# Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 from a uniform distribution. Figure 1.35 graphs the distribution density curve for a uniform distribution. Use areas under this density curve to answer the following questions. Define the random variable X to be the value that is generated by the computer. 0 1 X Figure 1.35 The density curve of a uniform distribution, for exercise (a) Why is the total area under this curve equal to 1? Since the figure is a defined as a density curve, then by definition it has a total area of 1 square unit. The area represents 100% of the population (b) What proportion of the observations lie above 0.75? To answer this question we need only to find the area above the curve corresponding to X > P(X > 0.75) = Height of yellow rectangle(width of yellow rectangle) = (1)(1 0.75) = 0.25 Keep in mind that because the author chose a uniform X distribution with endpoints 0 and 1, it is easy to see what the proportion should be without much thought. Make sure you learn the real lesson here, that in order to calculate proportions with density curves, the area underneath the curve is directly related to the corresponding proportion. (c) What proportion of the observations lie between 0.25 and 0.75? We need to calculate P(0.25 < X < 0.75). P(0.25 < X < 0.75) = 1( ) = X

2 1.81 Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the outcomes are to be distributed uniformly between 0 and 2. Then the density curve of the outcomes has constant height between 0 and 2, and height 0 elsewhere. Let the random variable Y be the value generated by the computer. (a) What is the height of the density curve between 0 and 2? Draw a graph of the density curve. The height of the density curve is ½, 0.5. Why? Because, a density curve, must have an area equal to 1 square unit. If you look at the dimensions of the rectangle we get ½ (2) = 1 square unit. ½ 0 2 Y (b) Use your graph from (a) and the fact that areas under the curve are proportions of outcomes to find the proportion of outcomes that are less than 1. It is very easy to see that the area is one, but to be complete I will run through the calculation. ½ 0 1 P(Y < 1) = ½ (1 0) = Y (c) Find the proportion of outcomes that lie between 0.5 and ½ Y P(0.5 < Y < 1.3) = ½ ( ) = 0.4

3 1. 82 What are the mean and the median of the uniform distribution from problem 1.80 (Figure 1.35)? What are the quartiles? Since this is a symmetric distribution, the median and the mean are the same value, the halfway point. Thus the mean is 0.5 as well as the median. To calculate the mean of any uniform distribution take the average of the two endpoints: (0 + 1)/2 = 0.5 Again, since the boundaries of the figure are 0 and 1, it is easy to see the position of the quartiles: Q 1 = 0.25 and Q 3 = Now while it is easy to see the quartile values, it is also easy to confuse what it is I am looking at. It just happens that the value of X also corresponds to the area it represents when we consider the frequency to the left of the number. That is, 1 P(X < 0.25) = P(X < Q 1 ) = 0.25 (area not value of X) P(X < 0.75) = P(X < Q 3 ) = 0.75 (area not value of X) X Q 1 Q 3 If you are unsure what the above notation means or how it is related to the picture on the left, see me quickly Figure 1.36 displays three density curves, each with three points marked on the axis. At which of these points on each curve do the mean and the median fall? A B C A B C A B C (a) (b) (c) In order to analyze these curves correctly, one needs to remember that for a density curve the median is the value that splits the area above exactly in half (the median is the point the cuts the ordered set of numbers in half); the mean is pulled by outliers. Thus for picture (a) The median appear to be B, which then makes the mean C. For picture (b), since we have a symmetric graph, the mean and median are represented by A. Lastly, for picture (c), the median appears to be B and thus, the mean is A, which is pulled by outliers.

4 1.84 The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 and standard deviation 16 days. Draw a density curve for this distribution on which the mean and standard deviation are correctly related. Let the random variable X denote the length of human pregnancies. µ 3σ µ 2σ µ σ µ µ+σ µ+2σ µ+3σ µ X 1.89 The height of women aged 20 to 29 are approximately normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches. What are the z-scores for a woman 6 feet tall and a man 6 feet tall? What information do the z-scores give that the actual heights do not? Women: {µ = 64 inches, σ = 2.7 inches} Men:{µ = 69.3 inches, σ = 2.8 inches} 0.75 Man: z = Woman: z = I can see that the six-foot tall woman is, among her peers, very tall, an extremely unusual height. (z = ). While the man is at six feet is above average but not as far away from the norm as the woman.

5 1.93 Using either Table A or your calculator or software, find the proportion of observations from a standard normal distribution that satisfies each of the following statements. In each case, sketch a standard normal curve and shade the area under the curve that is the answer to the question. (a) Z -2 (this is a cumulative proportion) If I looked this value up on a table then, I need to realize that -2, implies that my accuracy will be So I look up -2 on the column with the z-value and the first column gives you the rest of the accuracy P(Z -2) = z 0.00 Standard Normal Probabilities Z If I used Excel, the command would be =Normsdist(-2); this of course provides more accuracy in the result than the table.

6 (b) Z -2 What you need to keep in mind when looking up values on the table, is what area the table provides versus what you want. I want P(Z -2). The area underneath the whole curve is Thus, P(Z -2) = 1 - P(Z -2) = = Notice if I looked up Z = 2 on the table this is the associated value. On Excel the command is =1 normsdist(-2) Z 4. (c) Z > 1.67 P(Z > 1.67) = 1 P(Z < 1.67) = = On Excel the command would be = 1 normsdist(1.67) Standard Normal Probabilities Z 4. z

7 (d) -2 < Z < 1.67 To get this result I will use the previous information. I could look it up on the tables but it would most likely be the information I already have. Here is one way. P(-2 < Z < 1.67) = Z 4. The I got from problem (c). I note that P(Z > -1.67) = P(Z < 1.67). Now I need to subtract that little portion to the leftof 2, mainly the area Another way. P(-2 < Z < 1.67) = 1 ( ( )) Here I am using the fact that the entire area is one. I then calculate the two missing end points either directly or by another calculation. Subtract from one and I have the area I want. Using Excel = normsdist(1.67) normsdist(-2) Find the value of z of a standard normal variable Z that satisfies each of the following conditions. (If you use Table A, report the value of z that comes closest to satisfying the condition). In each case, sketch a standard normal curve with your value of z marked on the axis. (a) 20% of the observations fall below z. If I use table A, I find that P(Z < -0.84) =.2005 which is close to the Using software like Excel, I get z (=normsinv(0.2)) Z desired (b) 30% of the observations fall above z. 0.4 If I look at the table I see that P(Z > 0.52) = and P(Z > 0.53) = The value I want is about halfway between the two. So a good approximation of z is the average of 0.52 and 0.53 which is Z

8 Using software like Excel, I get z ; I entered =normsinv(0.7) The Wechsler Adult Intelligence Scale (WAIS) is the most common IQ test. The scale of scores is set separately for each age group and is approximately normal mean with mean 100 and standard deviation 15. The organization MENSA which calls itself the high IQ society, requires a WAIS score of 130 or higher for membership. What percent of adults would qualify for membership? Let the random variable X denote the WAIS score. We want to calculate P(X > 130). I notice that the value 130 is 2 standard deviations from the mean; by the rule then, P(X > 130) = 2.5%. So P(X > 130) = Notice if I use the tables or a computer by finding the z-score I will not get 2.5%. Z = 2 for X = 130. = less than 2.5% which is just an approximation. Using Excel, I type in =normsdist(2) and I get , which is the area to the right. The TI-83 command is normalcdf(2,10) 1.99 Jacob scores 16 on the ACT. Emily scores 670 on the SAT. Assuming that both tests measure the same thing, who has the highest score? SAT: µ = 1026 σ = 209 ACT: µ = 20.8 σ = 4.8 Emily: z = Jacob: z = = = -1 The z-scores tells us how far away each value is away from their respective means. So Emily is 1.7 standard deviations below the mean, and Jacob is only one standard deviation below the mean. Since Emily is much further below the mean than Jacob, Jacob has the higher score.

9 1.102 Reports on a student s ACT or SAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as percent: the percent of all scores that were lower than this one. Tonya scores 1318 on the SAT. What is the percentile? Let s see so far we have the words percentage, relative frequency, percentile, and soon to come probability. All are calculated exactly the same, but how we view it is slightly different, thus the name change. Basically I need to calculate the area to the left of 1318, for this normal distribution. µ = 1026 σ = 209 Let the random variable X denote an SAT score. P(X < 1318) The area above represents the frequency of the numbers found on the X-axis, (i.e. how often would I encounter a value less than 1318 for example). The z-score for 1318 is If I use Excel I would enter = normsdist(1.3971) which results in P(X < 1318) = which ranks Tonya very high, almost at the 92 percentile. If I were to use the table then instead of interpolating(the correct thing to do) to make it easier I will round (which does not give me as good of an approximation as interpolating, whatever that means). My z-score is then z = 1.40 P(Z < 1.40) = which essentially says the same thing as the other result, Tonya is almost at the 92 nd percentile Reports on a student s ACT or SAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as percent: the percent of all scores that were lower than this one. Jacob scores 16 on the ACT. What is his percentile? Since I know the distribution is normal ( a very important fact) then I will turn the value in question to a z-score, so I can look up the frequency on Table A, or input it into software such as Excel and get the required frequency. Z = = -1 Using the rule I calculate that P(Z < -1) 16%. Using table A, P(Z < -1) = 15.87%.

10 1.111 Middle-aged men are more susceptible to high cholesterol than the young women of Exercise The blood cholesterol levels of men aged 55 to 64 are approximately normal with mean 222 mg/dl and standard deviation 37 mg/dl. What percent of these men have high cholesterol (levels above 240 mg/dl)? What percent have borderline high cholesterol (between 200 and 240 mg/dl)? In order to do well in statistics, one needs to understand what information is available to them. This will be specially true in later chapters. So let us start good habits now. What is known? The distribution is normal (Knowing the distribution type is tremendously important). Do I know the parameters for the normal distribution? Yes, µ = 222 mg/dl and σ = 37 mg/dl. What do I want to know? How often I will see a value that is above 240 mg/dl, for one question and the other how often will I get a value between 200 mg/dl and 240 mg/dl. Since I know the distribution type and have all the necessary information I will find my z-scores so I can correlate the z-scores to the requested frequencies by looking on a table or using the computer. Let the random variable X denote the blood cholesterol levels. P(X > 240 mg/dl) = P Z > P(Z > 0.49) We would expect a reading above 240mg/dl about 31% of the time P(200 < X < 240 ) = P Z < P Z < P(Z < 0.49) P(Z < -0.59) of the time We would expect a reading between 200mg/dl and 240 mg/dl about 41%

### AP 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, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

### Section 1.3 Exercises (Solutions)

Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch

### 6 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

### Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

### Chapter 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

### Complement: 0.4 x 0.8 = =.6

Homework Chapter 5 Name: 1. Use the graph below 1 a) Why is the total area under this curve equal to 1? Rectangle; A = LW A = 1(1) = 1 b) What percent of the observations lie above 0.8? 1 -.8 =.2; A =

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

### Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5

ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has

### Chapter 3 Normal Distribution

Chapter 3 Normal Distribution Density curve A density curve is an idealized histogram, a mathematical model; the curve tells you what values the quantity can take and how likely they are. Example Height

### Density 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

### Chapter 3. The Normal Distribution

Chapter 3. The Normal Distribution Topics covered in this chapter: Z-scores Normal Probabilities Normal Percentiles Z-scores Example 3.6: The standard normal table The Problem: What proportion of observations

### Probability. Distribution. Outline

7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The

### 13.2 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

### MULTIPLE 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) 1-2 9 3-4 22 5-6

### The 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

### Key Concept. Density Curve

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

### Probability 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.

### 32 Measures of Central Tendency and Dispersion

32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency

### STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

### Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### Chapter 8 Homework ( ) -- Normal Distribution

Chapter 8 Homework (195-198) -- Normal Distribution Dr. McGahagan NOTE: I often abbreviate the text declaration that X is a random variable distributed normally with mean 8 and variance of 144 as " X is

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)

### Unit 8: Normal Calculations

Unit 8: Normal Calculations Summary of Video In this video, we continue the discussion of normal curves that was begun in Unit 7. Recall that a normal curve is bell-shaped and completely characterized

### 6.3 Applications of Normal Distributions

6.3 Applications of Normal Distributions Objectives: 1. Find probabilities and percentages from known values. 2. Find values from known areas. Overview: This section presents methods for working with normal

### First Midterm Exam (MATH1070 Spring 2012)

First Midterm Exam (MATH1070 Spring 2012) Instructions: This is a one hour exam. You can use a notecard. Calculators are allowed, but other electronics are prohibited. 1. [40pts] Multiple Choice Problems

### Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

### 6.4 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

### Chapter 5: The normal approximation for data

Chapter 5: The normal approximation for data Context................................................................... 2 Normal curve 3 Normal curve.............................................................

### Exercises - 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

### 7. Normal Distributions

7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

### Sampling 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

Using Your TI-NSpire 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

### HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

### Normal 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:

### 2.0 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

### Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

### EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!

STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.

### 4.3 Areas under a Normal Curve

4.3 Areas under a Normal Curve Like the density curve in Section 3.4, we can use the normal curve to approximate areas (probabilities) between different values of Y that follow a normal distribution Y

### Lesson 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

### 6.2 Normal distribution. Standard Normal Distribution:

6.2 Normal distribution Slide Heights of Adult Men and Women Slide 2 Area= Mean = µ Standard Deviation = σ Donation: X ~ N(µ,σ 2 ) Standard Normal Distribution: Slide 3 Slide 4 a normal probability distribution

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### FREQUENCY AND PERCENTILES

FREQUENCY DISTRIBUTIONS AND PERCENTILES New Statistical Notation Frequency (f): the number of times a score occurs N: sample size Simple Frequency Distributions Raw Scores The scores that we have directly

### Study 6.3, #87(83) 93(89),97(93)

GOALS: 1. Understand that area under a normal curve represents probabilities and percentages. 2. Find probabilities (percentages) associated with a normally distributed variable using SNC. 3. Find probabilities

### The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median

CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box

### Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

### Continuous Distributions, Mainly the Normal Distribution

Continuous Distributions, Mainly the Normal Distribution 1 Continuous Random Variables STA 281 Fall 2011 Discrete distributions place probability on specific numbers. A Bin(n,p) distribution, for example,

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### Using Your TI-NSpire Calculator: Normal Distributions Dr. Laura Schultz Statistics I

Using Your TI-NSpire 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),

### Unit 7: Normal Curves

Unit 7: Normal Curves Summary of Video Histograms of completely unrelated data often exhibit similar shapes. To focus on the overall shape of a distribution and to avoid being distracted by the irregularities

### Cents and the Central Limit Theorem Overview of Lesson GAISE Components Common Core State Standards for Mathematical Practice

Cents and the Central Limit Theorem Overview of Lesson In this lesson, students conduct a hands-on demonstration of the Central Limit Theorem. They construct a distribution of a population and then construct

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### Continuous 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

### MULTIPLE 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

### 4: Probability. What is probability? Random variables (RVs)

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### 3.4 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

### AP STATISTICS 2010 SCORING GUIDELINES

2010 SCORING GUIDELINES Question 4 Intent of Question The primary goals of this question were to (1) assess students ability to calculate an expected value and a standard deviation; (2) recognize the applicability

### CC 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

### 1) 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,

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### Lesson 7 Z-Scores and Probability

Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting

### Key Concept. February 25, 2011. 155S6.5_3 The Central Limit Theorem. Chapter 6 Normal Probability Distributions. Central Limit Theorem

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### Review the following from Chapter 5

Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

### Descriptive Statistics

Chapter 2 Descriptive Statistics 2.1 Descriptive Statistics 1 2.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Display data graphically and interpret graphs:

### Normal 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

### Section 2.4 Numerical Measures of Central Tendency

Section 2.4 Numerical Measures of Central Tendency 2.4.1 Definitions Mean: The Mean of a quantitative dataset is the sum of the observations in the dataset divided by the number of observations in the

### 4. 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

### Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics

Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),

### Rescaling and shifting

Rescaling and shifting A fancy way of changing one variable to another Main concepts involve: Adding or subtracting a number (shifting) Multiplying or dividing by a number (rescaling) Where have you seen

### Report 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

### . 58 58 60 62 64 66 68 70 72 74 76 78 Father s height (inches)

PEARSON S FATHER-SON DATA The following scatter diagram shows the heights of 1,0 fathers and their full-grown sons, in England, circa 1900 There is one dot for each father-son pair Heights of fathers and

### Mind on Statistics. Chapter 2

Mind on Statistics Chapter 2 Sections 2.1 2.3 1. Tallies and cross-tabulations are used to summarize which of these variable types? A. Quantitative B. Mathematical C. Continuous D. Categorical 2. The table

### AP * Statistics Review. Descriptive Statistics

AP * Statistics Review Descriptive Statistics Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Statistics Summary (prepared by Xuan (Tappy) He)

Statistics Summary (prepared by Xuan (Tappy) He) Statistics is the practice of collecting and analyzing data. The analysis of statistics is important for decision making in events where there are uncertainties.

### What Does the Normal Distribution Sound Like?

What Does the Normal Distribution Sound Like? Ananda Jayawardhana Pittsburg State University ananda@pittstate.edu Published: June 2013 Overview of Lesson In this activity, students conduct an investigation

### Descriptive statistics; Correlation and regression

Descriptive statistics; and regression Patrick Breheny September 16 Patrick Breheny STA 580: Biostatistics I 1/59 Tables and figures Descriptive statistics Histograms Numerical summaries Percentiles Human

### Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables 2

Lesson 4 Part 1 Relationships between two numerical variables 1 Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables

### Section 3 Part 1. Relationships between two numerical variables

Section 3 Part 1 Relationships between two numerical variables 1 Relationship between two variables The summary statistics covered in the previous lessons are appropriate for describing a single variable.

### Chapter 10 - Practice Problems 1

Chapter 10 - Practice Problems 1 1. A researcher is interested in determining if one could predict the score on a statistics exam from the amount of time spent studying for the exam. In this study, the

### Margin of Error When Estimating a Population Proportion

Margin of Error When Estimating a Population Proportion Student Outcomes Students use data from a random sample to estimate a population proportion. Students calculate and interpret margin of error in

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### Lesson 4 Measures of Central Tendency

Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central

### 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

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### given that among year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20...

Probability - Chapter 5 given that among 12-14 year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20... 5.6 What percentage of boys in this age range

### Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the

### Normal Probability Distribution

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

### Pie Charts. proportion of ice-cream flavors sold annually by a given brand. AMS-5: Statistics. Cherry. Cherry. Blueberry. Blueberry. Apple.

Graphical Representations of Data, Mean, Median and Standard Deviation In this class we will consider graphical representations of the distribution of a set of data. The goal is to identify the range of

### FINAL 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

### Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

### Describing Data. We find the position of the central observation using the formula: position number =

HOSP 1207 (Business Stats) Learning Centre Describing Data This worksheet focuses on describing data through measuring its central tendency and variability. These measurements will give us an idea of what