13. Sampling distributions
|
|
- Arnold Thompson
- 6 years ago
- Views:
Transcription
1 13. Sampling distributions The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company
2 Objectives (PSLS Chapter 13) Sampling distributions Parameter versus statistic Sampling distributions Sampling distribution of the sample mean The central limit theorem x Sampling distribution of the sample proportion The law of large numbers pˆ
3 Parameter versus statistic Population: the entire group of individuals in which we are interested but usually can t assess directly. A parameter is a number summarizing the population. Parameters are usually unknown. Sample: the part of the population we actually examine and for which we do have data. A statistic is a number summarizing a sample. We often use a statistic to estimate an unknown population parameter. Population Sample
4 Sampling distributions Different random samples taken from the same population will give different statistics. But there is a predictable pattern in the long run. A statistic computed from a random sample is a random variable. The sampling distribution of a statistic is the probability distribution of that statistic for samples of a given size n taken from a given population.
5 Sampling distribution of x (the sample mean) The mean of the sampling distribution of x is μ. There is no tendency for a sample average to fall systematically above or below μ, even if the population distribution is skewed. x is an unbiased estimate of the population mean μ. The standard deviation of the sampling distribution of x is σ/ n. The standard deviation of the sampling distribution measures how much the sample statistic x varies from sample to sample. Averages are less variable than individual observations.
6 For Normally distributed populations When a variable in a population is Normally distributed, the sampling distribution of the sample mean x is also Normally distributed. Sample means population N(m,s) sampling distribution N(m,s/ n) population
7 The blood cholesterols of 14-year-old boys is ~ N(µ = 170, σ = 30) mg/dl. The middle 99.7% of cholesterol levels in boys is 80 to 260 mg/dl Cholesterol level (mg/dl) We consider random samples of 25 boys. The sampling distribution of average cholesterol levels is ~ N(µ = 170, σ = 30/ 25 = 6) mg/dl. The middle 99.7% of average cholesterol levels in boys is 152 to 188 mg/dl Average cholesterol level (mg/dl)
8 Deer mice (Peromyscus maniculatus) have a body length (excluding the tail) known to vary Normally, with a mean body length µ = 86 mm, and standard deviation σ = 8 mm. For random samples of 20 deer mice, the distribution of the sample mean body length is A) Normal, mean 86, standard deviation 8 mm. B) Normal, mean 86, standard deviation 20 mm. C) Normal, mean 86, standard deviation mm. D) Normal, mean 86, standard deviation 3.9 mm.
9 Standardizing a Normal sampling distribution (z) When the sampling distribution is Normal, we can standardize the value of a sample mean x to obtain a z-score. This z-score can then be used to find areas under the sampling distribution from Table B. x N(µ, σ/ n) z x m s n z N(0,1) Here, we work with the sampling distribution, and s/ n is its standard deviation (indicative of spread). Remember that s is the standard deviation of the original population.
10 Hypokalemia is diagnosed when blood potassium levels are low, below 3.5mEq/dl. Let s assume that we know a patient whose measured potassium levels vary daily according to N(m = 3.8, s = 0.2). If only one measurement is made, what's the probability that this patient will be misdiagnosed hypokalemic? ( x m) s z P(z < 1.5) = % 1.5 If instead measurements are taken on four separate days, what is the probability of such a misdiagnosis? z ( x m) P(z < 1.5) = % s n 0.2 4
11 The central limit theorem Central limit theorem: When randomly sampling from any population with mean m and standard deviation s, when n is large enough, the sampling distribution of x is approximately Normal: N(m,s/ n). The larger the sample size n, the better the approximation of Normality. This is very useful in inference: Many statistical tests assume Normality for the sampling distribution. The central limit theorem tells us that, if the sample size is large enough, we can safely make this assumption even if the raw data appear non-normal.
12 How large a sample size? It depends on the population distribution. More observations are required if the population distribution is far from Normal. A sample size of 25 or more is generally enough to obtain a Normal sampling distribution from a skewed population, even with mild outliers in the sample. A sample size of 40 or more will typically be good enough to overcome an extremely skewed population and mild (but not extreme) outliers in the sample. In many cases, n = 25 isn t a huge sample. Thus, even for strange population distributions we can assume a Normal sampling distribution of the sample mean, and work with it to solve problems.
13 Population with strongly skewed distribution Sampling distribution of for n = 2 observations x Sampling distribution of for n = 10 observations x Sampling distribution of for n = 25 observations x Even though the population (a) is strongly skewed, the sampling distribution of when n = 25 (d) is approximately Normal, as expected from the central limit theorem. x
14 How do we know if the population is Normal or not? Sometimes we are told that a variable has an approximately Normal distribution (e.g. large studies on human height or bone density). Most of the time, we just don t know. All we have is sample data. We can summarize the data with a histogram and describe its shape. If the sample is random, the shape of the histogram should be similar to the shape of the population distribution. The central limit theorem can help guess whether the sampling distribution should look roughly Normal or not.
15 Number of subjects Frequency (a) Angle of big toe deformations in 38 patients: Symmetrical, one small outlier Population likely close to Normal Sampling distribution ~ Normal More HAV angle (b) Histogram of number of fruit per day for 74 adolescent girls Skewed, no outlier Population likely skewed Sampling distribution ~ Normal given large sample size
16 Atlantic acorn sizes (in cm 3 ) Sample of 28 acorns: Describe the distribution of the sample. What can you assume about the population distribution? Frequency M Acorn sizes What would be the shape of the sampling distribution: For samples of size 5? For samples of size 15? For samples of size 50?
17 Chapter 12 reminder: Sampling distribution of a count A population contains a proportion p of successes. If the population is much larger than the sample, the count X of successes in an SRS of size n has approximately the binomial distribution B(n, p) with mean m and standard deviation s: m np s npq np( 1 p) If n is large, and p is not too close to 0 or 1, this binomial distribution can be approximated by the Normal distribution: N m np, s np(1 p)
18 Sampling distribution of a proportion p When randomly sampling from a population with proportion p of successes, the sampling distribution of the sample proportion p [ p hat ] has mean and standard deviation: m pˆ p s pˆ p(1 n p) p is an unbiased estimator the population proportion p. Larger samples usually give closer estimates of the population proportion p.
19 Normal approximation The sampling distribution of p is never exactly Normal. But as the sample size increases, the sampling distribution of p becomes approximately Normal. The Normal approximation is most accurate for any fixed n when p is close to 0.5, and least accurate when p is near 0 or near 1. When n is large, and p is not too close to 0 or 1, the sampling distribution of p is approximately: N m p, s p(1 p) n
20 The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is about 8%. We wish to take a random sample of size 125 from this population. What is the probability that 10% or more in the sample are color blind? A sample size of 125 is enough to use of the Normal approximation (np = 10 and n(1 p) = 115). Normal approximation for p sampling distribution: Np 0.08, p(1 p) / n z = (p - p)/σ = ( )/0.024 = P(z 0.82) = from Table B Or P(p 0.10) = 1 NORM.DIST(0.10, 0.08, 0.024, 1) = (Excel) = normalcdf (0.10, 1E99, 0.08, 0.024) = (TI-83)
21 The law of large numbers Law of large numbers: As the number of randomly drawn observations (n) in a sample increases, the mean of the sample (x ) gets closer and closer to the population mean m (quantitative variable). the sample proportion ( ) gets closer and closer to the population proportion p (categorical variable). pˆ
22 Note: When sampling randomly from a given population: The law of large numbers describes what would happen if we took samples of increasing size n. A sampling distribution describes what would happen if we took all possible random samples of a fixed size n. Both are conceptual ideas with many important practical applications. We rely on their known mathematical properties, but we don t actually build them from data.
3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability
Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values
More information4/19/2009. Probability Distributions. Inference. Example 1. Example 2. Parameter versus statistic. Normal Probability Distribution N
Probability Distributions Normal Probability Distribution N Chapter 6 Inference It was reported that the 2008 Super Bowl was watched by 97.5 million people. But how does anyone know that? They certainly
More informationACMS Statistics for Life Sciences. Chapter 13: Sampling Distributions
ACMS 20340 Statistics for Life Sciences Chapter 13: Sampling Distributions Sampling We use information from a sample to infer something about a population. When using random samples and randomized experiments,
More informationChapter 7. Inference for Distributions. Introduction to the Practice of STATISTICS SEVENTH. Moore / McCabe / Craig. Lecture Presentation Slides
Chapter 7 Inference for Distributions Introduction to the Practice of STATISTICS SEVENTH EDITION Moore / McCabe / Craig Lecture Presentation Slides Chapter 7 Inference for Distributions 7.1 Inference for
More informationCHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10. Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Means Learning
More informationCHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.2 Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Means Learning
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 8 Estimating with Confidence n 8.1 Confidence Intervals: The Basics n 8.2
More information11. The Normal distributions
11. The Normal distributions The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 11) The Normal distributions Normal distributions The
More informationSampling Distribution Models. Chapter 17
Sampling Distribution Models Chapter 17 Objectives: 1. Sampling Distribution Model 2. Sampling Variability (sampling error) 3. Sampling Distribution Model for a Proportion 4. Central Limit Theorem 5. Sampling
More informationChapter 18. Sampling Distribution Models /51
Chapter 18 Sampling Distribution Models 1 /51 Homework p432 2, 4, 6, 8, 10, 16, 17, 20, 30, 36, 41 2 /51 3 /51 Objective Students calculate values of central 4 /51 The Central Limit Theorem for Sample
More informationChapter 18: Sampling Distribution Models
Chapter 18: Sampling Distribution Models Suppose I randomly select 100 seniors in Scott County and record each one s GPA. 1.95 1.98 1.86 2.04 2.75 2.72 2.06 3.36 2.09 2.06 2.33 2.56 2.17 1.67 2.75 3.95
More informationInference for Distributions Inference for the Mean of a Population
Inference for Distributions Inference for the Mean of a Population PBS Chapter 7.1 009 W.H Freeman and Company Objectives (PBS Chapter 7.1) Inference for the mean of a population The t distributions The
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE The One-Sample z Interval for a Population Mean In Section 8.1, we estimated the
More informationChapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.
Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,
More informationStatistic: a that can be from a sample without making use of any unknown. In practice we will use to establish unknown parameters.
Chapter 9: Sampling Distributions 9.1: Sampling Distributions IDEA: How often would a given method of sampling give a correct answer if it was repeated many times? That is, if you took repeated samples
More informationDescribing distributions with numbers
Describing distributions with numbers A large number or numerical methods are available for describing quantitative data sets. Most of these methods measure one of two data characteristics: The central
More information7.1: What is a Sampling Distribution?!?!
7.1: What is a Sampling Distribution?!?! Section 7.1 What Is a Sampling Distribution? After this section, you should be able to DISTINGUISH between a parameter and a statistic DEFINE sampling distribution
More informationCh18 links / ch18 pdf links Ch18 image t-dist table
Ch18 links / ch18 pdf links Ch18 image t-dist table ch18 (inference about population mean) exercises: 18.3, 18.5, 18.7, 18.9, 18.15, 18.17, 18.19, 18.27 CHAPTER 18: Inference about a Population Mean The
More informationCHAPTER 5 Probabilistic Features of the Distributions of Certain Sample Statistics
CHAPTER 5 Probabilistic Features of the Distributions of Certain Sample Statistics Key Words Sampling Distributions Distribution of the Sample Mean Distribution of the difference between Two Sample Means
More informationDescribing distributions with numbers
Describing distributions with numbers A large number or numerical methods are available for describing quantitative data sets. Most of these methods measure one of two data characteristics: The central
More informationChapter 7: Sampling Distributions
+ Chapter 7: Sampling Distributions Section 7.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE + Chapter 7 Sampling Distributions n 7.1 What is a Sampling Distribution? n 7.2 n
More information6.3 Use Normal Distributions. Page 399 What is a normal distribution? What is standard normal distribution? What does the z-score represent?
6.3 Use Normal Distributions Page 399 What is a normal distribution? What is standard normal distribution? What does the z-score represent? Normal Distribution and Normal Curve Normal distribution is one
More informationFrancine s bone density is 1.45 standard deviations below the mean hip bone density for 25-year-old women of 956 grams/cm 2.
Chapter 3 Solutions 3.1 3.2 3.3 87% of the girls her daughter s age weigh the same or less than she does and 67% of girls her daughter s age are her height or shorter. According to the Los Angeles Times,
More informationOne-sample categorical data: approximate inference
One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
More informationChapter 1. Looking at Data
Chapter 1 Looking at Data Types of variables Looking at Data Be sure that each variable really does measure what you want it to. A poor choice of variables can lead to misleading conclusions!! For example,
More informationInference for Proportions
Inference for Proportions Marc H. Mehlman marcmehlman@yahoo.com University of New Haven Based on Rare Event Rule: rare events happen but not to me. Marc Mehlman (University of New Haven) Inference for
More information1 Binomial Probability [15 points]
Economics 250 Assignment 2 (Due November 13, 2017, in class) i) You should do the assignment on your own, Not group work! ii) Submit the completed work in class on the due date. iii) Remember to include
More informationSections 3.4 and 3.5
Sections 3.4 and 3.5 Timothy Hanson Department of Statistics, University of South Carolina Stat 205: Elementary Statistics for the Biological and Life Sciences 1 / 20 Continuous variables So far we ve
More informationUnit Two Descriptive Biostatistics. Dr Mahmoud Alhussami
Unit Two Descriptive Biostatistics Dr Mahmoud Alhussami Descriptive Biostatistics The best way to work with data is to summarize and organize them. Numbers that have not been summarized and organized are
More informationNotice that these facts about the mean and standard deviation of X are true no matter what shape the population distribution has
7.3.1 The Sampling Distribution of x- bar: Mean and Standard Deviation The figure above suggests that when we choose many SRSs from a population, the sampling distribution of the sample mean is centered
More informationSections 3.4 and 3.5
Sections 3.4 and 3.5 Shiwen Shen Department of Statistics University of South Carolina Elementary Statistics for the Biological and Life Sciences (STAT 205) Continuous variables So far we ve dealt with
More informationContinuous Probability Distributions
1 Chapter 5 Continuous Probability Distributions 5.1 Probability density function Example 5.1.1. Revisit Example 3.1.1. 11 12 13 14 15 16 21 22 23 24 25 26 S = 31 32 33 34 35 36 41 42 43 44 45 46 (5.1.1)
More informationInference for Distributions Inference for the Mean of a Population. Section 7.1
Inference for Distributions Inference for the Mean of a Population Section 7.1 Statistical inference in practice Emphasis turns from statistical reasoning to statistical practice: Population standard deviation,
More informationChapter 7 Sampling Distributions
Statistical inference looks at how often would this method give a correct answer if it was used many many times. Statistical inference works best when we produce data by random sampling or randomized comparative
More informationInference for Proportions
Inference for Proportions Marc H. Mehlman marcmehlman@yahoo.com University of New Haven Based on Rare Event Rule: rare events happen but not to me. (University of New Haven) Inference for Proportions 1
More informationAP Statistics Cumulative AP Exam Study Guide
AP Statistics Cumulative AP Eam Study Guide Chapters & 3 - Graphs Statistics the science of collecting, analyzing, and drawing conclusions from data. Descriptive methods of organizing and summarizing statistics
More informationDescriptive Statistics-I. Dr Mahmoud Alhussami
Descriptive Statistics-I Dr Mahmoud Alhussami Biostatistics What is the biostatistics? A branch of applied math. that deals with collecting, organizing and interpreting data using well-defined procedures.
More informationPerformance of fourth-grade students on an agility test
Starter Ch. 5 2005 #1a CW Ch. 4: Regression L1 L2 87 88 84 86 83 73 81 67 78 83 65 80 50 78 78? 93? 86? Create a scatterplot Find the equation of the regression line Predict the scores Chapter 5: Understanding
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Boxplots and standard deviations Suhasini Subba Rao Review of previous lecture In the previous lecture
More information23. Inference for regression
23. Inference for regression The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 23) Inference for regression The regression model Confidence
More informationTastitsticsss? What s that? Principles of Biostatistics and Informatics. Variables, outcomes. Tastitsticsss? What s that?
Tastitsticsss? What s that? Statistics describes random mass phanomenons. Principles of Biostatistics and Informatics nd Lecture: Descriptive Statistics 3 th September Dániel VERES Data Collecting (Sampling)
More informationCounting principles, including permutations and combinations.
1 Counting principles, including permutations and combinations. The binomial theorem: expansion of a + b n, n ε N. THE PRODUCT RULE If there are m different ways of performing an operation and for each
More informationLecture 7: Confidence interval and Normal approximation
Lecture 7: Confidence interval and Normal approximation 26th of November 2015 Confidence interval 26th of November 2015 1 / 23 Random sample and uncertainty Example: we aim at estimating the average height
More informationInterpret Standard Deviation. Outlier Rule. Describe the Distribution OR Compare the Distributions. Linear Transformations SOCS. Interpret a z score
Interpret Standard Deviation Outlier Rule Linear Transformations Describe the Distribution OR Compare the Distributions SOCS Using Normalcdf and Invnorm (Calculator Tips) Interpret a z score What is an
More informationChapitre 3. 5: Several Useful Discrete Distributions
Chapitre 3 5: Several Useful Discrete Distributions 5.3 The random variable x is not a binomial random variable since the balls are selected without replacement. For this reason, the probability p of choosing
More informationStat 101 Exam 1 Important Formulas and Concepts 1
1 Chapter 1 1.1 Definitions Stat 101 Exam 1 Important Formulas and Concepts 1 1. Data Any collection of numbers, characters, images, or other items that provide information about something. 2. Categorical/Qualitative
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Probability and statistics: Module 25. Inference for means
1 Supporting Australian Mathematics Project 2 3 4 6 7 8 9 1 11 12 A guide for teachers Years 11 and 12 Probability and statistics: Module 2 Inference for means Inference for means A guide for teachers
More information10.1. Comparing Two Proportions. Section 10.1
/6/04 0. Comparing Two Proportions Sectio0. Comparing Two Proportions After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET
More information20 Hypothesis Testing, Part I
20 Hypothesis Testing, Part I Bob has told Alice that the average hourly rate for a lawyer in Virginia is $200 with a standard deviation of $50, but Alice wants to test this claim. If Bob is right, she
More informationMA 1125 Lecture 33 - The Sign Test. Monday, December 4, Objectives: Introduce an example of a non-parametric test.
MA 1125 Lecture 33 - The Sign Test Monday, December 4, 2017 Objectives: Introduce an example of a non-parametric test. For the last topic of the semester we ll look at an example of a non-parametric test.
More informationWhat is a parameter? What is a statistic? How is one related to the other?
Chapter Seven: SAMPLING DISTRIBUTIONS 7.1 Sampling Distributions Read 424 425 What is a parameter? What is a statistic? How is one related to the other? Example: Identify the population, the parameter,
More informationChapter 7 Discussion Problem Solutions D1 D2. D3.
Chapter 7 Discussion Problem Solutions D1. The agent can increase his sample size to a value greater than 10. The larger the sample size, the smaller the spread of the distribution of means and the more
More informationChapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
More informationDiscrete Multivariate Statistics
Discrete Multivariate Statistics Univariate Discrete Random variables Let X be a discrete random variable which, in this module, will be assumed to take a finite number of t different values which are
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationChapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67
Chapter 6 The Standard Deviation as a Ruler and the Normal Model 1 /67 Homework Read Chpt 6 Complete Reading Notes Do P129 1, 3, 5, 7, 15, 17, 23, 27, 29, 31, 37, 39, 43 2 /67 Objective Students calculate
More informationMath/Stat 352 Lecture 10. Section 4.11 The Central Limit Theorem
Math/Stat 352 Lecture 10 Section 4.11 The Central Limit Theorem 1 Summing random variables Summing random variables Summing random variables Generally summation changes the shape of the distribution: range
More informationChapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 1 Activity 9A, pp. 486-487 2 We ve just begun a sampling distribution! Strictly speaking, a sampling distribution is: A theoretical distribution of the values of a statistic
More informationChapter. The Normal Probability Distribution 7/24/2011. Section 7.1 Properties of the Normal Distribution
Chapter The Normal Probability Distribution 3 7 Section 7.1 Properties of the Normal Distribution 2010 Pearson Prentice Hall. All rights 2010 reserved Pearson Prentice Hall. All rights reserved 7-2 Illustrating
More informationIV. The Normal Distribution
IV. The Normal Distribution The normal distribution (a.k.a., a the Gaussian distribution or bell curve ) is the by far the best known random distribution. It s discovery has had such a far-reaching impact
More informationDETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET a confidence interval to compare two proportions.
Section 0. Comparing Two Proportions Learning Objectives After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET a confidence
More informationIntroduction to Estimation. Martina Litschmannová K210
Introduction to Estimation Martina Litschmannová martina.litschmannova@vsb.cz K210 Populations vs. Sample A population includes each element from the set of observations that can be made. A sample consists
More informationLecture Topic 4: Chapter 7 Sampling and Sampling Distributions
Lecture Topic 4: Chapter 7 Sampling and Sampling Distributions Statistical Inference: The aim is to obtain information about a population from information contained in a sample. A population is the set
More informationMeasures of Central Tendency. Mean, Median, and Mode
Measures of Central Tendency Mean, Median, and Mode Population study The population under study is the 20 students in a class Imagine we ask the individuals in our population how many languages they speak.
More informationSign test. Josemari Sarasola - Gizapedia. Statistics for Business. Josemari Sarasola - Gizapedia Sign test 1 / 13
Josemari Sarasola - Gizapedia Statistics for Business Josemari Sarasola - Gizapedia 1 / 13 Definition is a non-parametric test, a special case for the binomial test with p = 1/2, with these applications:
More informationChapter 10: Comparing Two Populations or Groups
Chapter 10: Comparing Two Populations or Groups Sectio0.1 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 10 Comparing Two Populations or Groups 10.1 10.2 Comparing Two Means
More information5-1. For which functions in Problem 4-3 does the Central Limit Theorem hold / fail?
Ismor Fischer, 8/1/008 Stat 541 / 5-9 5.3 Problems 5-1. For which functions in Problem 4-3 does the Central Limit Theorem hold / fail? 5-. Refer to Problem 4-9. (a) Suppose that a random sample of n =
More informationElementary Statistics
Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q:
More informationBIOS 2041: Introduction to Statistical Methods
BIOS 2041: Introduction to Statistical Methods Abdus S Wahed* *Some of the materials in this chapter has been adapted from Dr. John Wilson s lecture notes for the same course. Chapter 0 2 Chapter 1 Introduction
More informationCHAPTER 7 Sampling Distributions
CHAPTER 7 Sampling Distributions 7.3 Sample Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Sample Means Learning Objectives After this section,
More informationCHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.1 Comparing Two Proportions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Proportions
More informationACMS Statistics for Life Sciences. Chapter 11: The Normal Distributions
ACMS 20340 Statistics for Life Sciences Chapter 11: The Normal Distributions Introducing the Normal Distributions The class of Normal distributions is the most widely used variety of continuous probability
More informationThe Components of a Statistical Hypothesis Testing Problem
Statistical Inference: Recall from chapter 5 that statistical inference is the use of a subset of a population (the sample) to draw conclusions about the entire population. In chapter 5 we studied one
More informationDifference Between Pair Differences v. 2 Samples
1 Sectio1.1 Comparing Two Proportions Learning Objectives After this section, you should be able to DETERMINE whether the conditions for performing inference are met. CONSTRUCT and INTERPRET a confidence
More informationChapters 1 & 2 Exam Review
Problems 1-3 refer to the following five boxplots. 1.) To which of the above boxplots does the following histogram correspond? (A) A (B) B (C) C (D) D (E) E 2.) To which of the above boxplots does the
More informationLecture 10A: Chapter 8, Section 1 Sampling Distributions: Proportions
Lecture 10A: Chapter 8, Section 1 Sampling Distributions: Proportions Typical Inference Problem Definition of Sampling Distribution 3 Approaches to Understanding Sampling Dist. Applying 68-95-99.7 Rule
More informationChapter 2 Solutions Page 15 of 28
Chapter Solutions Page 15 of 8.50 a. The median is 55. The mean is about 105. b. The median is a more representative average" than the median here. Notice in the stem-and-leaf plot on p.3 of the text that
More informationInferential Statistics
Inferential Statistics Part 1 Sampling Distributions, Point Estimates & Confidence Intervals Inferential statistics are used to draw inferences (make conclusions/judgements) about a population from a sample.
More informationTHE SAMPLING DISTRIBUTION OF THE MEAN
THE SAMPLING DISTRIBUTION OF THE MEAN COGS 14B JANUARY 26, 2017 TODAY Sampling Distributions Sampling Distribution of the Mean Central Limit Theorem INFERENTIAL STATISTICS Inferential statistics: allows
More informationSampling Distributions. Introduction to Inference
Sampling Distributions Introduction to Inference Parameter A parameter is a number that describes the population. A parameter always exists but in practice we rarely know it s value because we cannot examine
More informationSTA Why Sampling? Module 6 The Sampling Distributions. Module Objectives
STA 2023 Module 6 The Sampling Distributions Module Objectives In this module, we will learn the following: 1. Define sampling error and explain the need for sampling distributions. 2. Recognize that sampling
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationChapter 15 Sampling Distribution Models
Chapter 15 Sampling Distribution Models 1 15.1 Sampling Distribution of a Proportion 2 Sampling About Evolution According to a Gallup poll, 43% believe in evolution. Assume this is true of all Americans.
More informationLecture 6: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries)
Lecture 6: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries) Summarize with Shape, Center, Spread Displays: Stemplots, Histograms Five Number Summary, Outliers, Boxplots Cengage Learning
More informationDescriptive statistics
Patrick Breheny February 6 Patrick Breheny to Biostatistics (171:161) 1/25 Tables and figures Human beings are not good at sifting through large streams of data; we understand data much better when it
More informationStatistical Intervals (One sample) (Chs )
7 Statistical Intervals (One sample) (Chs 8.1-8.3) Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to normally distributed with expected value µ and
More informationA C E. Answers Investigation 4. Applications
Answers Applications 1. 1 student 2. You can use the histogram with 5-minute intervals to determine the number of students that spend at least 15 minutes traveling to school. To find the number of students,
More informationHarvard University. Rigorous Research in Engineering Education
Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected
More informationy = a + bx 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation Review: Interpreting Computer Regression Output
12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation y = a + bx y = dependent variable a = intercept b = slope x = independent variable Section 12.1 Inference for Linear
More informationThe Normal Distribution. Chapter 6
+ The Normal Distribution Chapter 6 + Applications of the Normal Distribution Section 6-2 + The Standard Normal Distribution and Practical Applications! We can convert any variable that in normally distributed
More informationChapter 3. Measuring data
Chapter 3 Measuring data 1 Measuring data versus presenting data We present data to help us draw meaning from it But pictures of data are subjective They re also not susceptible to rigorous inference Measuring
More informationWhat is statistics? Statistics is the science of: Collecting information. Organizing and summarizing the information collected
What is statistics? Statistics is the science of: Collecting information Organizing and summarizing the information collected Analyzing the information collected in order to draw conclusions Two types
More informationChapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
More informationStatistical Inference. Section 9.1 Significance Tests: The Basics. Significance Test. The Reasoning of Significance Tests.
Section 9.1 Significance Tests: The Basics Significance Test A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess.
More informationChapter 23. Inference About Means
Chapter 23 Inference About Means 1 /57 Homework p554 2, 4, 9, 10, 13, 15, 17, 33, 34 2 /57 Objective Students test null and alternate hypotheses about a population mean. 3 /57 Here We Go Again Now that
More informationWhat is Statistics? Statistics is the science of understanding data and of making decisions in the face of variability and uncertainty.
What is Statistics? Statistics is the science of understanding data and of making decisions in the face of variability and uncertainty. Statistics is a field of study concerned with the data collection,
More informationØ Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.
Statistical Tools in Evaluation HPS 41 Fall 213 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific
More informationChapter 10: Comparing Two Populations or Groups
Chapter 10: Comparing Two Populations or Groups Sectio0.1 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 10 Comparing Two Populations or Groups 10.1 10.2 Comparing Two Means
More informationContinuous Probability Distributions
1 Chapter 5 Continuous Probability Distributions 5.1 Probability density function Example 5.1.1. Revisit Example 3.1.1. 11 12 13 14 15 16 21 22 23 24 25 26 S = 31 32 33 34 35 36 41 42 43 44 45 46 (5.1.1)
More informationChapter 7: Sampling Distributions
Chapter 7: Sampling Distributions Section 7.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 7 Sampling Distributions 7.1 What is a Sampling Distribution? 7.2 7.3 Sample
More information