Normal Probability Distribution


 Neal Stone
 1 years ago
 Views:
Transcription
1 Normal Probability Distribution The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve. Using this function returns the ycoordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invnorm( inv = Inverse Normal Probability Distribution Function This function returns the xvalue given the probability region to the left of the xvalue. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invnorm (probability, mean, standard deviation) Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90 th percentile for this distribution. (answers will be rounded to the nearest thousandth) 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax:normalcdf(lower bound, upper bound, mean, standard deviation) : The probability is 1 P a g e
2 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x Type 1 EE 99. Enter the EE by pressing 2nd, comma  only one E will show on the screen.) : The probability is 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is 1 x Type 1 EE 99. Enter the EE by pressing 2nd, comma  only one E will show on the screen.) : The probability is 1d: Find the 90 th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine the value using invnorm.) : The xvalue is Example 3: Graph and examine a situation where the mean score is 46 and the standard deviation is 8.5 for a normally distributed set of data. Go to Y=. Adjust the window. GRAPH. 2 P a g e
3 1. Practice and Homework (Independent # odd and HW even #) The amount of mustard dispensed from a machine at The Hotdog Emporium is normally distributedwith a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times, approximately how many times will it be expected to dispense 1 or more ounces of mustard. Choose: Professor Halen has 184 students in his college mathematics lecture class. The scores on the midterm exam are normally distributed with a mean of 72.3 and a standard deviation of 8.9. How many students in the class can be expected to receive a score between 82 and 90? Express answer to the nearest student. 3. A machine is used to fill soda bottles. The amount of soda dispensed into each bottle varies slightly. Suppose the amount of soda dispensed into the bottles is normally distributed. If at least 99% of the bottles must have between 585 and 595 milliliters of soda, find the greatest standard deviation, to the nearest hundredth, that can be allowed. 4. Residents of upstate New York are accustomed to large amounts of snow with snowfalls often exceeding 6 inches in one day. In one city, such snowfalls were recorded for two seasons and are as follows (in inches): 8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5, 7.7, 6.8, 6.1, 8.5, 14.4, 6.1, 8.0, 9.2, P a g e
4 What are the mean and the population standard deviation for this data, to the nearest hundredth? 5. Neesha's scores in Chemistry this semester were rather inconsistent: 100, 85, 55, 95, 75, 100. For this population, how many scores are within one standard deviation of the mean? 6. Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To the nearest percent, what percent of batteries have lifetimes longer than 561 days? 7. The number of children of each of the first 41 United States presidents is given in the accompanying table. For this population, determine the mean and the standard deviation to the nearest tenth. How many of these presidents fall within one standard deviation of the mean? 8. 4 P a g e From 1984 to 1995, the winning scores for a golf tournament were 276, 279, 279, 277, 278, 278, 280, 282, 285, 272, 279, and 278. Using the standard deviation for
5 this sample, Sx, find the percent of these winning scores that fall within one standard deviation of the mean. 9. A shoe manufacturer collected data regarding men's shoe sizes and found that the distribution of sizes exactly fits the normal curve. If the mean shoe size is 11 and the standard deviation is 1.5, find: a. the probability that a man's shoe size is greater than or equal to 11. b. the probability that a man's shoe size is greater than or equal to c. 10. Five hundred values are normally distributed with a mean of 125 and a standard deviation of 10. a. What percent of the values lies in the interval , to the nearest percent? b. What percent of the values is in the interval , to the nearest percent? c. What interval about the mean includes 95% of the data? d. What interval about the mean includes 50% of the data? 5 P a g e
6 11. A group of 625 students has a mean age of 15.8 years with a standard deviation of 0.6 years. The ages are normally distributed. How many students are younger than 16.2 years? Express answer to the nearest student? Binomial Probability "Exactly", "At Most", "At Least" Example 4 Problem used for demonstration: A fair coin is tossed 100 times. What is the probability that: a. heads will appear exactly 52 times? b. there will be at most 52 heads? c. there will be at least48 heads? Dealing with "Exactly": A fair coin is tossed 100 times. What is the probability that: part a. heads will appear exactly 52 times? We have seen that the formula used with Bernoulli trials (binomial probability) computes 6 P a g e
7 the probability of obtaining exactly "r" events in "n" trials: where n = number of trials, r = number of specific events you wish to obtain, p = probability that the event will occur, and q = probability that the event will not occur (q = 1  p, the complement of the event). We have also seen that the builtin command binompdf(binomial probability density function) can also be used to quickly determine "exactly". (Remember, the function binompdf is found under DISTR (2nd VARS), arrow down to #0 binompdf and the parameters are: binompdf (number of trials, probability of occurrence, number of specific events) binompdf (n, p, r) Here is our answer to part a. Dealing with"at Most": A fair coin is tossed 100 times. What is the probability that: part b. there will be at most 52 heads? The formula needed for answering part b is : There is a builtin command binomcdf(binomial cumulative density function) that can be used to quickly determine "at most". Because this is a "cumulative" function, it will find the sum of all of the probabilities up to, and including, the given value of P a g e (The function binomcdf is found under DISTR (2nd VARS), arrow down to #A binomcdf
8 and the parameters are: binomcdf (number of trials, probability of occurrence, number of specific events) binomcdf (n, p, r) Here is our answer to part b. Dealing with "At Least": A fair coin is tossed 100 times. What is the probability that: part c. there will be at least48 heads? The formula needed for answering part b is : Keep in mind that "at least" 48 is the complement of "at most" 47. In a binomial distribution,. While there is no builtin command for "at least", you can quickly find the result by creating this complement situation by subtracting from 1. Just remember to adjust the value to 47. The adjusted formula for "at least" is 1  binomcdf (n, p, r  1).Here is our answer to part c. 8 P a g e
9 The fact that this answer is the same as the "at most" answer for the number 52, is due to the symmetric nature of the distribution about its mean of 50. Practice (independent odd # HW even #) 1. The probability that Kyla will score above a 90 on a mathematics test is 4/5. What is the probability that she will score above a 90 on exactly three of the four tests this quarter? Choose: 2. Which fraction represents the probability of obtaining exactly eight heads in ten tosses of a fair coin? Choose: 45/ / / / Experience has shown that 1/200 of all CDs produced by a certain machine are defective. If a 9 P a g e
10 quality control technician randomly tests twenty CDs, compute each of the following probabilities: P(exactly one is defective) P(half are defective) P( no more than two are defective) 4. A fair coin is tossed 5 times. What is the probability that it lands tails up exactly 3 times? Choose: 5. After studying a couple's family history, a doctor determines that the probability of any child born to this couple having a gene for disease X is 1 out of 4. If the couple has three children, what is the probability that exactly two of the children have the gene for disease X? 6. If a binomial experiment has seven trials in which the probability of success is p and the probability of failure is q, write an expression that could be used to compute each of the following probabilities: P(exactly five successes) P(at least five successes) 10 P a g e P(at most five successes) 7. On any given day, the probability that the entire Watson family eats dinner together is 2/5. Find the probability that, during any 7day period, the Watson's each dinner together at least six times.
11 8. When Joe bowls, he can get a strike (knock down all of the pins) 60% of the time. How many times more likely is it for Joe to bowl at least three strikes out of four times as it is for him to bowl zero strikcs out of four tries? Round answer to the nearest whole number. 9. A board game has a spinner on a circle that has five equal sectors, numbered 1, 2, 3, 4, and 5, respectively. If a player has four spins, find the probability that the player spins an even number no more than two times on those four spins. 10. Give an example of an experiment where it is appropriate to use a normal distribution as an approximation for a binomial probability. Explain why in this example an approximation of the probability is a better approach than finding the exact probability. Key 11 P a g e
12 3. The 99% implies a distribution within 3 standard deviations of the mean. The difference from 585 milliliters to 595 milliliters is 10 milliliters. Symmetrically divided, there are 5 milliliters used to create 3 standard deviations on one side of the mean. Dividing 5 by 3, we get the standard deviation to be 1.67 milliliters, to the nearest hundredth. 4. Snowfall: 8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5, 7.7, 6.8, 6.1, 8.5, 14.4, 6.1, 8.0, 9.2, 7.1 The mean = 9.46 inches. Standard deviation = Scores: 100, 85, 55, 95, 75, 100. From the graphing calculator we have the mean = 85 and the population standard deviation = = = 101 and = 69 All but one of the scores falls in the range from 69 to 101. : 5 scores 12 P a g e
13 8. Scores: 276, 279, 279, 277, 278, 278, 280, 282, 285, 272, 279, and 278. Mean = Sample Standard Deviation = One standard deviation above and below the mean creates a range from to (275 < score < 281). Of the 12 scores, 9 fall into this range, which is 75% of the scores. 7. Use your graphing calculator. Be sure to remember that you are working with grouped data (frequency table). Mean = 3.6 Standard deviation = 2.9 One standard deviation from the mean creates a range from 0.7 to 6.5 which includes presidents with from 1 to 6 children. There are a total of 31 such presidents. 13 P a g e
14 9. Mean = 11 and standard deviation = 1.5 a. 50% In a normal distribution, the mean divides the data into two equal areas. Since 11 is the mean, 50% of the data is above 11 and 50% is below 11. b is exactly one standard deviation above the mean. Examining the normal distribution chart shows that 15.9%will fall above one standard deviation. Probability is.159. c and 8 are exactly one standard deviation above the mean and 2 standard deviations below the mean respectively. Using the chart we know: scores, mean 125, standard deviation 10 a. What percent of the values is in the interval ? mean + one standard deviation = 135 mean  one standard deviation = % % % + 15% = 68.2% (from chart) Percent within one standard deviation of the mean = 68.2% = 68% b. What percent of the values is in the interval ? mean standard deviations = 150 mean standard deviations = 100 2(1.7% + 4.4% + 9.2% + 15% %) = 98.8% (from chart) Percent with 2.5 standard deviations of the mean = 98.8% = 99% c. What interval about the mean includes approximately 95% of the data? 2 standard deviations about the mean for a total interval size of 40, with the mean in the center. mean + 2 standard deviations = 145 mean  2 standard deviations = 105 Interval: [105,145] d. What interval about the mean includes 50% of the data?.50% of the distribution lies within standard deviations about the mean for a total interquartile range (size) of , with the mean in the center. mean standard deviations = mean standard deviations = Interval: [ , ] 14 P a g e
15 Binomial 3. a.) P(exactly one is defective) b.) P(half are defective) = P(exactly 10 are defective) c.) P( no more than two are defective) = P( at most two are defective). None, one, or two could be defective. Sum = which is approximately 1 or 100% P a g e
16 6. a.) b.) c.) 7. At least means 6 times or 7 times. 8. At least three strikes: Exactly zero strikes: 19 times more likely. 16 P a g e
17 9. There are 2 even numbers out of the 5 numbers. No more than = at most. 10. One possible answer: Find the probability of getting at most 250 heads when flipping a fair coin 300 times. When dealing with very large samples, it can become very tedious to compute certain probabilities. In such cases, the normal distribution can be used to more quickly approximate the probabilities that otherwise would have been obtained through laborious computations. 17 P a g e
Stats on the TI 83 and TI 84 Calculator
Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and
More informationYou flip a fair coin four times, what is the probability that you obtain three heads.
Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More informationProbability Distributions
CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,
More informationChapter 6 Random Variables
Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:
More information6.1 Graphs of Normal Probability Distributions. Normal Curve aka Probability Density Function
Normal Distributions (Page 1 of 23) 6.1 Graphs of Normal Probability Distributions Normal Curve aka Probability Density Function Normal Probability Distribution TP TP µ! " µ µ +! x xaxis Important Properties
More informationCopyright 2013 by Laura Schultz. All rights reserved. Page 1 of 6
Using Your TINSpire Calculator: Binomial Probability Distributions Dr. Laura Schultz Statistics I This handout describes how to use the binompdf and binomcdf commands to work with binomial probability
More information3.4 The Normal Distribution
3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous
More informationBinomial Probability Distribution
Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0.
Ch. 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 1 Find Areas Under the Standard Normal Curve 1) Find the area under the standard normal
More information13.2 Measures of Central Tendency
13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers
More informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More information1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700
Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,
More informationUniversity of California, Los Angeles Department of Statistics. Normal distribution
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Normal distribution The normal distribution is the most important distribution. It describes
More informationChapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.
Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,
More informationFINAL EXAM REVIEW  Fa 13
FINAL EXAM REVIEW  Fa 13 Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. 1) The temperatures of eight different plastic spheres. 2) The sample
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
More informationMAT 155. Key Concept. September 22, 2010. 155S5.3_3 Binomial Probability Distributions. Chapter 5 Probability Distributions
MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance, and Standard
More informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationChapter 4 The Standard Deviation as a Ruler and the Normal Model
Chapter 4 The Standard Deviation as a Ruler and the Normal Model The standard deviation is the most common measure of variation; it plays a crucial role in how we look at data. Z scores measure standard
More informationAP STATISTICS 2009 SCORING GUIDELINES
2009 SCORING GUIDELINES Question 2 Intent of Question The primary goals of this question were to assess a student s ability to (1) calculate a percentile value from a normal probability distribution; (2)
More informationEstimating and Finding Confidence Intervals
. Activity 7 Estimating and Finding Confidence Intervals Topic 33 (40) Estimating A Normal Population Mean μ (σ Known) A random sample of size 10 from a population of heights that has a normal distribution
More informationMATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem
MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Sample Final Exam Spring 2008 DeMaio Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the given degree of confidence and sample data to construct
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationSTATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS
STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationChapter 5: Discrete Probability Distributions
Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions As a reminder, a variable or what will be called the random variable from now on, is represented by the letter
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
More informationThursday, November 13: 6.1 Discrete Random Variables
Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.
More informationChapter 6: Continuous Probability Distributions
Chapter 6: Continuous Probability Distributions Chapter 5 dealt with probability distributions arising from discrete random variables. Mostly that chapter focused on the binomial experiment. There are
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 41/42 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
More informationAP Statistics Solutions to Packet 8
AP Statistics Solutions to Packet 8 The Binomial and Geometric Distributions The Binomial Distributions The Geometric Distributions 54p HW #1 1 5, 7, 8 8.1 BINOMIAL SETTING? In each situation below, is
More informationSTA201 Intermediate Statistics Lecture Notes. Luc Hens
STA201 Intermediate Statistics Lecture Notes Luc Hens 15 January 2016 ii How to use these lecture notes These lecture notes start by reviewing the material from STA101 (most of it covered in Freedman et
More informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.18.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationX: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001
Tuesday, January 17: 6.1 Discrete Random Variables Read 341 344 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.
More informationChapter 5  Practice Problems 1
Chapter 5  Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 (b) 1
Unit 2 Review Name Use the given frequency distribution to find the (a) class width. (b) class midpoints of the first class. (c) class boundaries of the first class. 1) Miles (per day) 12 9 34 22 56
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will get observations
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More information2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table
2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations
More information6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.
6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationThe normal approximation to the binomial
The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There
More informationTopic 5 Review [81 marks]
Topic 5 Review [81 marks] A foursided die has three blue faces and one red face. The die is rolled. Let B be the event a blue face lands down, and R be the event a red face lands down. 1a. Write down
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationLesson 20. Probability and Cumulative Distribution Functions
Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic
More informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 68 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More informationTEST 2 STUDY GUIDE. 1. Consider the data shown below.
2006 by The Arizona Board of Regents for The University of Arizona All rights reserved Business Mathematics I TEST 2 STUDY GUIDE 1 Consider the data shown below (a) Fill in the Frequency and Relative Frequency
More informationHypothesis Testing. Learning Objectives. After completing this module, the student will be able to
Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and
More informationNumber of observations is fixed. Independent observations  knowledge of the outcomes of earlier trials does not affect the
Binomial Probability Frequently used in analyzing and setting up surveys Our interest is in a binomial random variable X, which is the count of successes in n trials. The probability distribution of X
More informationProbability distributions
Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.142.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationTImath.com. Statistics. Areas in Intervals
Areas in Intervals ID: 9472 TImath.com Time required 30 minutes Activity Overview In this activity, students use several methods to determine the probability of a given normally distributed value being
More informationChapter 6 ATE: Random Variables Alternate Examples and Activities
Probability Chapter 6 ATE: Random Variables Alternate Examples and Activities [Page 343] Alternate Example: NHL Goals In 2010, there were 1319 games played in the National Hockey League s regular season.
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationProblem sets for BUEC 333 Part 1: Probability and Statistics
Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are backofchapter exercises from
More informationMCQ S OF MEASURES OF CENTRAL TENDENCY
MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ No 3.1 Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: (a) Skewness (b)
More informationUsing Your TINSpire Calculator: Normal Distributions Dr. Laura Schultz Statistics I
Using Your TINSpire Calculator: Normal Distributions Dr. Laura Schultz Statistics I Always start by drawing a sketch of the normal distribution that you are working with. Shade in the relevant area (probability),
More informationChapter 6 Continuous Probability Distributions
Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1342 (Elementary Statistics) Test 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the indicated probability. 1) If you flip a coin
More informationMAT 118 DEPARTMENTAL FINAL EXAMINATION (written part) REVIEW. Ch 13. One problem similar to the problems below will be included in the final
MAT 118 DEPARTMENTAL FINAL EXAMINATION (written part) REVIEW Ch 13 One problem similar to the problems below will be included in the final 1.This table presents the price distribution of shoe styles offered
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationNonparametric Statistics
1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables
More informationSampling Distribution of a Sample Proportion
Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given
More informationContinuous Random Variables and the Normal Distribution
CHAPTER 6 Continuous Random Variables and the Normal Distribution CHAPTER OUTLINE 6.1 The Standard Normal Distribution 6.2 Standardizing a Normal Distribution 6.3 Applications of the Normal Distribution
More informationMental Questions. Day 1. 1. What number is five cubed? 2. A circle has radius r. What is the formula for the area of the circle?
Mental Questions 1. What number is five cubed? KS3 MATHEMATICS 10 4 10 Level 8 Questions Day 1 2. A circle has radius r. What is the formula for the area of the circle? 3. Jenny and Mark share some money
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationConfidence Intervals. Chapter 11
. Chapter 11 Confidence Intervals Topic 22 covers confidence intervals for a proportion and includes a simulation. Topic 23 addresses confidence intervals for a mean. Differences between two proportions
More informationI know when I have written a number backwards and can correct it when it is pointed out to me I can arrange numbers in order from 1 to 10
Mathematics Targets Moving from Level W and working towards level 1c I can count from 1 to 10 I know and write all my numbers to 10 I know when I have written a number backwards and can correct it when
More informationExample: Find the expected value of the random variable X. X 2 4 6 7 P(X) 0.3 0.2 0.1 0.4
MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:
More informationNormal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.
Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e (y µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:
More informationReport of for Chapter 2 pretest
Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every
More informationComputing Binomial Probabilities
The Binomial Model The binomial probability distribution is a discrete probability distribution function Useful in many situations where you have numerical variables that are counts or whole numbers Classic
More informationThe practice test follows this cover sheet. It is very similar to the real Chapter Test.
AP Stats Unit IV (Chapters 1417) TakeHome Test Info The practice test follows this cover sheet. It is very similar to the real Chapter 1417 Test. The real test will consist of 20 multiplechoice questions
More informationModels for Discrete Variables
Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationSection 53 Binomial Probability Distributions
Section 53 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial
More informationProbability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationLESSON 5  DECIMALS INTRODUCTION
LESSON 5  DECIMALS INTRODUCTION Now that we know something about whole numbers and fractions, we will begin working with types of numbers that are extensions of whole numbers and related to fractions.
More informationProbability QUESTIONS Principles of Math 12  Probability Practice Exam 1 www.math12.com
Probability QUESTIONS Principles of Math  Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationInstructions for : TI83, 83Plus, 84Plus for STP classes, Ela Jackiewicz
Computing areas under normal curves: option 2 normalcdf(lower limit, upper limit, mean, standard deviation) will give are between lower and upper limits (mean=0 and St.dev=1 are default values) Ex1 To
More informationCHAPTER 3 CENTRAL TENDENCY ANALYSES
CHAPTER 3 CENTRAL TENDENCY ANALYSES The next concept in the sequential statistical steps approach is calculating measures of central tendency. Measures of central tendency represent some of the most simple
More informationZtable pvalues: use choice 2: normalcdf(
Pvalues with the Ti83/Ti84 Note: The majority of the commands used in this handout can be found under the DISTR menu which you can access by pressing [ nd ] [VARS]. You should see the following: NOTE:
More informationEngineering Problem Solving and Excel. EGN 1006 Introduction to Engineering
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More informationExercises  The Normal Curve
Exercises  The Normal Curve 1. Find e following proportions under e Normal curve: a) P(z>2.05) b) P(z>2.5) c) P(1.25
More informationTImath.com. F Distributions. Statistics
F Distributions ID: 9780 Time required 30 minutes Activity Overview In this activity, students study the characteristics of the F distribution and discuss why the distribution is not symmetric (skewed
More informationCC Investigation 5: Histograms and Box Plots
Content Standards 6.SP.4, 6.SP.5.c CC Investigation 5: Histograms and Box Plots At a Glance PACING 3 days Mathematical Goals DOMAIN: Statistics and Probability Display numerical data in histograms and
More informationINTRODUCTION TO PROBABILITY AND STATISTICS
INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a,,, or 4 on the
More information