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Course: STAT 051, Fall 2010
School: GWU
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2 Descriptive Chapter Statistics Contents Introduction Qualitative Data Frequency Distribution Graphical Methods Quantitative Data Graphical Methods Numerical Summary Box Plot 2 Qualitative Data Measurements that cannot be measured on a natural numerical scale Observations can be classified into one of a group of categories Examples: a)Political affiliation (Democrat, Republican, etc) b)Rating of a...

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2 Descriptive Chapter Statistics Contents Introduction Qualitative Data Frequency Distribution Graphical Methods Quantitative Data Graphical Methods Numerical Summary Box Plot 2 Qualitative Data Measurements that cannot be measured on a natural numerical scale Observations can be classified into one of a group of categories Examples: a)Political affiliation (Democrat, Republican, etc) b)Rating of a service (excellent, good, poor etc.) c)Race d)Gender 3 Summarizing Qualitative Data Tabular Summary Frequency Distribution Relative Frequency Distribution Graphical Summary Bar Chart Pie Chart 4 Frequency Distribution A FREQUENCY DISTRIBUTION is a tabular summary of data showing different classes and the frequency (or number) of items in each of several non-overlapping classes. It shows how the observations in the sample are distributed The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data. 5 Example 1.0 Students taking a distance learning course were asked to rate the quality of the course as being excellent, good, average, below average, or poor. The ratings provided by a sample of 20 students are shown below. Average Good Excellent Excellent Below Average Good Average Average Good Average Good Good Good Poor Average Good Below average Good Good Poor 6 Avg Below Good Good Avg Good Good Good Poor Avg Good Below Good Good Poor Example 1.0 Frequency Distribution: Shows 5 different classes(from 5 levels of the rating variable) and frequency for each class. Good Exclnt Avg Exclnt Avg Rating Poor Below Average Average Good Excellent Total Frequency 2 2 5 9 2 20 7 Relative Frequency Distribution RELATIVE FREQUENCY of a class is the proportion of the total number of observations belonging to the class. Tabular Summary Relative Frequency Distribution Frequency Distribution Rating Poor Poor Avg Below Bverage Aelow Avg Average Good Gxcellent E ood Excellent Total Total Relative Frequency F 2 requency 0.10 2/20 0.10 2/20 5 0.25 5/20 9 0.45 9/20 2 0/20 20 .10 1.00 20/20 8 Bar graphical Chart A method for depicting qualitative data (from a frequency, relative frequency distribution.) n the horizontal axis the levels of the qualitative O variable are specified. ertical axis represents frequency or relative V frequency. ar of fixed width are drawn for each class, with B height being proportional to the frequency. he bars are separated T 9 Example 1.0 (Bar Chart) Rating Rating Poor Below Average Average Good Rreq F el. Freq 2 0.10 2.10 0 Relative Frequency Bar Chart Frequency Bar Chart 5.25 0 9.45 0 Excellent 2.10 0 Total 20 1.0 10 Pie Chart The PIE CHART is a graphical method used for presenting relative frequency distributions for qualitative data. A circle represents the entire dataset. It is partitioned so that each sector represents a class. Area of the sectors represent the relative frequency of class it is representing. Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume (. 25)(360) = 90 degrees of the circle. 11 Example 1.0 (contd) Rating Poor Below Average Average Good Rel. Freq 0.10 0.10 9 0o Poor Below Avg Average Good Excellent 0.25 0.45 Excellent 0.10 Total 1.0 12 Example 2.0 The following Dataset refers to a report published by Forbes magazine. In addition to income data Forbes collected and reported personal data on the CEOs, including level of education. The following data represent the highest degree obtained for the top 25 highest paid CEOs of 1999. 13 CEODEGREES DATA: Bachelors Bachelors Bachelors none Doctorate Masters Doctorate Masters JD Masters Masters Masters Masters Masters Bachelors Bachelors Bachelors Masters Masters Bachelors Masters LLB JD LLB Masters 14 Example 2.0 Variable: CEO degree We want see the distribution of the highest degree of CEOs of different companies. IN SPSS open the dataset: Then, Select: Graphs followed by BAR / PIE 15 Using SPSS : Dataset 16 Bar Chart: SPSS 12 10 Degrees of CEOs COUNT 8 6 4 2 0 Bachelors Doctorate JD LLB Masters None DEGREE 17 Pie Chart : SPSS None Bachelors Masters Doctorate JD LLB 18
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GWU - STAT - 051
Comparing Mean, Median and ModeFor negatively (left) skewed distributions Mean &lt; Median &lt; ModeMean MedianMode1Skewed RightFor positively skewed distributions Mean &gt; Median &gt; ModeModeMean Median2Symmetric DistributionFor symmetric (not skewed) d
GWU - STAT - 051
A Complete ExampleThe dataset gives ammonia level near an exit ramp of a tunnel tunnel for 8 different days cfw_1.53, 1.50, 1.37, 1.51, 1.55, 1.42, 1.41, 1.48Mean and SDObs 1.53 1.50 1.37 1.51 1.55 1.42 1.41 1.48 11.772 Deviation Deviation2.05875 .02
GWU - STAT - 051
Chapter 1IntroductionContents Whatis Statistics? Applications Fundamental Elements Types of Data Data CollectionWhat is Statistics?Statistics is the Science of collection and analysis of Data.Various aspects: Classifying Summarizing Analyzing Inte
GWU - STAT - 051
The Empirical RuleSometimes the distribution is symmetric and bell shaped. For these kind of distribution mean and sd together can describe the distribution fairly well. Most of the observations lie near the center or mean of the data. EMPIRICAL RULE sum
GWU - STAT - 051
Descriptive StatisticsNumerical SummarySummation NotationObservations in a dataset are denoted by cfw_ x1,x2,x3,x4,.xn ; n = sample size x1 is the first observation, x2 is 2nd obs. and so on. We use xi to denote x1 + x2 + x3 + x4 +.+ xn In particular,
GWU - STAT - 051
Frequency DistributionDataset is summarized in a tabular form. Range of the dataset is partitioned into a number of classes of equal width. Frequency distribution table is constructed by counting number of observations (called frequency) in each class an
GWU - STAT - 051
Descriptive StatisticsQuantitative DataQuantitative DataMeasurements that can be measured on a natural and meaningful numerical scale Examples:a) b) c)SAT score of students The current unemployment rate for 50 states Number of calls made over last we
SUNY Buffalo - CSE - 566
CSE 566: Wireless Networks Security (Spring 2010) Project 1a Date: 02/08/2010 Due Date: 02/22/2010 This is the first part of a simulation project. The second part will be assigned later in the semester. This part is to familiarize you with the ns2-network
Rutgers - GEO - 101
1 Exam 1 Review Chapter 1: Introduction to Earth System Earth as a System Geography 5 Themes of Geography (Location, Place, Region, Movement, Human-Earth Interaction) Physical Geography Cultural/Human Geography Universe (Big bang theory); Milky Way Galaxy
University of Texas - MATH - 408D
beck (kab3335) HW #1 Antoniewicz (57420) This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 10.0 points Assuming that 67.9% of the Earths surface is covered with
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4301 1,2,3,n , n , * W , * W , .) p ? (, W , * W , k), * W , . ,k , * W , , (, : 1 2 4 n = 7,8,9,100 , 4302 ,E :* n D x * ( 1) W * ( 2) W (3) 4303 x * x W3 x + 4 y + 5 z E :*= n D * n W n W Ip n WC60 24E48 30x36DAB4304 * W x k H y, 3, zx xx xx
U. Houston - MATH - 1313
Section 2.5 Multiplication of Matrices If A is a matrix of size mxn and B is a matrix of size nxp then the product AB is defined and is a matrix of size mxp. So, two matrices can be multiplied if and only if the number of columns in the first matrix is eq
U. Houston - MATH - 1313
Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on the curve?y 1
U. Houston - MATH - 1313
Lesson 2 One-Sided Limits and Continuity One-Sided Limits Sometimes we are only interested in the behavior of a function when we look from one side and not from the other. Example 1: Consider the function f ( x) =x x . Find lim f ( x ).x 0Now suppose w
U. Houston - MATH - 1313
Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point.We need a way to find the
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x ) f ' ( x) dx Example 1: Use the produ
U. Houston - MATH - 1313
Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = (3 x 2 5 x + 6 ) into functions f ( x) and g ( x ) such that h( x) = ( f o g )( x).4Rule 10: The Cha
U. Houston - MATH - 1313
Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ' ' (
U. Houston - MATH - 1313
Lesson 8 Some Applications of the Derivative Equations of Tangent Lines The first applications of the derivative involve finding the slope of the tangent line and writing equations of tangent lines. Example 1: Find the equation of the line tangent to f (
U. Houston - MATH - 1313
Lesson 9 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know how much it costs to produce one more unit of this pr
U. Houston - MATH - 1313
Lesson 10 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f A function is increasing on an interval (a, b) if, for any two numbers x1 and x 2 in (a, b), f ( x1 ) &lt; f ( x 2 )
U. Houston - MATH - 1313
Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is increasing or decr
U. Houston - MATH - 1313
Math 1314 Lesson 12 Curve Sketching One of our objectives in this part of the course is to be able to graph functions. In this lesson, well add to some tools we already have to be able to sketch an accurate graph of each function. From prerequisite materi
U. Houston - MATH - 1313
Lesson 13 Absolute Extrema In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will learn how to find absolute extrema, that
U. Houston - MATH - 1313
Lesson 14 Optimization Now youll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. In many problems, youll state the domain before you wo
U. Houston - MATH - 1313
Lesson 15 Exponential Functions as Mathematical Models In this lesson, we will look at a few applications involving exponential functions. Well first consider some word problems having to do with money. Next, well consider exponential growth and decay pro
U. Houston - MATH - 1313
Lesson 16 Antiderivatives So far in this course, we have been interested in finding derivatives and in the applications of derivatives. In this chapter, we will look at the reverse process. Here we will be given the answer and well have to find the proble
U. Houston - MATH - 1313
Lesson 17 Integration by Substitution Sometimes the rules from the last lesson arent enough. In this lesson, you will learn to integrate using substitution. This is related to the chain rule that you used in finding derivatives. Using Substitution to Inte
U. Houston - MATH - 1313
Lesson 18 Area and the Definite Integral We are now ready to tackle the second basic question of calculus the area question. We can easily compute the area under the graph of a function so long as the shape of the region conforms to something for which we
U. Houston - MATH - 1313
Lesson 19 The Fundamental Theorem of Calculus In the last lesson, we approximated the area under a curve by drawing rectangles, computing the area of each rectangle and then adding up their areas. We saw that the actual area was found as we let the number
U. Houston - MATH - 1313
Lesson 20 Evaluating Definite Integrals We will sometimes need these properties when computing definite integrals. Properties of Definite Integrals Suppose f and g are integrable functions. Then: 1. aaf ( x)dx = 0 f ( x)dx = f ( x)dxb a2. 3.ba cf
U. Houston - MATH - 1313
Lesson 21 Area Between Two Curves Two advertising agencies are competing for a major client. The rate of change of the clients revenues using Agency As ad campaign is approximated by f(x) below. The rate of change of the clients revenues using Agency Bs a
U. Houston - MATH - 1313
Lesson 22 Functions of Several Variables So far, we have looked at functions of a single variable. In this section, we will consider functions of more than one variable. You are already familiar with some examples of these.P ( x, y ) = 2 x + 2 y A( P, i,
U. Houston - MATH - 1313
Lesson 23 Partial Derivatives When we are asked to find the derivative of a function of a single variable, f ( x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. With a function of two variables, we
U. Houston - MATH - 1313
Lesson 24 Maxima and Minima of Functions of Several VariablesWe learned to find the maxima and minima of a function of a single variable earlier in the course. Although we did not use it much, we had a second derivative test to determine whether a critic
U. Houston - MATH - 1313
1. Given the following graph of a function, f(x).y6 5 4 3 2 1x9 8 7 6 5 4 3 2 1 1 2 3 4 1 2 3 4 5 6 7 8 9 10Find: a. lim f ( x)x 2b.lim f ( x)x 1xc.xlim f ( x)5d.limf ( x) 2+e.lim f ( x)x 1f. f(-2) h. f(2)g. f(1)2. Given the follow
U. Houston - MATH - 1313
Math 1314 Test 1 Review Review Properties of Exponents 1. a 0 = 12. a n =1 n1 an3. a = n am4. a n = n a m 5. a m a n = a m + n am 6. n = a m n , a 0 a 7 . ( a m ) n = a mn 8. (ab) n = a n b nan a 9. = n , b 0 b b 10. For b 1 , b x = b y if and only
U. Houston - MATH - 1313
Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on the curve?y 1
U. Houston - MATH - 1313
Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point.We need a way to find the
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x ) f ' ( x) dx Example 1: Use the produ
U. Houston - MATH - 1313
Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = (3 x 2 5 x + 6 ) into functions f ( x) and g ( x ) such that h( x) = ( f o g )( x).4Rule 10: The Cha
U. Houston - MATH - 1313
Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ' ' (
U. Houston - MATH - 1313
Math 1314 Test 2 ReviewIn numbers 1 7, find the limit. 1.xlim x3 + 2x 2 1 2 x + 32. limxx3 + x 1 x 13. limxx3 x 1 x 14. Given the following graph of a function f, find lim f ( x) if it exists.x 4Math 1314 Test 2 Review1Limits at Infinity: I
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CUNY Queens - ECON - 247
Bus 247 Homework Set 1 Summer 2009 Queens College Professor Bradbury This assignment is due at the beginning of class on Monday June 15. The assignment must be typed and stapled in order to receive credit. (though graphical solutions may be handwritten in
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
Home Work II Exam II Preparation Assignment This assignment is due in class Tuesday April 29. The assignment will not be accepted if it is turned in late. The assignment will not be accepted unless it is typed. 1) Consider the following Game: Compete Comp
CUNY Queens - ECON - 247
HW III Bus 247 Spring 2008 Professor Bradbury Please answer each of the following questions. This assignment must be turned in typed together with your final paper by May 20th. The assignment will be graded for accuracy.1) When firms are able to achieve
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 1-Fall 2008 Prof. Bradbury 1. The follow equation represents the Total Cost Function for a representative F i rm: TC= C (q) = 480 + 4q^2 a. Fixed Cost = 480 Variable Cost=4q^2b. Expression for Average To
CUNY Queens - ECON - 247
Bus 247 Business Economics Homework I Fall 2008 Prof. Bradbury To the best of your abilities, please answer each question in its entirety. This assignment will be due at the beginning of class on Thursday October, 2. All assignments must be typed and stab
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 2-Fall 2008 Prof. Bradbury 1. The behavior of average cost which we associate with natural monopoly is falling cost 2. I t makes more sense for one fi rm to satisfy all market demand rather than m any fi
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
1.Profit q2=320 Maximization Reaction Function, 2 firms competing by quantity:q1=R1 q2= 480- q22| q2=R2 q1= 480- q12A. WAY # 1: B. WAY # 2 :q2R1 q2=R2 q1 480- q22 = 480- q12 2q2 32)q2=48023 =12q2=960-480 480 (480 - q22 ) 0 = 480 240-q24 240 = q2 24
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
Steven Kordvani Business Economics (BUS 247) HW # 4-Fall 2008 Prof. Bradbury 1. When fi rms are able to achieve perfect price discrimination: P1= 3 5 10 a. The value of consumer surplus is zero because all the consumer surplus is t ransferredto the fi rm