting 3. Hyperparameters and Validation Sets 4. Estimators, Bias and Variance 5. Maximum Likelihood Estimation 6. Bayesian Statistics 7. Supervised Learning Algorithms 8. Unsupervised Learning Algorithms 9. Stochastic Gradient Descent 10. Building a Machine Learning Algorithm 11. Challenges Motivating Deep Learning 2 Deep Learning Srihari Topics in Estimators, Bias, Variance 0. Statistical tools useful for generalization 1. Point estimation 2. Bias 3. Variance and Standard Error 4. Bias-Variance tradeoff to minimize MSE 5. Consistency 3 Deep Learning Srihari Statistics provides tools for ML • The field of statistics provides many tools to achieve the ML goal of solving a task not only on the training set but also to generalize • Foundational concepts such as – Parameter estimation – Bias – Variance • They characterize notions of generalization, over- and under-fitting 4 Deep Learning Srihari Point Estimation • Point Estimation is the attempt to provide the single best prediction of some quantity of interest – Quantity of interest can be: • A single parameter • A vector of parameters – E.g., weights in linear regression • A whole function 5 Deep Learning Srihari Point estimator or Statistic • To distinguish estimates of parameters
    [Show full text]
  • Population and Sample Practice 1. for Each Statement, Identify Whether
    Population and Sample Practice 1. For each statement, identify whether the numbers underlined are statistics or parameters. a. Of all U.S. kindergarten teachers, 32% say that knowing the alphabet is an essential skill. b. Of the 800 U.S. kindergarten teachers polled, 34% say that knowing the alphabet is an essential skill. 2. Of the U.S. adult population, 36% has an allergy. A sample of 1200 randomly selected adults resulted in 33.2% reporting an allergy. a. Who is the population? b. What is the sample? c. Identify the statistic and give its value. d. Identify the parameter and give its value. 3. In your own words, explain why the parameter is fixed and the statistic varies. 4. Select 90 students currently enrolled at NCSU and ask how many years they’ve attended the university, how old they are, and if they live on campus. a. What is the population? b. What is the sample? 5. Suppose a 12 year old asked you to explain the difference between a sample and a population, how would you explain it to him/her? How might you explain why you would want to take a sample, rather than surveying every member of the population? 6. In your own words, explain the difference between a statistic and a parameter. 7. A study reveals that there are exactly 100 Senators in the 109th Congress of the United States, and 55% of them are Republicans. a. Do the data comprise a sample or a population? b. Do the results represent a statistic or a parameter? 8.
    [Show full text]
  • Clarity in Reporting Parameter Variance Needed to Improve Use Of
    Clarity in Reporting Parameter Variance Needed to Improve Use of Published Models for Simulation Applications Clary, James1; Fiedler-Kelly, Jill2; Owen, Joel2 1Union University College of Pharmacy, Jackson, TN; 2Cognigen Corporation a Simulations Plus company, Buffalo, NY ABSTRACT METHODS a Background: Since published pharmacokinetic and pharmacodynamic models are often used by others for To give context to the frequency of equivocal reporting of ω2 parameter estimates, we performed a survey of the purpose of simulations, enhanced clarity in reporting and clear statements regarding assumptions will models (PK, PD, PBPK, mathematical) published in the journal CPT: Pharmacometrics & Systems improve the reproducibility of modeling and simulation results and allow for accurate re-use of models and Pharmacology (CPT:PSP) from the first online issue (September 2012) until December 2018. Each manuscript modeling findings. was examined to see whether the authors reported: 1) either the method used to compute %CV values or the ω2 parameter estimates themselves, or 2) neither the equation used to compute %CV nor the ω2 Methods: To illustrate the importance of this issue, simulations were performed using the pharmacokinetic parameter estimates. Three time ranges were considered: 2012-2015, 2016-2017, and 2018. This choice of model for paroxetine by Feng et. al.1, a manuscript that did not report which method was used for %CV range and stratification was intended to give both a broad scope of the literature and a look at current calculation. This paper was selected in part because reported variability was in excess of 70 %CV for several practice in a core pharmacometrics journal. parameters.
    [Show full text]