...Lecture 7. Sampling Distributions. Statistical Inference: Using statistics calculated from samples to estimate the values of population parameters. Select Random Sample Sample for (statistic) Calculate to estimate Becomes Population Parameter. BASIC Example: Soft Drink Bottler μ=600, σ=10. Normal Distribution. What is P(X>598)? p(x<598) . Sampling Dist.of the Mean – Distribution of all Possible Sample Means if you select a sample of a certain size. μX= μ. μ = i=1NXiN (formula for mean) . σ = i=1N(Xi-μ)2N Although you do not know how close the sample mean of any particular sample selected comes to the pop mean, you know that the mean of all possible sample means that could have been selected = the pop mean. Standard error is calculating the probability of a certain amount of error. EXAMPLE: Standard error is . As n increases decreases. CENTRAL LIMIT THEOREM: Regardless of shape of individual values in distribution; as long as sample size is large enough the sampling distribution of the mean will be approximately normally distributed with μX= μ and σX= σ . For most population distributions n ≥ 30 will be large enough. For symmetric population distributions, n ≥ 5 is sufficient. For normal population distributions, the sampling distribution of the mean is always normally distributed EXAMPLE: SAMPLING DISTRIBUTION OF THE PROPORTION. π is the proportion of items in the population with a characteristic of interest. p is the sample proportion...
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...perfectly comfortable with formal mathematical presentation of material. Others, who have had less technical training, may easily be overwhelmed by mathematical formalism. Most students, however, will benefit from some coaching to make the study of investment easier and more efficient. If you had a good introductory quantitative methods course, and like the text that was used, you may want to refer to it whenever you feel in need of a refresher. If you feel uncomfortable with standard quantitative texts, this reference is for you. Our aim is to present the essential quantitative concepts and methods in a self-contained, nontechnical, and intuitive way. Our approach is structured in line with requirements for the CFA program. The material included is relevant to investment management by the ICFA, the Institute of Chartered Financial Analysts. We hope you find this appendix helpful. Use it to make your venture into investments more enjoyable. 1006 Appendix A 1007 A.1 PROBABILITY DISTRIBUTIONS Statisticians talk about “experiments,” or “trials,” and refer to possible outcomes as “events.” In a roll of a die, for example, the “elementary events” are the numbers 1 through 6. Turning up one side represents the most disaggregate mutually exclusive outcome. Other events are compound, that is, they consist of more than one elementary event, such as the result “odd number” or “less than 4.” In this case “odd” and “less than 4” are not mutually exclusive. Compound events can...
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...Part I (Chapters 1 – 11) MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. A. Review of Basic Statistics (Chapters 1-11) Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain events. We use statistical methods and statistical analysis to make decisions in uncertain environment. Population: Sample: A population is the complete set of all items in which an investigator is interested. A sample is a subset of population values. & Example: Population - High school students - Households in the U.S. Sample - A sample of 30 students - A Gallup poll of 1,000 consumers - Nielson Survey of TV rating Random Sample: A random sample of n data values is one selected from the population in such a way that every different sample of size n has an equal chance of selection. & Example: Random Selection - Lotto numbers - Random numbers Random Variable: A variable takes different possible values for a given subject of study. Numerical Variable: A numerical variable takes some countable finite numbers or infinite numbers. Categorical Variable: A categorical variable takes values that belong to groups or categories. Data: Data are measured values of the variable. There are two types of data: quantitative data and qualitative data. 1 Part I (Chapters 1 – 11) Quantitative Data: Qualitative Data: & Example: 1. 2. 3. 3. 4. 5. 6. 7. 8. Statistics: Quantitative data are data measured on a numerical scale. Qualitative data are non-numerical...
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...Newsvendor Model Chapter 11 1 utdallas.edu/~metin Learning Goals Determine the optimal level of product availability – Demand forecasting – Profit maximization Service measures such as a fill rate utdallas.edu/~metin 2 Motivation Determining optimal levels (purchase orders) – Single order (purchase) in a season – Short lifecycle items 1 month: Printed Calendars, Rediform 6 months: Seasonal Camera, Panasonic 18 months, Cell phone, Nokia Motivating Newspaper Article for toy manufacturer Mattel Mattel [who introduced Barbie in 1959 and run a stock out for several years then on] was hurt last year by inventory cutbacks at Toys “R” Us, and officials are also eager to avoid a repeat of the 1998 Thanksgiving weekend. Mattel had expected to ship a lot of merchandise after the weekend, but retailers, wary of excess inventory, stopped ordering from Mattel. That led the company to report a $500 million sales shortfall in the last weeks of the year ... For the crucial holiday selling season this year, Mattel said it will require retailers to place their full orders before Thanksgiving. And, for the first time, the company will no longer take reorders in December, Ms. Barad said. This will enable Mattel to tailor production more closely to demand and avoid building inventory for orders that don't come. - Wall Street Journal, Feb. 18, 1999 utdallas.edu/~metin For tax (in accounting), option pricing (in finance)...
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...4 problem set 1: Normal Probability Distributions Page.285 Ex 6,8,10,12 6. x = 80, z=80-10015 = -1.33 z= 0.0918 1-0.0918 = 0.9082 8. x = 110, z=110-10015 = 0.67 z= 0.7486 z= 75-10015 = -1.67 z= 0.0475 0.7486-0.0475= 0.7011 (shaded area) 10. z= 0.84 (shaded) z= -0.84 x= 100+(-0.84∙15) = 87 (rounded) 12. . z= 2.33 x= 100+(2.33∙15) = 135 (rounded) Page 288 Ex 34 34.Appendix B Data Set: Duration of Shuttle Flights a. Find the mean and standard deviation, and verify that the data have a distribution that is roughly normal. Mean= 25317115 = 220.15 Standard Deviation=115253172-(25317)2115(115-1) = 86 (rounded) The normal distribution is 115 b. Treat the statistics from part (a) as if they are population parameters and assume a normal distribution to find the values of the quartiles 1,2 and 3. Mean= 220.15 Standard Deviation= 86 Q1 = 220.5 + (-0.67 ∙ 86)= 162.53 Q2= 220.5 + (0.00 ∙ 86) = 220.5 Q3=220.5 + (0.67 ∙ 86) = 277.77 Page.300 Ex 20 Quality Control: Sampling Distribution of Proportion after constructing a new manufacturing machine. 5 prototype integrated circuit chips are produced and it is found that 2 are defective (D) and 3 are acceptable (A). Assume that two if the chips are randomly selected with replacement from this population a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions...
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...Z-Scores A z-score, which is symbolized as z, is the statistic that relates the distance a score is relative to its mean when measured in standard deviation (SD) units (Heiman, 2012). The importance of a z-score is that it enables one to analyze data relative to scores. Z-Scores allow us to determine whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores, and how far a particular score is away from the mean. Evaluating the scores of Eric’s to determine different z-scores, we use the following computations that he computed where it takes a mean of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to his job. , Next we determine the z-score relative to the mean and the SD, to analyze the difference in time to accomplish these steps on different days. One day it took Eric 21 minutes to get to work, and computing the means from the minutes by the SD results in a z of +1.33, which tells you that Eric's time to get to work is 1.33 standard deviations from the mean. The z is positive because it is above the mean, and demonstrates that it took longer for Eric to leave home, park his car and walk to his job when the raw value of time was 21 minutes. On another day in which it took Eric 12 minutes to get to work, the z value resulted -1.66, demonstrating that the time it took Eric to get to work was below the mean, or it took less time. Observing the time in comparison to the mean, it is obvious...
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...CHAPTER 6: THE NORMAL DISTRIBUTION AND OTHER CONTINUOUS DISTRIBUTIONS 1. In its standardized form, the normal distribution a) has a mean of 0 and a standard deviation of 1. b) has a mean of 1 and a variance of 0. c) has an area equal to 0.5. d) cannot be used to approximate discrete probability distributions. ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: standardized normal distribution, properties 2. Which of the following about the normal distribution is NOT true? a) Theoretically, the mean, median, and mode are the same. b) About 2/3 of the observations fall within 1 standard deviation from the mean. c) It is a discrete probability distribution. d) Its parameters are the mean, , and standard deviation, . ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 3. If a particular batch of data is approximately normally distributed, we would find that approximately a) 2 of every 3 observations would fall between 1 standard deviation around the mean. b) 4 of every 5 observations would fall between 1.28 standard deviations around the mean. c) 19 of every 20 observations would fall between 2 standard deviations around the mean. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 4. For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770. The value of Z is a) 0.18. b) 0.81. c) 1.16. d) 1...
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...Week Three Quiz Comparing Variations: 1. For the following exercise, complete the following by using the data sets below (there are two separate sets of values, one shows the ages of Presidents from Washington through Jackson and the other shows the ages of the seven most recent Presidents): A. Find the mean, median, and range for each of the two data sets. First set of numbers: Mean: 58.28571 Median: 57 Range 4 Second set of numbers: Mean: 56.14286 Median: 54 Range 23 B. Find the standard deviation using the range rule of thumb for each of the data sets. Please show your work. (Please see Chapter 4, Section 4.3, page 173 of the text).Standard deviation = 61 – 57 = 4 4 / 4 = 1 1 is the standard deviation using the range rule of thumb. Standard deviation = 69 – 46 = 23 23 / 4 = 5.75 5.75 is the standard deviation using the range rule of thumb. C. Compare the two sets and describe what you discover. You could interpret that younger men are being elected president from past trends.The standard deviation shows an increase. The following data sets shows the ages of the first seven presidents (President Washington through President Jackson) and the seven most recent presidents including President Obama. Age is given at time of inauguration. First 7: 57 61 57 57 58 57 61 Second 7: 61 52 69 64 46 54 47 2. Any given data set consists of a set of numerical values. Please indicate by stating yes or no for each of the following statements whether or not it could be correct for...
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...University of Phoenix Material Distribution, Hypothesis Testing, and Error Worksheet Answer the following questions. Questions that are answered without the work will not receive full credit. When a question says explain or describe, please DO NOT copy word for word from a reference. You need to explain the concept so I know you understand what it means. For questions requiring material from Statdisk, make sure to turn labels on, take a screen capture (CTRL-Print Screen on most Windows-based computers), and paste the image into the worksheet. Crop the image as appropriate. 1. Describe a normal distribution in no more than 100 words (5 point). Answer: A normal distribution is a continuous random variable distribution with a bell shape, and has only two parameters: the mean, and the variance. A normal distribution can be represented by the formula: y=e^(-1/2)(x-μ/σ)/(σ√2pi). The mean can be any positive number and variance can be any positive number, so there are an infinite number of normal distributions. The shape of the distribution when graphed is symmetrical and bell-shaped. Use this information to answer questions 2-4. Following a brushfire, a forester takes core samples from the ten surviving Bigcone Douglas-fir trees in a test plot within the burn area, and a dendrochronologist determines the age of the source trees to be as follows (in years): 15 38 48 67 81 83 94 102 135 167 2. Construct a normal quantile plot in Statdisk, show the regression...
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...MAT 540 Quiz Answers 1) Deterministic techniques assume that no uncertainty exists in model parameters. Answer: TRUE Diff: 1 Page Ref: 489 Main Heading: Types of Probability Key words: deterministic techniques 2) Probabilistic techniques assume that no uncertainty exists in model parameters. Answer: FALSE Diff: 1 Page Ref: 489 Main Heading: Types of Probability Key words: probabilistic techniques 3) Objective probabilities that can be stated prior to the occurrence of an event are classical or a priori. Answer: TRUE Diff: 2 Page Ref: 489 Main Heading: Types of Probability Key words: objective probabilities, classical probabilities 4) Objective probabilities that are stated after the outcomes of an event have been observed are relative frequencies. Answer: TRUE Diff: 2 Page Ref: 489 Main Heading: Types of Probability Key words: relative frequencies 5) Relative frequency is the more widely used definition of objective probability. Answer: TRUE Diff: 1 Page Ref: 490 Main Heading: Types of Probability Key words: relative frequencies 6) Subjective probability is an estimate based on personal belief, experience, or knowledge of a situation. Answer: TRUE Diff: 2 Page Ref: 490 Main Heading: Types of Probability Key words: subjective probability 7) An experiment is an activity that results in one of several possible outcomes. Answer: TRUE Diff: 1 Page Ref: 491 Main Heading: Fundamentals of Probability Key words: experiment ...
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...Case Study#2 The XYZ Company Katharine Rally is the vice president of operations for the XYZ Company. She oversees operations at a plant that manufactures components for hydraulic systems. Katharine is concerned about the plant’s present production capability. She has reduced the decision situation to three alternatives. The first alternative, which is fully automation, would result in significant changes in present operations. The second alternative, which is semi-automation, involves fewer changes in present operations. The third alternative is to make no changes (do nothing). As a manager of the plant management team, you have been assigned the task of analyzing the alternatives and recommending a course of action. The capital investment and annual revenue for the first two alternatives are shown in the following table: |Alternative |Capital Investment |Future Sales |Annual Revenue | |A |$300,000 |Good |$250,000 | | | |Average |$100,000 | | | |Poor |$50,000 | |B |$85,000 |Good |$100,000 | | | ...
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...→ Calculate but don’t use in ANOVA * Also need RSSBLOCKS, RSSPP, and RSSM (CT and RSSTOTAL) * F values are calculated using the error from the same block * For t-test * Standard errors: * Error (b)n for interaction 9.78583 * 2 × Error (b)n for Factor M 2 × 9.78586 * 2 × Error (a)n for Factor PP 2 × 4.96756 * 2 critical-t values → t at 2 and t at 4 df i.e. 4.303 and 2.776 * Could ask: do ANOVA and t-test, or ANOVA and interpret results from F; Standard error for the difference (a or b); Conclusion: levels differ/do not differ at 1% etc. NS 13 – Non-parametric tests * Parametric tests for data with normal distribution (t, F or X2 distribution) * Non-parametric tests for * Categorical data, * Quantitative data divided into class intervals, * Small data sets, * Data sets without repetition of the TMTs. * Non- parametric tests * Medians, not Means * Usually rank your data * Single sample: * Sign test (No assumptions about distribution) * Rank test (assumes data comes from symmetrical distribution) * Wilcoxon’s symmetry test * For 2 independent samples * Mann-Whitney U test (Assumes distributions have same shape and equal...
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...1. (20 pt) Go to http://www.stat.tamu.edu/~west/ph/sampledist.html. Use a skewed distribution. Take 1000 samples of sizes 2, 10 and 100. Construct a mean for each sample and look at the distribution of the sample means (third row). Record the mean and SD in the population for the original random variable. Make a table and record the mean of the means, and the standard deviation of the mean for each sample size (2, 10 and 100). For a bell-shaped distribution we showed that the mean of the sample means is very similar to the mean in the population for all population sizes. This is the property of unbiasedness. From your results, does the mean appear to be unbiased when the original distribution of the data is skewed? For a bell-shaped distribution we showed that the standard deviation of the mean, or the standard error of the mean, decreases with sample size. Is this true for skewed distributions? Calculate the standard error of the mean that is expected for each sample size and record in the table. How did the observed result compared to your calculation? * * Mean of the Population: 15.5297 * Median of the Population: 12.2861 * Standard Deviation of the Population: 12.5078 * * Sample Size | * Mean of the means | * Standard Deviation | * Standard Error of the Mean | * 2 | * 15.7741 | * 8.8115 | * 6.23067 | * 10 | * 15.4568 | * 3.847 | * 1.21653 | * 100 | * 15.5223 | * 1.2636 | * 0.12636 | * ...
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...Some basic relationships of Probability Read ASW Chapter 4 or LR Chapter 4 | 2 | Theories of Probability - Classical theory, Relative Frequency theory, Axioms, Addition rule, Multiplication rule, Rule of at least one, Concept of Expected number of Success – Numerical Problems & Applications Case Problem: Hamilton County JudgesSIP: Morton International - Chicago, Illinois Read ASW Chapter 4 or LR Chapter 4 | 3 | Bayes Theorem – Theory, Problems & Applications, Probability revision using tabular approach Read ASW Chapter 4 or LR Chapter 4 | 4 | Probability Distribution - Meaning of Probability Distribution, Type of Probability Distribution, Need (Application) for Probability DistributionRead ASW Chapter 5 or LR Chapter 5 | 5 | Discrete Probability Distribution - Binomial Distribution – Applications, Numerical Problems, Excel & SPSS functions, Poisson Distribution...
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...Statistics for Business [pic] Discrete and Continuous Probability Distributions Business Statistics With Canadian Applications Hummelbrunner Rak Gray Third Edition Week6 Pages 261-263 chapter 8 Pages 288-314, 320-325 chapter 9 Arranged by: Neiloufar Aminneia Probability distribution A probability distribution is a list of all events of an experiment together with the probability associated with each event in a tabular form. It is used for business and economic problems. We learned frequency distribution to classify data, relating to actual observations and experiments but probability distribution describes how outcomes are expected to vary. Probability distribution for rolling a true die x P(x) Events Frequencies Probability 1 1 1/6 2 1 1/6 3 1 1/6 4 1 1/6 5 1 1/6 6 1 1/6 Total: 1 [pic] Probability distribution for tossing 3 coins x P(x) Events(# of Heads) Frequencies Probability 0 1 (TTT))...
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