Math 227 Elementary Statistics. Bluman 5 th edition
|
|
- Conrad Singleton
- 5 years ago
- Views:
Transcription
1 Math 227 Elementary Statistics Bluman 5 th edition
2 CHAPTER 6 The Normal Distribution 2
3 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find the area under the standard normal distribution, given various z values. Find the probabilities for a normally distributed variable by transforming it into a standard normal variable. 3
4 Objectives (cont.) Find specific data values for given percentages using the standard normal distribution. Use the central limit theorem to solve problems involving sample means for large samples. Use the normal approximation to compute probabilities for a binomial variable. 4
5 Introduction Many continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables. A normal distribution is also known as the bell curve or the Gaussian distribution. 5
6 Normal and Skewed Distributions The normal distribution is a continuous, bellshaped distribution of a variable. If the data values are evenly distributed about the mean, the distribution is said to be symmetrical. If the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed. 6
7 Left Skewed Distributions When the majority of the data values fall to the right of the mean, the distribution is said to be negatively or left skewed. The mean is to the left of the median, and the mean and the median are to the left of the mode. 7
8 Right Skewed Distributions When the majority of the data values fall to the left of the mean, the distribution is said to be positively or right skewed. The mean falls to the right of the median and both the mean and the median fall to the right of the mode. 8
9 6.1 Normal Distribution I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use to represent a probability density function. Unfortunately, does not give us the probability that the value x will be observed. To understand how a probability density function for a continuous random variable enables us to find probabilities, it is important to understand the relationship between probability and area. For the following given histogram, what is the probability that x is in between 2.5 to 5.5? A. B. Frequency Histogram Relative Frequency Histogram C C1 9
10 Use the given frequency histogram to calculate P(2.5 < x < 5.5) : A: P (2.5 < x < 5.5) = ( ) / ( ) = 13 / 25 = 52% Use the corresponding relative frequency histogram to calculate P(2.5 < x < 5.5) : B: P(2.5 < x < 5.5) = 16% + 20% + 16% = 52% which is the same as the area of the three middle bars of the relative frequency histogram. The width of each bar is one and the height is the given percentage. For a continuous probability distribution, 1) for all values x of the random variable; 2) the total area under the graph of is 1; 3) P (a < x < b) can be approximated by the area under the graph of for a < x < b. 10
11 Note : P (x = a) = 0 for continuous random variables. This implies P(a x b) = P(a < x < b); P(x a) = P(x > a); and P(x a) = P(x < a). 11
12 II. The Normal Distribution Continuous probability distributions can assume a variety of shapes. However, the most important distribution of continuous random variables in statistics is the normal distribution, that is approximately moundshaped. Many naturally occurring random variables such as IQs, height of humans, weights, times, etc. have nearly normal distributions. 12
13 The mathematical equation for a normal distribution is mean where e = population mean = population standard deviation The mean is located at the center of distribution. The distribution is symmetric about its mean. 13
14 There is a correspondence between area and probability. Since the total area under the normal probability distribution is equal to 1, the symmetry implies that the area to the right of is 0.5 and the area to the left of is also 0.5. Large values of reduce the height of the curve and increase the spread. Small values of increase the height of the curve and reduce the spread. Almost all values of a normal random variable lie in the interval 14
15 III. Properties of the Normal Distribution The shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation. Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable s mean and standard deviation. 15
16 Normal Distribution Properties The normal distribution curve is bell-shaped. The mean, median, and mode are equal and located at the center of the distribution. The normal distribution curve is unimodal (i.e., it has only one mode). The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center. 16
17 Normal Distribution Properties (cont.) The curve is continuous i.e., there are no gaps or holes. For each value of X, here is a corresponding value of Y. The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis but it gets increasingly closer. 17
18 IV. The Standard Normal Distribution Since each normally distributed variable has its own mean and standard deviation, the shape and location of these curves will vary. In practical applications, one would have to have a table of areas under the curve for each variable. To simplify this, statisticians use the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. 18
19 Recall: z Values The z value is the number of standard deviations that a particular X value is away from the mean. The formula for finding the z value is: z value mean standard deviation or z X 19
20 Finding Areas Under the Standard Normal Distribution Curve Area To the Left of Any z Value Look up the z value in the table and use the area given. 0 z z 0 20
21 To the Right of Any z Value Look up the z value in the table to get the area. Subtract the area from 1. - z 0 21
22 Between Any Two z Values Look up both z values to get the areas. Subtract the smaller area from the larger area. z 0 +z 22
23 Between Any Two z Values Look up both z values to get the areas. Subtract the smaller area from the larger area. 0 z2 z1 23
24 Area Under the Curve The area under the curve is more important than the frequencies because the area corresponds to the probability! Note: In a continuous distribution, the probability of any exact Z value is 0 since area would be represented by a vertical line above the value. But vertical lines in theory have no area. So 24
25 Example 1 : (a) Find P ( z < 1.63) From table E (b) Find P (-2.48 < z < 0) P (z < 1.63) = From table E Area for z=0? 0.5 P (-2.48 < z < 0) = =
26 (c) Find P (-2.02 < z < 1.74) From Table E P (-2.02 < z < 1.74) = =
27 (d) Find the probability that z is larger than From Table E P (z >1.76) = =
28 Example 2 : Assume the standard normal distribution. Fill in the blanks. (a) P (0 < z < 1.46 ) = Add 0.5 to the given area of to get the cumulative area of z? z = 1.46 (b) P (0 < z < 3.09 ) = Add 0.5 to the given area of to get the cumulative area of z? z
29 (c) P ( < z < 0) = = z? 0 z = because the z-value is to the left of the mean. (d) P (z < 2.16 ) = From Table E 0 z? z =
30 (e) Find the z value to the left of the mean so that 71.90% of the area under the distribution curve lies to the right of it % = = From Table E z? z =
31 (f) Find two z values, one positive and one negative, so that the areas in the two tails total to 12% = 0.06 (one tailed area) From Table E 0 z = ±
32 6.2 Applications of the Normal I. Calculating Probabilities for a Non-Standard Normal Distribution Consider a normal variable x with mean and standard deviation. 1. Standardize from x to z. Distribution 2. Use Table E to find the central area corresponding to z. 3. Adjust the area to answer the question. 32
33 Example 1 : Let x be a normal random variable with mean 80 and standard deviation 12. What percentage of values are (a) larger than 56? P (x > 56) Standardize from x to z: P (x > 56) = P (z > -2) From Table E P (z > -2) = z 2 0 =
34 (b) less than 62? P (x < 62) Standardize from x to z: P (x < 62) = P (z < -1.5) From Table E z
35 (c) Between 85 and 98? P (85 x 98) P (0.42 z 1.5) From Table E P (0.42 z 1.5) = =
36 (d) outside of 1.5 standard deviations of the mean What is outside of 1.5 standard deviation of the mean? From Table E P (-1.5 < z < 1.5) = =
37 Example 2 : (Ref: General Statistics by Chase/Bown, 4 th ed.) The length of times it takes for a ferry to reach a summer resort from the mainland is approximately normally distributed with mean 2 hours and standard deviation of 12 minutes. Over many past trips, what proportion of times has the ferry reached the island in (a) less than 1 hour 45 minutes? P (z < -1.25) From Table E
38 (b) more than 2 hours, 5 minutes? P (z > 0.42) From Table E P (z > 0.42) = =
39 (c) between 1 hour, 50 minutes and 2 hours, 20 minutes? P (110 x 140) P (-0.83 z 1.67) From Table E P (-0.83 z 1.67) = =
40 II. Calculating a Cutoff Value Backward steps for calculating probabilities of a non-standard normal distribution. 1. Adjust to the corresponding central area. 2. Use Table E to find the corresponding z cutoff value. 3. Non-standardize from z to x. 40
41 Example 1 : Employees of a company are given a test that is distributed normally with mean 100 and variance 25. The top 5% will be awarded top positions with the company. What score is necessary to get one of the top positions? 2 Normal distribution, 100, = z? cutoff From Table E z = Non-standardize 41
42 Example 2 : Quiz scores were normally distributed with = 14 and = 2.8, the lower 20% should receive tutorial service. Find the cutoff score. Normal distribution, 14, From Table E z? 0 z = Non-standardize 42
43 Section 6 3 The Central Limit Theorem I. Sampling Distribution of Sample Mean Example 1 : Population Distribution Table (a) Find the population mean and population standard deviations of the population distribution table. 43
44 (b) Construct a probability histogram for x P(x) x 44
45 Example 2 : From the population distribution of example 1, 2 random variables are randomly selected. (a) List out all possible combinations (sample place) and for each combination
46 (b) Construct a probability distribution table for. (c) Construct a probability histogram for. P(x) 4/16 3/16 2/16 1/ x 46
47 (d) Find the mean of sampling distribution of. 47
48 (e) Find the standard deviation of the sampling distribution of. 48
49 (f) Compare with From (a) of Example 1, From (d) of Example 2, This shows that (g) Compare with From (a) of Example 1, From (e) of Example 2, This shows that ; however 49
50 Population parameter Sample statistics Mean Standard deviation Population Distribution Sampling Distribution P(x) P(x) 4/10 3/ x 2/10 1/ x 50
51 II. Central Limit Theorem If the population distribution is normally distributed, the sampling distribution of will be normally distributed for any size of n. P(x) P(x) x x If the population distribution is not normally distributed, the sampling distribution x of will be normally distributed for any size of n 30. P(x) P(x) x x 51
52 Example 1 : Population distribution P(x) Given : x (a) Find and for n = 4 (b) Is the sampling distribution normally distributed? According to central limit theory, will NOT be normally distributed because the population distribution is NOT normally distributed and n is NOT greater than 30. (c) If n is changed from 4 to 36, is the sampling distribution Yes, because n is greater than or equal to 30. normal distributed? 52
53 Example 2 : (Ref: General Statistics by Chase/Bown 4 th ed.) A population has mean 325 and variance 144. Suppose the distribution of sample mean is generated by random samples of size 36. (a) Find and (b) Find Recall : Standardize Now use :
54 (c) Find
55 Example 3 : The average number of days spent in a North Carolina hospital for a coronary bypass in 1992 was 9 days and the standard deviation was 4 days (North Carolina Medical Database Commission, Consumer s Guide to Hospitalization Charges in North Carolina Hospitals, August 1994). What is the probability that a random sample of 30 patients will have an average stay longer than 9.5 days?
56 Example 4 : Suppose the test scores for an exam are normally distributed with = 75, = 8 (a) What percentage of the students has a score greater than 85?
57 (b) What is the probability that 4 randomly selected students will have a mean score higher than 85?
58 Section 6-4 Normal Approximation to the Binomial Distribution I. When to use a Normal distribution to approximate a Binomial distribution? Recall that a binomial distribution is determined by n and p. When p is approximately 0.5, and as n increases, the shape of the binomial distribution becomes similar to the normal distribution. In order to use a normal distribution to approximate a binomial distribution, n must be sufficiently large. It is known n will be sufficiently large if np 5 and nq 5. When using a normal distribution to approximate a binomial distribution, the mean and standard deviation of the normal distribution is the same as the binomial distribution. Now recall the formulas for finding the mean and standard deviation. np, npq 58
59 II. Continuity Correction In addition to the condition np 5 and nq 5, a correction for continuity is used in employing a continuous distribution (Normal distribution) to approximate a discrete distribution (Binomial distribution). Warning : The continuity correction should be used only when approximating the Binomial probability with a normal probability. Don t use the continuity correction with other normal probability problems. Continuity correction x ±
60 Example 1 : Use the continuity correction to rewrite each expression : (a) Binomial Distribution Normal Distribution P (x > 6) P ( x > 6.5) (b) Binomial Distribution Normal Distribution P (x 3) P ( x 3.5) (c) Binomial Distribution Normal Distribution P (x 9) P ( x 9.5) 60
61 (d) Binomial Distribution Normal Distribution P (1< x < 7) P ( 1.5 < x < 6.5) (e) Binomial Distribution Normal Distribution P (5 x 10) P (4.5 x 10.5) (f) Binomial Distribution Normal Distribution P (4 < x 6) P (4.5 < x 6.5) 61
62 III. Using a Normal Distribution to approximate a Binomial Distribution Step 1 : Check whether the normal distribution can be used. ( np 5 and nq 5) Step 2 : Find the mean and standard deviation. Step 3 : Write the problem in probability notation, using x. Step 4 : Step 5 : Rewrite the problem by using the continuity correction factor. Continuity correction x ± 0.5 Find the corresponding z value(s) Step 6 : Use the z table to find the center area and adjust the center area to answer the question. 62
63 Example 1 : (Ref: General Statistics by Chase/Bown, 4 th ed.) Assume that the experiment is a binomial experiment. Find the probability of 10 or more successes, where n = 13 and p = 0.4. (a) Use the Binomial table P (x 10) = P (x = 10) + P (x = 11) + P (x = 12) + P (x = 13) = = (b) Use the normal approximate to the binomial Step 1 : Check : np nq Step 2 : Find and 63
64 Step 3 : Step 4 : P (x 10) Binomial Distribution P (x 10) Normal Distribution P (x > 9.5) Step 5 : Step 6 :
65 Example 2 : A dealer states that 90% of all automobiles sold have air conditioning. If the dealer sells 250 cars, find the probability that fewer than 5 of them will not have air conditioning. p = 0.10, q = 0.9 n = 250 Step 1 : Check : np nq Step 2 : Find and Step 3 : P (x < 5) Step 4 : Binomial Distribution Normal Distribution P (x < 5) P (x < 4.5) 65
66 Step 5 : Step 6 :
67 Example 3 : In a corporation, 30% of the people elect to enroll in the financial investment program offered by the company. Find the probability that of 800 randomly selected people, between 260 and 300 inclusive have enrolled in the program. p = 0.3, q = 0.7 n = 800 Step 1 : Check : np nq Step 2 : Find and Step 3 : P (260 x 300) Step 4 : Binomial Distribution Normal Distribution P (260 x 300) P (259.5 x 300.5) 67
68 Step 5 : Step 6 :
69 Summary The normal distribution can be used to describe a variety of variables, such as heights, weights, and temperatures. The normal distribution is bell-shaped, unimodal, symmetric, and continuous; its mean, median, and mode are equal. Mathematicians use the standard normal distribution which has a mean of 0 and a standard deviation of 1. 69
70 Summary (cont.) The normal distribution can be used to describe a sampling distribution of sample means. These samples must be of the same size and randomly selected with replacement from the population. The central limit theorem states that as the size of the samples increases, the distribution of sample means will be approximately normal. 70
71 Summary (cont.) The normal distribution can be used to approximate other distributions, such as the binomial distribution. For the normal distribution to be used as an approximation to the binomial distribution, the conditions np 5 and nq 5 must be met. A correction for continuity may be used for more accurate results. 71
72 Conclusions The normal distribution can be used to approximate other distributions to simplify the data analysis for a variety of applications. 72
A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More informationThe Central Limit Theorem
Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table 2 4 6 8 P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationChapter 4. The Normal Distribution
Chapter 4 The Normal Distribution 1 Chapter 4 Overview Introduction 4-1 Normal Distributions 4-2 Applications of the Normal Distribution 4-3 The Central Limit Theorem 4-4 The Normal Approximation to the
More informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationChapter Seven. The Normal Distribution
Chapter Seven The Normal Distribution 7-1 Introduction Many continuous variables have distributions that are bellshaped and are called approximately normally distributed variables, such as the heights
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More information11.5: Normal Distributions
11.5: Normal Distributions 11.5.1 Up to now, we ve dealt with discrete random variables, variables that take on only a finite (or countably infinite we didn t do these) number of values. A continuous random
More informationNormal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
More informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More information8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions 8-1
8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions For Example: On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community.
More informationThese Statistics NOTES Belong to:
These Statistics NOTES Belong to: Topic Notes Questions Date 1 2 3 4 5 6 REVIEW DO EVERY QUESTION IN YOUR PROVINCIAL EXAM BINDER Important Calculator Functions to know for this chapter Normal Distributions
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More informationChapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
More informationChapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
More informationChapter ! Bell Shaped
Chapter 6 6-1 Business Statistics: A First Course 5 th Edition Chapter 7 Continuous Probability Distributions Learning Objectives In this chapter, you learn:! To compute probabilities from the normal distribution!
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationMath 14 Lecture Notes Ch The Normal Approximation to the Binomial Distribution. P (X ) = nc X p X q n X =
6.4 The Normal Approximation to the Binomial Distribution Recall from section 6.4 that g A binomial experiment is a experiment that satisfies the following four requirements: 1. Each trial can have only
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationExample. Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables
Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables You are dealt a hand of 5 cards. Find the probability distribution table for the number of hearts. Graph
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationIOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
More informationProbability Distribution Unit Review
Probability Distribution Unit Review Topics: Pascal's Triangle and Binomial Theorem Probability Distributions and Histograms Expected Values, Fair Games of chance Binomial Distributions Hypergeometric
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationFound under MATH NUM
While you wait Edit the last line of your z-score program : Disp round(z, 2) Found under MATH NUM Bluman, Chapter 6 1 Sec 6.2 Bluman, Chapter 6 2 Bluman, Chapter 6 3 6.2 Applications of the Normal Distributions
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationSTAT:2010 Statistical Methods and Computing. Using density curves to describe the distribution of values of a quantitative
STAT:10 Statistical Methods and Computing Normal Distributions Lecture 4 Feb. 6, 17 Kate Cowles 374 SH, 335-0727 kate-cowles@uiowa.edu 1 2 Using density curves to describe the distribution of values of
More informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationExpected Value of a Random Variable
Knowledge Article: Probability and Statistics Expected Value of a Random Variable Expected Value of a Discrete Random Variable You're familiar with a simple mean, or average, of a set. The mean value of
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationStatistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
More informationProb and Stats, Nov 7
Prob and Stats, Nov 7 The Standard Normal Distribution Book Sections: 7.1, 7.2 Essential Questions: What is the standard normal distribution, how is it related to all other normal distributions, and how
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 The binomial: mean and variance Recall that the number of successes out of n, denoted
More informationChapter Seven: Confidence Intervals and Sample Size
Chapter Seven: Confidence Intervals and Sample Size A point estimate is: The best point estimate of the population mean µ is the sample mean X. Three Properties of a Good Estimator 1. Unbiased 2. Consistent
More informationWeek 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics.
Week 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics. Convergent validity: the degree to which results/evidence from different tests/sources, converge on the same conclusion.
More informationECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationExamples of continuous probability distributions: The normal and standard normal
Examples of continuous probability distributions: The normal and standard normal The Normal Distribution f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More information4: Probability. Notes: Range of possible probabilities: Probabilities can be no less than 0% and no more than 100% (of course).
4: Probability What is probability? The probability of an event is its relative frequency (proportion) in the population. An event that happens half the time (such as a head showing up on the flip of a
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationChapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 6 The Normal Distribution And Other Continuous Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 6 The Normal Distribution And Other Continuous Distributions 1999 Prentice-Hall, Inc. Chap. 6-1 Chapter Topics The Normal Distribution The Standard
More informationStatistics for Business and Economics: Random Variables:Continuous
Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive
More informationChapter 5 Normal Probability Distributions
Chapter 5 Normal Probability Distributions Section 5-1 Introduction to Normal Distributions and the Standard Normal Distribution A The normal distribution is the most important of the continuous probability
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More informationChapter 15: Sampling distributions
=true true Chapter 15: Sampling distributions Objective (1) Get "big picture" view on drawing inferences from statistical studies. (2) Understand the concept of sampling distributions & sampling variability.
More informationLecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
More informationSTAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
STAT 203 - Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More informationCHAPTER 5 Sampling Distributions
CHAPTER 5 Sampling Distributions 5.1 The possible values of p^ are 0, 1/3, 2/3, and 1. These correspond to getting 0 persons with lung cancer, 1 with lung cancer, 2 with lung cancer, and all 3 with lung
More informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationSTAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
More informationThe Normal Approximation to the Binomial Distribution
7 6 The Normal Approximation to the Binomial Distribution Objective 7. Use the normal approximation to compute probabilities for a binomial variable. The normal distribution is often used to solve problems
More informationModule Tag PSY_P2_M 7. PAPER No.2: QUANTITATIVE METHODS MODULE No.7: NORMAL DISTRIBUTION
Subject Paper No and Title Module No and Title Paper No.2: QUANTITATIVE METHODS Module No.7: NORMAL DISTRIBUTION Module Tag PSY_P2_M 7 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Properties
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name The bar graph shows the number of tickets sold each week by the garden club for their annual flower show. ) During which week was the most number of tickets sold? ) A) Week B) Week C) Week 5
More informationName PID Section # (enrolled)
STT 315 - Lecture 3 Instructor: Aylin ALIN 04/02/2014 Midterm # 2 A Name PID Section # (enrolled) * The exam is closed book and 80 minutes. * You may use a calculator and the formula sheet that you brought
More informationRandom variables The binomial distribution The normal distribution Sampling distributions. Distributions. Patrick Breheny.
Distributions September 17 Random variables Anything that can be measured or categorized is called a variable If the value that a variable takes on is subject to variability, then it the variable is a
More informationThe topics in this section are related and necessary topics for both course objectives.
2.5 Probability Distributions The topics in this section are related and necessary topics for both course objectives. A probability distribution indicates how the probabilities are distributed for outcomes
More informationSampling Distributions For Counts and Proportions
Sampling Distributions For Counts and Proportions IPS Chapter 5.1 2009 W. H. Freeman and Company Objectives (IPS Chapter 5.1) Sampling distributions for counts and proportions Binomial distributions for
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
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 informationStatistics, Measures of Central Tendency I
Statistics, Measures of Central Tendency I We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom
More informationCS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 12: Continuous Distributions Uniform Distribution Normal Distribution (motivation) Discrete vs Continuous
More informationReview. What is the probability of throwing two 6s in a row with a fair die? a) b) c) d) 0.333
Review In most card games cards are dealt without replacement. What is the probability of being dealt an ace and then a 3? Choose the closest answer. a) 0.0045 b) 0.0059 c) 0.0060 d) 0.1553 Review What
More informationUNIT 4 NORMAL DISTRIBUTION: DEFINITION, CHARACTERISTICS AND PROPERTIES
f UNIT 4 NORMAL DISTRIBUTION: DEFINITION, CHARACTERISTICS AND PROPERTIES Normal Distribution: Definition, Characteristics and Properties Structure 4.1 Introduction 4.2 Objectives 4.3 Definitions of Probability
More informationStandard Normal, Inverse Normal and Sampling Distributions
Standard Normal, Inverse Normal and Sampling Distributions Section 5.5 & 6.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy
More informationContinuous Probability Distributions
Continuous Probability Distributions Chapter 7 McGraw-Hill/Irwin Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. GOALS 1. Understand the difference between discrete and continuous
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More informationStatistics (This summary is for chapters 18, 29 and section H of chapter 19)
Statistics (This summary is for chapters 18, 29 and section H of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x n =
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
More informationSTAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
STAT 203 - Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good
More information5.1 Mean, Median, & Mode
5.1 Mean, Median, & Mode definitions Mean: Median: Mode: Example 1 The Blue Jays score these amounts of runs in their last 9 games: 4, 7, 2, 4, 10, 5, 6, 7, 7 Find the mean, median, and mode: Example 2
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION
In Inferential Statistic, ESTIMATION (i) (ii) is called the True Population Mean and is called the True Population Proportion. You must also remember that are not the only population parameters. There
More informationStatistics (This summary is for chapters 17, 28, 29 and section G of chapter 19)
Statistics (This summary is for chapters 17, 28, 29 and section G of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationBoth the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need.
Both the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need. For exams (MD1, MD2, and Final): You may bring one 8.5 by 11 sheet of
More informationNOTES: Chapter 4 Describing Data
NOTES: Chapter 4 Describing Data Intro to Statistics COLYER Spring 2017 Student Name: Page 2 Section 4.1 ~ What is Average? Objective: In this section you will understand the difference between the three
More informationguessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
More informationFall 2011 Exam Score: /75. Exam 3
Math 12 Fall 2011 Name Exam Score: /75 Total Class Percent to Date Exam 3 For problems 1-10, circle the letter next to the response that best answers the question or completes the sentence. You do not
More information