ACMS Statistics for Life Sciences. Chapter 11: The Normal Distributions
|
|
- Aileen Hodge
- 5 years ago
- Views:
Transcription
1 ACMS Statistics for Life Sciences Chapter 11: The Normal Distributions
2 Introducing the Normal Distributions The class of Normal distributions is the most widely used variety of continuous probability distributions. Normal density curves are symmetric, single-peaked, and bell-shaped. The are not normal in the sense of typical or boring, but they are actually quite special.
3 !"#$%&'(#)*+,#-+./0 3*455#0/6/,78/0#2 9"7+:*,#,*;#7<#/++7 2/,8#+*>.7"*,.?/#>2/ 7<#>2/#:/>270#7<#,/ Why Normal?!"#$%&'()*+,- In 1809 Carl Friedrich Gauss developed his normal law of errors to help rationalize the use of the method of least squares.!!
4 Why Normal? Many years ago I called the Laplace-Gaussian curve the normal curve, which... has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another abnormal.!"#$%&'()*+,-!"#$%&%'#()&#*+&,&-#..'/& 01'&2#3.#-'45#6))7#$& -6(8'&01'&$+(9#.&-6(8':& ;17-1<<<1#)&01'& /7)#/8#$0#*'&+=&.'#/7$*& 3'+3.'&0+&>'.7'8'&01#0&#..& +01'(&/7)0(7>607+$)&+=& =('?6'$-%&#('&7$&+$'& B#(.&C'#()+$&DEFGHI -Karl Pearson (1920)
5 !"#$%&'()#*+,$-'(./#$(0#%1 Francis Galton s Bean Machine The first generator of Normal random variables.!!
6 !"#$%"&'#$()$*(+,&-./0 The Shape of Normality!!
7 !"#$%"&'#$()$*(+,&-./0 The Shape of Normality 7&03#+)(/&$2/(&) 7&03#+)(/&$2/(&)!"#$%#&'()*$+,-.#$/0$1$21-)(+,31-$4/-513$ The density curve of a particular Normal distribution is described by %(')-(6,)(/&$('$%#'+-(6#%$6*$()'$5#1&$!$1&%$()'$ its mean µ and its stand deviation σ. ')1&%1-%$%#.(1)(/&$"# The equation of the density curve is!! f (x) = 1 2π e 1 2( x µ σ ) 2.
8 Mean and Standard Deviation!"#$%#$&%'(#$&#)&%*"+,#(,-$!"#$%&$%'(")'*)#$'*)+),-'."#$%)/'0")+)' (")'.1+2)'&/'.)$()+)34 Changing the mean, µ, merely changes where the curve is centered.!"#"$%&"$'&"(%' )"#"* :5 :6 :7 :8 : ;5!! Here are Normal curves with µ = 10, 15, and 20, and σ = 3.
9 Mean and Standard Deviation!"#$%#$&%'(#$&#)&%*"+,#(,-$ Changing the standard deviation, σ, changes the spread of the!"#$%&$%'(")'*(#$+#,+'+)-&#(&.$'/"#$%)*' curve. (")'*0,)#+'.1'(")'/2,-)3 This also changes the height (since the area = 1). 4.()'("&*'5&66'#6*.'/"#$%)'(")'")&%"(''''' 78,)#'9':;!! Here are curves with µ = 15 and σ = 2, 4, and 6.
10 Why Care About Normal Distributions? 1. They provide good descriptions of real data, including many biological characteristics, such as blood pressure, bone density, heights, and yields of corn. 2. They provide good approximations of many chance outcomes, such as the proportion of boys over many hospital births. 3. Many statistical inference methods rely on Normal distributions. (We ll see this in chapter 13 and beyond.)
11 Warning! Do not assume that every variable has a Normal distribution!! For example, the guinea pig survival times are skewed to the right. ''()*# -.&/0*#+&'# 0#."$33!"#$%$&' 60*7# %.)*'#&$2#.:+%'#&-*# %"#%+*#!
12 A Few Words on Notation There is a common shorthand for Normal distributions. A Normal distribution with mean µ and standard deviation σ is denoted N(µ, σ).
13 The Rule For a Normal distribution with mean µ and standard deviation σ, (i.e. N(µ, σ)), approximately 68% of observations fall within σ of µ; approximately 95% of observations fall within 2σ of µ; and approximately 99.7% of observations fall within 3σ of µ. This rule holds for all Normal distributions.
14 The Rule!"#$%&'()'((*+$,-.#!!
15 !"#$%&'()*+$,-$.,/0($1,2&0 Heights of Young Women!"#$%&'()*(+),-$(.)/"-(0$"1(23(&)(45(06"( The heights of young women between ages 18 to 24 are 0776)8#/0&"9+(:)6/099+(1#'&6#;,&"1(.#&%((((((((((((((( approximately Normally distributed with µ = 64.5 in. and σ = 2.5 in.!"#"<5=>(#-=(0-1($"#"%&'"()&!!
16 !"#$%&'()*+$,-$.,/0($1,2&0 Heights of Young Women!"#$%&'%$"(%)*+,#,&-&$.%$"#$%#%/+0#1%&'% What is the probability that a young woman is taller than 62 inches? $#--(*%$"#1%23%&14"('5 (% between 62 and 67) + (% above 67) = (% above 62)!! 68% + 16% = 84%
17 Standard Normal Distribution There are many possible Normal distributions (one for every µ and positive σ). By sliding and stretching the curve, we can transform any Normal distribution to any other Normal distribution. Not only do Normal distributions share common properties. We single out the Normal distribution N(0, 1), and call it the standard Normal distribution. And we call the transformation of an arbitrary Normal distribution to the standard one standardizing.
18 Standardizing If x is an observation from the Normal distribution N(µ, σ), the standardized value of x is z = x µ σ The standardized values are often called z-scores.
19 z-scores z-scores measure how many standard deviations an observation is away from the mean. A positive z-score indicates the observation is greater than the mean. A negative z-score indicates the observation is less than the mean.
20 Heights of Young Women Recall the height distribution of young women is N(64.5, 2.5). The standardized height is z = height A woman 70 inches tall has the z-score z = = 2.2. A woman 5 feet (60 inches) tall has the z-score z = = 1.8.
21 !"#$"#%&'()*+,&-)(.+.","/"01 Finding Normal Probabilities!"#$"#%&'()*+&(,-$./%#&,%&$/01#(2&3,%4/1& 5%,0/0)1)$)#(&/%#&+)6#*&/(&!"#"$%&'()7 Whether using software or tables, Normal probabilities are given as cumulative probabilities. 8"#&9'4'1/$)6#&5%,0/0)1)$:&-,%&/&6/1'#&;&)(&$"#& The cumulative probability for a value x is the proportion of 5%,5,%$),*&,-&,0(#%6/$),*(&/$&,%&0#1,.&;7 observations less than or equal to x.!!
22 Tips for Finding Normal Probabilities We use the addition rule and the complement rule to find probabilities. Recall that the probability of any individual value is 0. So, P(X 40) = P(X < 40) + P(X = 40) = P(X < 40). Sketching a picture of the area you want can be very helpful.
23 Methods of Finding Normal Probabilities Normal Curve applet on the website CrunchIt! distribution calculator Standard Normal Tables
24 The Standard Normal Table!"#$%&'()'*)$+,*-'.$!'/.#!!
25 The Standard Normal Table.0062 is the area under N(0,1) left of z = -2.50!
26 The Standard Normal Table.0062 is the area under N(0,1) left of z = is the area under N(0,1) left of z = -2.51!
27 The Standard Normal Table.0062 is the area under N(0,1) left of z = is the area under N(0,1) left of z = is the area under N(0,1) left of z = -2.51!
28 Heights of Young Women 1 What is the probability that a randomly selected young woman measures between 60 and 68 inches tall? Recall, our distribution is N(64.5, 2.5). First, let s sketch the density curve to find the cumulative probabilities we need.
29 Heights of Young Women!"#$%&'()*+$,-$.,/0($1,2&0 We need to find the area less than 68 inches and subtract the area less than 60 inches.!!
30 Heights of Young Women 2 First we find the z-scores by standardizing: If then it follows that x < 68, x < Thus if we set z = x , we have 1.8 z < 1.4.
31 !"#$%&'()*+$,-$.,/0($1,2&0 Heights of Young Women 3!"#$#%$&'%$()%$(*+,%$("$-./0$()%$*1%*'2 Second, we use the table to find the areas.!
32 Heights of Young Women 4 Lastly, we finish the calculation. Area between -1.8 and 1.4 = (area left of 1.4) (area left of -1.8) = = The probability that a randomly selected young woman measures between 60 and 68 inches tall is about 0.88 or 88%.
33 Question 7 Using N(1, 5), for what value of a do we have P(X < a) = 1 4? Standardize a: z a = a 1 5 Use table to find z a such that P(Z < z a ) = 1 4. P(Z < 0.68) = P(Z < 0.67) = Choose z a = 0.67 since that is closer to Now solve for a: z a = 0.67 = a 1 5 = a = (5)( 0.67) + 1 = 2.35
34 The Other Quartile Using N(1, 5), for what value of a do we have P(X < a) = 1 4? a = 2.35 Now, for what value of a do we have P(X > a) = 0.25? P(X < a) = = z a = a 1 5 The table gives P(Z < 0.67) = a = (5)(0.67) + 1 = 4.35
35 General Procedure To solve P(X < a) = p for some normal distribution N(µ, σ): Find the corresponding z a value using the table for the standard normal distribution: P(Z < z a ) = p. Use algebra to solve for a: a = µ + σz a.
11. 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 informationChapter 3: The Normal Distributions
Chapter 3: The Normal Distributions http://www.yorku.ca/nuri/econ2500/econ2500-online-course-materials.pdf graphs-normal.doc / histogram-density.txt / normal dist table / ch3-image Ch3 exercises: 3.2,
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 informationContinuous random variables
Continuous random variables A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The total area under a density
More informationStatistics Lecture 3
Statistics 111 - Lecture 3 Continuous Random Variables The probable is what usually happens. (Aristotle ) Moore, McCabe and Craig: Section 4.3,4.5 Continuous Random Variables Continuous random variables
More informationThe empirical ( ) rule
The empirical (68-95-99.7) rule With a bell shaped distribution, about 68% of the data fall within a distance of 1 standard deviation from the mean. 95% fall within 2 standard deviations of the mean. 99.7%
More informationThe normal distribution
The normal distribution Patrick Breheny September 29 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/28 A common histogram shape The normal curve Standardization Location-scale families A histograms
More informationMath/Stat 352 Lecture 9. Section 4.5 Normal distribution
Math/Stat 352 Lecture 9 Section 4.5 Normal distribution 1 Abraham de Moivre, 1667-1754 Pierre-Simon Laplace (1749 1827) A French mathematician, who introduced the Normal distribution in his book The doctrine
More informationLet us think of the situation as having a 50 sided fair die; any one number is equally likely to appear.
Probability_Homework Answers. Let the sample space consist of the integers through. {, 2, 3,, }. Consider the following events from that Sample Space. Event A: {a number is a multiple of 5 5, 0, 5,, }
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 informationLecture 3. The Population Variance. The population variance, denoted σ 2, is the sum. of the squared deviations about the population
Lecture 5 1 Lecture 3 The Population Variance The population variance, denoted σ 2, is the sum of the squared deviations about the population mean divided by the number of observations in the population,
More informationSTA 218: Statistics for Management
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Simple Example Random Experiment: Rolling a fair
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 informationAnswers Part A. P(x = 67) = 0, because x is a continuous random variable. 2. Find the following probabilities:
Answers Part A 1. Woman s heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the probability that a single randomly selected woman will be 67 inches
More informationThe Standard Deviation as a Ruler and the Normal Model
The Standard Deviation as a Ruler and the Normal Model Al Nosedal University of Toronto Summer 2017 Al Nosedal University of Toronto The Standard Deviation as a Ruler and the Normal Model Summer 2017 1
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 information(i) The mean and mode both equal the median; that is, the average value and the most likely value are both in the middle of the distribution.
MATH 382 Normal Distributions Dr. Neal, WKU Measurements that are normally distributed can be described in terms of their mean µ and standard deviation σ. These measurements should have the following properties:
More informationEQ: What is a normal distribution?
Unit 5 - Statistics What is the purpose EQ: What tools do we have to assess data? this unit? What vocab will I need? Vocabulary: normal distribution, standard, nonstandard, interquartile range, population
More informationNormal Distribution and Central Limit Theorem
Normal Distribution and Central Limit Theorem Josemari Sarasola Statistics for Business Josemari Sarasola Normal Distribution and Central Limit Theorem 1 / 13 The normal distribution is the most applied
More informationChapter 2. Continuous random variables
Chapter 2 Continuous random variables Outline Review of probability: events and probability Random variable Probability and Cumulative distribution function Review of discrete random variable Introduction
More informationNormal Distribution. Distribution function and Graphical Representation - pdf - identifying the mean and variance
Distribution function and Graphical Representation - pdf - identifying the mean and variance f ( x ) 1 ( ) x e Distribution function and Graphical Representation - pdf - identifying the mean and variance
More informationMath 2311 Sections 4.1, 4.2 and 4.3
Math 2311 Sections 4.1, 4.2 and 4.3 4.1 - Density Curves What do we know about density curves? Example: Suppose we have a density curve defined for defined by the line y = x. Sketch: What percent of observations
More informationProbability Distributions: Continuous
Probability Distributions: Continuous INFO-2301: Quantitative Reasoning 2 Michael Paul and Jordan Boyd-Graber FEBRUARY 28, 2017 INFO-2301: Quantitative Reasoning 2 Paul and Boyd-Graber Probability Distributions:
More information9/19/2012. PSY 511: Advanced Statistics for Psychological and Behavioral Research 1
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 The aspect of the data we want to describe/measure is relative position z scores tell us how many standard deviations above or below
More informationSection 3.4 Normal Distribution MDM4U Jensen
Section 3.4 Normal Distribution MDM4U Jensen Part 1: Dice Rolling Activity a) Roll two 6- sided number cubes 18 times. Record a tally mark next to the appropriate number after each roll. After rolling
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 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 informationRecall that the standard deviation σ of a numerical data set is given by
11.1 Using Normal Distributions Essential Question In a normal distribution, about what percent of the data lies within one, two, and three standard deviations of the mean? Recall that the standard deviation
More informationII. The Normal Distribution
II. 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 informationIV. The Normal Distribution
IV. The Normal Distribution The normal distribution (a.k.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 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 informationNotation: X = random variable; x = particular value; P(X = x) denotes probability that X equals the value x.
Ch. 16 Random Variables Def n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variable is a random variable that assumes separate values. # of people
More information6/25/14. The Distribution Normality. Bell Curve. Normal Distribution. Data can be "distributed" (spread out) in different ways.
The Distribution Normality Unit 6 Sampling and Inference 6/25/14 Algebra 1 Ins2tute 1 6/25/14 Algebra 1 Ins2tute 2 MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or measurement
More informationSTA 111: Probability & Statistical Inference
STA 111: Probability & Statistical Inference Lecture Four Expectation and Continuous Random Variables Instructor: Olanrewaju Michael Akande Department of Statistical Science, Duke University Instructor:
More informationChapter 3 - The Normal (or Gaussian) Distribution Read sections
Chapter 3 - The Normal (or Gaussian) Distribution Read sections 3.1-3.2 Basic facts (3.1) The normal distribution gives the distribution for a continuous variable X on the interval (-, ). The notation
More informationSolutions to Additional Questions on Normal Distributions
Solutions to Additional Questions on Normal Distributions 1.. EPA fuel economy estimates for automobile models tested recently predicted a mean of.8 mpg and a standard deviation of mpg for highway driving.
More informationThe Normal Distribuions
The Normal Distribuions Sections 5.4 & 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 15-3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationACMS Statistics for Life Sciences. Chapter 9: Introducing Probability
ACMS 20340 Statistics for Life Sciences Chapter 9: Introducing Probability Why Consider Probability? We re doing statistics here. Why should we bother with probability? As we will see, probability plays
More informationPercentile: Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included:
AP Statistics Chapter 2 Notes 2.1 Describing Location in a Distribution Percentile: The pth percentile of a distribution is the value with p percent of the observations (If your test score places you in
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 information6 THE NORMAL DISTRIBUTION
CHAPTER 6 THE NORMAL DISTRIBUTION 341 6 THE NORMAL DISTRIBUTION Figure 6.1 If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described
More informationStats Review Chapter 6. Mary Stangler Center for Academic Success Revised 8/16
Stats Review Chapter Revised 8/1 Note: This review is composed of questions similar to those found in the chapter review and/or chapter test. This review is meant to highlight basic concepts from the course.
More informationSampling Distribution Models. Central Limit Theorem
Sampling Distribution Models Central Limit Theorem Thought Questions 1. 40% of large population disagree with new law. In parts a and b, think about role of sample size. a. If randomly sample 10 people,
More informationLooking at data: distributions - Density curves and Normal distributions. Copyright Brigitte Baldi 2005 Modified by R. Gordon 2009.
Looking at data: distributions - Density curves and Normal distributions Copyright Brigitte Baldi 2005 Modified by R. Gordon 2009. Objectives Density curves and Normal distributions!! Density curves!!
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Lecture 9 (MWF) Calculations for the normal distribution Suhasini Subba Rao Evaluating probabilities
More information2. The Normal Distribution
of 5 7/6/2009 5:55 AM Virtual Laboratories > 5 Special Distributions > 2 3 4 5 6 7 8 9 0 2 3 4 5 2 The Normal Distribution The normal distribution holds an honored role in probability and statistics, mostly
More informationChapter 3 Probability Distributions and Statistics Section 3.1 Random Variables and Histograms
Math 166 (c)2013 Epstein Chapter 3 Page 1 Chapter 3 Probability Distributions and Statistics Section 3.1 Random Variables and Histograms The value of the result of the probability experiment is called
More informationSpecial distributions
Special distributions August 22, 2017 STAT 101 Class 4 Slide 1 Outline of Topics 1 Motivation 2 Bernoulli and binomial 3 Poisson 4 Uniform 5 Exponential 6 Normal STAT 101 Class 4 Slide 2 What distributions
More informationEssential Statistics Chapter 6
1 Essential Statistics Chapter 6 By Navidi and Monk Copyright 2016 Mark A. Thomas. All rights reserved. 2 Continuous Probability Distributions chapter 5 focused upon discrete probability distributions,
More informationStatistics and Sampling distributions
Statistics and Sampling distributions a statistic is a numerical summary of sample data. It is a rv. The distribution of a statistic is called its sampling distribution. The rv s X 1, X 2,, X n are said
More informationMath 223 Lecture Notes 3/15/04 From The Basic Practice of Statistics, bymoore
Math 223 Lecture Notes 3/15/04 From The Basic Practice of Statistics, bymoore Chapter 3 continued Describing distributions with numbers Measuring spread of data: Quartiles Definition 1: The interquartile
More informationObjective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode.
Chapter 3 Numerically Summarizing Data Chapter 3.1 Measures of Central Tendency Objective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode. A1. Mean The
More informationContinuous distributions
Continuous distributions In contrast to discrete random variables, like the Binomial distribution, in many situations the possible values of a random variable cannot be counted. For example, the measurement
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 information(i) The mean and mode both equal the median; that is, the average value and the most likely value are both in the middle of the distribution.
MATH 183 Normal Distributions Dr. Neal, WKU Measurements that are normally distributed can be described in terms of their mean µ and standard deviation!. These measurements should have the following properties:
More informationSampling, Frequency Distributions, and Graphs (12.1)
1 Sampling, Frequency Distributions, and Graphs (1.1) Design: Plan how to obtain the data. What are typical Statistical Methods? Collect the data, which is then subjected to statistical analysis, which
More informationThe Central Limit Theorem
- The Central Limit Theorem Definition Sampling Distribution of the Mean the probability distribution of sample means, with all samples having the same sample size n. (In general, the sampling distribution
More informationLecture 10/Chapter 8 Bell-Shaped Curves & Other Shapes. From a Histogram to a Frequency Curve Standard Score Using Normal Table Empirical Rule
Lecture 10/Chapter 8 Bell-Shaped Curves & Other Shapes From a Histogram to a Frequency Curve Standard Score Using Normal Table Empirical Rule From Histogram to Normal Curve Start: sample of female hts
More informationCh. 7: Estimates and Sample Sizes
Ch. 7: Estimates and Sample Sizes Section Title Notes Pages Introduction to the Chapter 2 2 Estimating p in the Binomial Distribution 2 5 3 Estimating a Population Mean: Sigma Known 6 9 4 Estimating a
More information13. Sampling distributions
13. Sampling distributions The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 13) Sampling distributions Parameter versus statistic Sampling
More informationMODULE 9 NORMAL DISTRIBUTION
MODULE 9 NORMAL DISTRIBUTION Contents 9.1 Characteristics of a Normal Distribution........................... 62 9.2 Simple Areas Under the Curve................................. 63 9.3 Forward Calculations......................................
More information68% 95% 99.7% x x 1 σ. x 1 2σ. x 1 3σ. Find a normal probability
11.3 a.1, 2A.1.B TEKS Use Normal Distributions Before You interpreted probability distributions. Now You will study normal distributions. Why? So you can model animal populations, as in Example 3. Key
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Review of previous lecture We showed if S n were a binomial random variable, where
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 information11/16/2017. Chapter. Copyright 2009 by The McGraw-Hill Companies, Inc. 7-2
7 Chapter Continuous Probability Distributions Describing a Continuous Distribution Uniform Continuous Distribution Normal Distribution Normal Approximation to the Binomial Normal Approximation to the
More information6.2A Linear Transformations
6.2 Transforming and Combining Random Variables 6.2A Linear Transformations El Dorado Community College considers a student to be full time if he or she is taking between 12 and 18 credits. The number
More informationUnit 4 Probability. Dr Mahmoud Alhussami
Unit 4 Probability Dr Mahmoud Alhussami Probability Probability theory developed from the study of games of chance like dice and cards. A process like flipping a coin, rolling a die or drawing a card from
More informationUnderstanding Inference: Confidence Intervals I. Questions about the Assignment. The Big Picture. Statistic vs. Parameter. Statistic vs.
Questions about the Assignment If your answer is wrong, but you show your work you can get more partial credit. Understanding Inference: Confidence Intervals I parameter versus sample statistic Uncertainty
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a
More informationWhat is a parameter? What is a statistic? How is one related to the other?
7.1 Sampling Distributions Read 424 425 What is a parameter? What is a statistic? How is one related to the other? Alternate Example: Identify the population, the parameter, the sample, and the statistic:
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 informationEssential Statistics. Gould Ryan Wong
Global Global Essential Statistics Eploring the World through Data For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects
More informationUNIVERSITY OF MASSACHUSETTS Department of Biostatistics and Epidemiology BioEpi 540W - Introduction to Biostatistics Fall 2004
UNIVERSITY OF MASSACHUSETTS Department of Biostatistics and Epidemiology BioEpi 50W - Introduction to Biostatistics Fall 00 Exercises with Solutions Topic Summarizing Data Due: Monday September 7, 00 READINGS.
More informationCommon Core State Standards for Mathematical Practice (6-8)
Common Core State Standards for Mathematical Practice (6-8) Lesson 3.1 - Investigating Energy PLTW Standards and Alignment Compute fluently with multi-digit numbers and find common factors and multiples.
More informationChapter 8 Sampling Distributions Defn Defn
1 Chapter 8 Sampling Distributions Defn: Sampling error is the error resulting from using a sample to infer a population characteristic. Example: We want to estimate the mean amount of Pepsi-Cola in 12-oz.
More information1. Poisson distribution is widely used in statistics for modeling rare events.
Discrete probability distributions - Class 5 January 20, 2014 Debdeep Pati Poisson distribution 1. Poisson distribution is widely used in statistics for modeling rare events. 2. Ex. Infectious Disease
More informationTheoretical Foundations
Theoretical Foundations Sampling Distribution and Central Limit Theorem Monia Ranalli monia.ranalli@uniroma3.it Ranalli M. Theoretical Foundations - Sampling Distribution and Central Limit Theorem Lesson
More informationWe're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation, Y ~ BIN(n,p).
Sampling distributions and estimation. 1) A brief review of distributions: We're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation,
More informationUNIVERSITY OF MASSACHUSETTS Department of Biostatistics and Epidemiology BE540w - Introduction to Biostatistics Fall 2004
Page 1 of 9 UNIVERSITY O MASSACHUSETTS Department of Biostatistics and Epidemiology BE540w - Introduction to Biostatistics all 2004 Exercises with Solutions Topic 5 Normal Distribution Due: Monday November
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 informationReview. A Bernoulli Trial is a very simple experiment:
Review A Bernoulli Trial is a very simple experiment: Review A Bernoulli Trial is a very simple experiment: two possible outcomes (success or failure) probability of success is always the same (p) the
More informationAP Statistics Ch 7 Random Variables
Ch 7.1 Discrete and Continuous Random Variables Introduction A random variable is a variable whose value is a numerical outcome of a random phenomenon. If an experiment or sample survey is repeated, different
More informationChapter 6: SAMPLING DISTRIBUTIONS
Chapter 6: SAMPLING DISTRIBUTIONS Read Section 1.5 Graphical methods may not always be sufficient for describing data. Numerical measures can be created for both populations and samples. Definition A numerical
More informationBiostatistics: Correlations
Biostatistics: s One of the most common errors we find in the press is the confusion between correlation and causation in scientific and health-related studies. In theory, these are easy to distinguish
More informationLecture 10: Normal RV. Lisa Yan July 18, 2018
Lecture 10: Normal RV Lisa Yan July 18, 2018 Announcements Midterm next Tuesday Practice midterm, solutions out on website SCPD students: fill out Google form by today Covers up to and including Friday
More informationMath 1040 Sample Final Examination. Problem Points Score Total 200
Name: Math 1040 Sample Final Examination Relax and good luck! Problem Points Score 1 25 2 25 3 25 4 25 5 25 6 25 7 25 8 25 Total 200 1. (25 points) The systolic blood pressures of 20 elderly patients in
More informationIntroduction to Probability and Statistics Twelfth Edition
Introduction to Probability and Statistics Twelfth Edition Robert J. Beaver Barbara M. Beaver William Mendenhall Presentation designed and written by: Barbara M. Beaver Introduction to Probability and
More informationAre data normally normally distributed?
Standard Normal Image source Are data normally normally distributed? Sample mean: 66.78 Sample standard deviation: 3.37 (66.78-1 x 3.37, 66.78 + 1 x 3.37) (66.78-2 x 3.37, 66.78 + 2 x 3.37) (66.78-3 x
More informationSection 7.1 Properties of the Normal Distribution
Section 7.1 Properties of the Normal Distribution In Chapter 6, talked about probability distributions. Coin flip problem: Difference of two spinners: The random variable x can only take on certain discrete
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 informationTest 3 SOLUTIONS. x P(x) xp(x)
16 1. A couple of weeks ago in class, each of you took three quizzes where you randomly guessed the answers to each question. There were eight questions on each quiz, and four possible answers to each
More informationIn this chapter, you will study the normal distribution, the standard normal, and applications associated with them.
The Normal Distribution The normal distribution is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost
More informationCh7 Sampling Distributions
AP Statistics Name: Per: Date: Ch7 Sampling Distributions 7.1 What Is a Sampling Distribution? Read 424 425 Parameters and Statistics Vocab: parameter, statistic How is one related to the other? Alternate
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Review Our objective: to make confident statements about a parameter (aspect) in
More informationComplement: 0.4 x 0.8 = =.6
Homework The Normal Distribution Name: 1. Use the graph below 1 a) Why is the total area under this curve equal to 1? Rectangle; A = LW A = 1(1) = 1 b) What percent of the observations lie above 0.8? 1
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 informationIntro to Confidence Intervals: A estimate is a single statistic based on sample data to estimate a population parameter Simplest approach But not always very precise due to variation in the sampling distribution
More informationInterval estimation. October 3, Basic ideas CLT and CI CI for a population mean CI for a population proportion CI for a Normal mean
Interval estimation October 3, 2018 STAT 151 Class 7 Slide 1 Pandemic data Treatment outcome, X, from n = 100 patients in a pandemic: 1 = recovered and 0 = not recovered 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0
More informationSTATISTICS 1 REVISION NOTES
STATISTICS 1 REVISION NOTES Statistical Model Representing and summarising Sample Data Key words: Quantitative Data This is data in NUMERICAL FORM such as shoe size, height etc. Qualitative Data This is
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 information