For a continued random variable the probability of observing one particular value is zero.

Size: px
Start display at page:

Download "For a continued random variable the probability of observing one particular value is zero."

Transcription

1 Chapter 7The Normal Probability Distribution Chapter 7.1 Uniform and Normal Distribution Objective A: Uniform Distribution A1. Introduction Recall: Discrete random variable probability distribution Special case: Binomial distribution Finding the probability of obtaining a success in n independent trials of a binomial experiment is calculated by plugging the value of a into the binomial formula as shown below: a P( x a) C p (1 p) Continuous Random variable For a continued random variable the probability of observing one particular value is zero. i.e. P( x a) 0 n a n a Continuous Probability Distribution We can only compute probability over an interval of values. Since P( x a) 0 and P( x b) 0 fora continuous variable, P( a x b) P( a x b) To find probabilities for continuous random variables, we use probability density functions. 1

2 Two common types of continuous random variable probability distribution: 1) Uniform distribution. 2) Normal distribution. A2. Uniform distribution 1 b a a b Note: The area under a probability density function is 1. Area of rectangle Height Width 1 Height ( b a) 1 Height for a uniform distribution ( b a) Example 1: A continuous random variable x is uniformly distributed with10 x 50. (a) Draw a graph of the uniform density function. Height = 1 (b a) Area of rectangle = Height x Width 1 = Height x(b - a) = 1 = 1 (50 10) 40 (b) What is P(20 x 30)? Area of rectangle = Height x Width = 1 * (30-20) 40 = 1 * = 1 =

3 (c) What is Px ( 15)? Area of rectangle = Height x Width P (x< 15) = P (x 15) = 1 = P (10 x 15) = 1 40 * 5 40 * (15 10) 1 40 = 1 40 = Objective B: Normal distribution Bell-shaped Curve 3

4 Example 1: Graph of a normal curve is given. Use the graph to identify the value of and X 730 Example 2: The lives of refrigerator are normally distributed with mean 14 years and standard deviation 2.5 years. (a) Draw a normal curve and the parameters labeled X (b) Shade the region that represents the proportion of refrigerator that lasts for more than 17 years. 4

5 (c) Suppose the area under the normal curve to the right x 17 is Provide twointerpretations of this result. Notation: P (x 17) = The area under the normal curve for any interval of values of the random variable x represents either: The proportions of the population with the characteristic described by the interval of values % of all refrigerators are kept for at least 17 years. the probability that a randomly selected individual from the population will have the characteristic described by the interval of values. The probability that a randomly selected refrigerator will be kept for at least 17 years is 11.51%. Chapter 7.2 Applications of the Normal Distribution Objective A: Area under the Standard Normal Distribution The standard normal distribution Bell shaped curve =0 and = Z The random variable for the standard normal distribution is Z. Negative Z Positive Z Use the Z table (Table V) to find the area under the standard normal distribution. Each value in the body of the table is a cumulative area from the left up to a specific Z -score. Probability is the area under the curve over an interval. The total area under the normal curve is 1. Z 0 Z 5

6 Under the standard normal distribution, (a) What is the area to the right 0? 0.5 (b) What is the area to the left 0?0.5 Example 1: Draw the standard normal curve with the appropriate shaded area and use StatCrunch to determine the shaded area. Open StatCrunch select Stat Calculators Normal select Standard select Input desired value for X compute record results (a) Find the shaded area that lies to the left of Z P( Z 1.38) (b) Find the shaded area that lies to the right of Similar steps as in part (a) except you want to select and input value, compute and record results Z P (Z> 0.56) =

7 (c) Find the shaded area that lies in between 1.85 and Open StatCrunch select Stat Calculators Normal select Between Input desired values for X range compute record results P (1.85 Z 2.47) = Objective B: Finding the Z-score for a given probability Z Area 0.5 Area 0. 5 Area 0. 5 Example 1: Draw the standard normal curve and the z -score such that the area to the left of the z -score is Use StatCrunch to find the z -score. Open StatCrunch Select Stat Calculator Normal Standard Input the value for P (x ) = Compute and record the results P (Z <-1.73) = Z? 0 7

8 Example 2: Draw the standard normal curve and the Z-score such that the area to the right of the Z-score is 0.18.Use StatCrunch to find the Z-score. Similar to example 1, input P (x ) = 0.18, compute P (Z > ) = Z? Example 3: Draw the standard normal curve and two Z-scores such that the middle area of the standard normal curve is Use StatCrunch to find the two Z-scores. If the middle area is 0.70, the total tailed areas is 0.30 (1-0.70) and the left tailed area is 0.15 (0.30/2). We will use StatCrunch to find the z score for the lower bound then use the symmetric concept to find the z score for the upper bound. Open StatCrunch Select Stat Calculator Normal Standard Input the value for P (x ) = 0.15 Compute and record the results % 0.15 P(-1.04< Z <1.04 ) = 0.70 Z 1.04 Z 1.04 (Due to symmetry) Objective C: Probability under a Normal Distribution Step 1: Draw a normal curve and shade the desired area. X Step 2: Convert the values X to Z -scores using Z. Step 3: Use StatCrunch to find the desired area. 8

9 Example 1: Assume that the random variable X is normally distributed with mean 50 and a standard deviation 7. (Note: this is not the standard normal curve because 0 and 1.) (a) PX ( 58) Z X X P( Z 1.14) Z (b) P(45 X 63) X 45 Z X Z 0.71 X 63 Z X Z 1.86 P (-0.71 Z 1.86) =

10 Example 2: Redo Example 1 Use StatCrunch and random variable X directly without converting to Z first. (a) PX ( 58) Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean, Std. Dev. and P (x 58) = Compute P (x 58) = (b) P(45 X 63) Open StatCrunch Select Stat Calculator Normal Between Input the values for Mean, Std. Dev. and P (45 x 63) = Compute P (45 x 63) =

11 Example 3: GE manufactures a decorative Crystal Clear 60-watt light bulb that it advertises will last 1,500 hours. Suppose that the lifetimes of the light bulbs are approximately normal distributed, with a mean of 1,550 hours and a standard deviation of 57 hours, use StatCrunch to find what proportion of the light bulbs will last more than 1650 hours? Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean, Std. Dev. and P (x 1650) = Compute P( X 1650) Objective D: Finding the Value of a Normal Random Variable Step 1: Draw a normal curve and shade the desired area. Step 2: Use StatCrunch to find the appropriate cutoff Z -score. X Step 3: Obtain X from Z by the formula Z or X Z. Example 1: The reading speed of 6th grade students is approximately normal (bell-shaped) with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the reading speed of a 6th graderwhose reading speed is at the 90 percentile? Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = 0.90 Compute and record the results 11

12 X Z X (24) X (b) Determine the reading rates of the middle 95 percentile. 95% in the middle means each tail is 5% divided by 2 = 2.5% = Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = Compute and record the results X Z X 125 ( ) (24) X Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = Compute and record the results X Z X 125 ( ) (24) X The middle 95% reading speed are between words per minute to words per minute. 12

13 Example 2: Redo Example 1 Use StatCrunch to find X directly without converting from Z to X. Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean = 125, Std. Dev. = 24, and P (x ) = 0.90 Compute (a) What is the reading speed of a 6th grader whose reading speed is at the 90 percentile? X (b) Determine the reading rates of the middle 95% percentile. If the middle area is 0.95, the total tailed areas is 0.05 and the left tailed area is (0.05/2). We will use StatCrunch to find the X score for the lower bound then change the inequality sign to find the X score for the upper bound. Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 125, Std. Dev. = 24, and P (x ) = > Compute X = words per minute 13

14 Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 125, Std. Dev. = 24, and P (x ) = > Compute X = words per minute The middle 95% reading speed are between words per minute to words per minute. Chapter 7.3 Normality Plot Recall: A set of raw data is given, how would we know the data has a normal distribution? Use histogram or stem leaf plot. Histogram is designed for a large set of data. For a very small set of data it is not feasible to use histogram to determine whether the data hasa bell-shaped curve or not. We will use the normal probability plot to determine whether the data were obtained from a normal distribution or not. If the data were obtained from a normal distribution, the data distribution shape is guaranteed to be approximately bell-shaped for n is less than

15 Perfect normal curve. The curve is aligned with the dots. Almost a normal curve. The dots are within the boundaries. Not a normal curve. Data is outside the boundaries. 15

16 Example 1: Determine whether the normal probability plot indicates that the sample data could have come from a population that is normally distributed. (a) Not a normal curve. The sample data do not come from a normally distributed population. There is no guarantee that this sample data set is normally distributed. (b) A normal curve. The sample data come from a normally distribute population. There is a guarantee that this sample data set is approximately normally distributed. 16

Example: Uniform Distribution. Chapter 6. Continuous Random Variables. Density Curve. Example: Uniform Distribution. Using Area to Find Probability

Example: Uniform Distribution. Chapter 6. Continuous Random Variables. Density Curve. Example: Uniform Distribution. Using Area to Find Probability Chapter 6. Continuous Random Variables Reminder: Continuous random variable takes infinitely many values Example: Uniform Distribution A continuous random variable has a uniform distribution if its values

More information

Continuous Random Variables and the Normal Distribution

Continuous Random Variables and the Normal Distribution Overview Continuous Random Variables and the Normal Distribution Dr Tom Ilvento Department of Food and Resource Economics Most intro stat class would have a section on probability - we don t But it is

More information

1 Probability Distributions

1 Probability Distributions 1 Probability Distributions In the chapter about descriptive statistics samples were discussed, and tools introduced for describing the samples with numbers as well as with graphs. In this chapter models

More information

Chapter 6: Continuous Random Variables & the Normal Distribution. 6.1 Continuous Probability Distribution. Area under the Curve as Probability

Chapter 6: Continuous Random Variables & the Normal Distribution. 6.1 Continuous Probability Distribution. Area under the Curve as Probability Chapter 6: Continuous Random Variables & the Normal Distribution 6.1 Continuous Probability Distribution and the Normal Probability Distribution 6.2 Standardizing a Normal Distribution 6.3 Applications

More information

Continuous Random Variables & Probability Distributions

Continuous Random Variables & Probability Distributions Continuous Random Variables & Probability Distributions What is a Random Variable? It is a quantity whose values are real numbers and are determined by the number of desired outcomes of an experiment.

More information

CHAPTERS 5 & 6: CONTINUOUS RANDOM VARIABLES

CHAPTERS 5 & 6: CONTINUOUS RANDOM VARIABLES CHAPTERS 5 & 6: CONTINUOUS RANDOM VARIABLES DISCRETE RANDOM VARIABLE: The variable can take on only certain specified values. There are gaps between possible data values. Values may be counting numbers

More information

Math 1313 The Normal Distribution. Basic Information

Math 1313 The Normal Distribution. Basic Information Math 1313 The Normal Distribution Basic Information All of the distributions of random variables we have discussed so far deal with distributions of finite discrete random variables. Their probability

More information

5.1 Introduction to Normal Distributions

5.1 Introduction to Normal Distributions 5.1 Introduction to Normal Distributions Properties of a Normal Distribution The mean, median, and mode are equal Bell shaped and is symmetric about the mean The total area that lies under the curve is

More information

0.1 Normal Distribution

0.1 Normal Distribution 0.1 Normal Distribution By now: Description of the distribution of a quantitative variable: 1. Obtain a plot 2. Examine the plot for patterns and deviations from it 3. Calculate an appropriate numerical

More information

Properties of a Normal Distribution

Properties of a Normal Distribution Properties of a Normal Distribution 5.1 Introduction to Normal Distributions The mean, median, and mode are equal Bell shaped and is symmetric about the mean The total area that lies under the curve is

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Lecture 8 (MWF) The binomial and introducing the normal distribution Suhasini Subba Rao The binomial:

More information

Probability Distributions for Continuous Random Variables

Probability Distributions for Continuous Random Variables 1 Probability Distributions for Continuous Random Variables Section 5.1 A continuous random variable is one that has an infinite number of possible outcomes. Its possible values can form an interval. The

More information

Chapter 6 Random Variables and the Normal Distribution

Chapter 6 Random Variables and the Normal Distribution 1 Chapter 6 Random Variables and the Normal Distribution Random Variable o A random variable is a variable whose values are determined by chance. Discrete and Continuous Random Variables o A discrete random

More information

Chapter 2. The Normal Distribution

Chapter 2. The Normal Distribution Chapter 2 The Normal Distribution Lesson 2-1 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve

More information

Normal Random Variables

Normal Random Variables 1 5.1 Using the Z Table Continuous Probability Distributions Normal Random Variables In this section, we work with random variables that are continuous. A continuous random variable can assume any numerical

More information

Displaying data Pick and mix revision cards. What information should be included in graphs, charts or diagrams? Give two measures of dispersion.

Displaying data Pick and mix revision cards. What information should be included in graphs, charts or diagrams? Give two measures of dispersion. Give two measures of dispersion. What information should be included in graphs, charts or diagrams? Range Interquartile range Standard deviation Scale or key Labels (with units) on axes Title When do you

More information

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS A random variable(rv) is a variable (typically represented by x) that has a single numerical value for each outcome of an experiment. A discrete random variable

More information

The probability that a randomly selected person will be exactly 5.00 feet tall is 0%.

The probability that a randomly selected person will be exactly 5.00 feet tall is 0%. Continuous Random Variables and the Normal Distribution Last time: We looked at discrete random variables. We looked at what a discrete probability distribution looks like. (a table or a bar graph) We

More information

Chapter 7. 1 P a g e Hannah Province Mathematics Department Southwest Tennessee Community College

Chapter 7. 1 P a g e Hannah Province Mathematics Department Southwest Tennessee Community College Chapter 7 Models of Distributions - Often a data set has a distribution that can be idealized by a mathematical description called a mathematical model. Often we can approximate a histogram by a smooth

More information

Chapter. The Normal Probability Distribution. Pearson Prentice Hall. All rights reserved

Chapter. The Normal Probability Distribution. Pearson Prentice Hall. All rights reserved Chapter 37 The Normal Probability Distribution 2010 Pearson Prentice Hall. All rights 2010 reserved Pearson Prentice Hall. All rights reserved 7-2 Relative frequency histograms that are symmetric and bell-shaped

More information

Minitab Guide for Math 355

Minitab Guide for Math 355 Minitab Guide for Math 355 8 7 6 Heights of Math 355 Students Spring, 2002 Frequency 5 4 3 2 1 0 60 62 64 66 68 70 72 74 76 78 80 Height Heights by Gender 80 Height 70 60 Female Gender Male Descriptive

More information

Chapter 6: The Normal Distribution

Chapter 6: The Normal Distribution January 5, 2010 Chapter Outline 6.1 Introducing Normally Distributed Variables 6.2 Areas Under Standard Normal Distribution 6.3 Working With Any Normal Distribution 6.4 Assessing Normality: Normal Probability

More information

Ch. 4 Continuous Random Variables and Probability Distributions

Ch. 4 Continuous Random Variables and Probability Distributions Ch. 4 Continuous Random Variables and Probability Distributions 4.1 Probability Density Functions A continuous random variable is a random variable with an interval of real numbers for its range. Probability

More information

Section 6-2 The Standard Normal Distribution

Section 6-2 The Standard Normal Distribution Section 6-2 The Standard Normal Distribution 6.1-1 Continuous Random Variables Continuous random variable A random variable X takes infinitely many values, and those values can be associated with measurements

More information

Lecture 3: Normal Distribution (Continued); Two useful Discrete Distributions: Binomial and Poisson. Chapter 1

Lecture 3: Normal Distribution (Continued); Two useful Discrete Distributions: Binomial and Poisson. Chapter 1 Lecture 3: Normal Distribution (Continued); Two useful Discrete Distributions: Binomial and Poisson Chapter 1 Back to Standard Normal (Z): backwards? If I give you a probability, can you find the corresponding

More information

b. Plot the histogram of the data, in the space below, using the above frequency distribution.

b. Plot the histogram of the data, in the space below, using the above frequency distribution. MATH 150-768 Elementary Statistics Final Review (Spring 2012) Chapter 2. Summarizing and Graphing Data 1. For the data given below, 70 61 74 80 58 66 53 70 65 71 63 60 75 62 68 a. Complete constructing

More information

Chapter 5: Continuous Probability Distributions

Chapter 5: Continuous Probability Distributions Department of Mathematics Izmir University of Economics Week 7-8 2014-2015 Introduction In this chapter we will focus on continuous random variables, cumulative distribution functions and probability density

More information

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: Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

More information

(i) The mean and mode both equal the median; that is, the average value and the most likely value are both in the middle of the distribution.

(i) The mean and mode both equal the median; that is, the average value and the most likely value are both in the middle of the distribution. MATH 203 Normal Distributions Dr. Neal, Spring 2009 Measurements that are normally distributed can be described in terms of their mean and standard deviation. These measurements should have the following

More information

P(66 height 69) = The probability distribution graph (histogram) would look like this:

P(66 height 69) = The probability distribution graph (histogram) would look like this: 111, section 8.5 The Normal Distribution notes by Tim Pilachowski Suppose we measure the heights of 25 people to the nearest inch and get the following results: height (in.) 64 65 66 67 68 69 70 frequency

More information

This section will focus on ways to organize categorical data and numerical data into tables, charts, and graphs.

This section will focus on ways to organize categorical data and numerical data into tables, charts, and graphs. Section 2.2 Frequency Distributions This section will focus on ways to organize categorical data and numerical data into tables, charts, and graphs. Frequency: is how often a categorical or quantitative

More information

Chapter 5. Chapter 5: Normal Distributions. Continuous Distribution. Continuous Distribution. Uniform Distribution

Chapter 5. Chapter 5: Normal Distributions. Continuous Distribution. Continuous Distribution. Uniform Distribution Chapter 5: Normal Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 5.2 Normal Distributions: Finding Probabilities 5.3 Normal Distributions: Finding Values 5.4

More information

Final Exam Review. Topics

Final Exam Review. Topics Final Exam Review The final exam is comprehensive, but focuses more on the material from chapters 5-8. The final will take place during the last class session Thursday 7/18. That is the only thing we will

More information

(a) (i) Use StatCrunch to simulate 1000 random samples of size n 10

(a) (i) Use StatCrunch to simulate 1000 random samples of size n 10 Chapter 8 Sampling Distribution Ch 8.1 Distribution of Sample Mean Objective A: Shape, Center, and Spread of the Distributions of A1. Sampling Distributions of Mean A1.1 Sampling Distribution of the Sample

More information

Chapter 5: The Normal Distribution

Chapter 5: The Normal Distribution Chapter 5: The Normal Distribution MODELING CONTINUOUS VARIABLES Histogram 6.1 Proportion 5 0.08 0.06 0.04 0.02 30 35 40 45 50 55 Age If we draw a curve through the tops of the bars in Histogram 6.1 and

More information

NORMAL DISTRIBUTION. Key words: mean, standard deviation, standard normal, inverse normal, continuity correction.

NORMAL DISTRIBUTION. Key words: mean, standard deviation, standard normal, inverse normal, continuity correction. NORMAL DISTRIBUTION Achievement Standard: 90646 (2.6) (part) external; credits 4 Key words: mean, standard deviation, standard normal, inverse normal, continuity correction. 1. The Normal Distribution

More information

Chapter 5 Continuous Random Variables. otherwise. The probability density function (PDF) is given by: 0 for all values of x.

Chapter 5 Continuous Random Variables. otherwise. The probability density function (PDF) is given by: 0 for all values of x. Chapter 5 Continuous Random Variables A continuous random variable can take any numerical value in some interval. Assigning probabilities to individual values is not possible. Probabilities can be measured

More information

Business Statistics: A First Course

Business Statistics: A First Course Business Statistics: A First Course 5 th Edition Chapter 6 The Normal Distribution Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc. Chap 6-1 Learning Objectives In this chapter, you learn:

More information

STA 291 Lecture 15. Normal Distributions (Bell curve) STA Lecture 15 1

STA 291 Lecture 15. Normal Distributions (Bell curve) STA Lecture 15 1 STA 291 Lecture 15 Normal Distributions (Bell curve) STA 291 - Lecture 15 1 Distribution of Exam 1 score STA 291 - Lecture 15 2 Mean = 80.98 Median = 82 SD = 13.6 Five number summary: 46 74 82 92 100 STA

More information

Source: D. Morton, et al., Lead Absorption in Children of Employees in a Lead-Related Industry, American Journal of Epidemiology 155 (1982).

Source: D. Morton, et al., Lead Absorption in Children of Employees in a Lead-Related Industry, American Journal of Epidemiology 155 (1982). STAT E-50 - Introduction to Statistics The Normal Model 1. Researchers have investigated lead absorption in children of parents who worked in a factory where lead is used to make batteries. Shown below

More information

Chapter 7 Normal Probability Distribution. Section 7.1 and 7.2

Chapter 7 Normal Probability Distribution. Section 7.1 and 7.2 Chapter 7 Normal Probability Distribution. When the data values are evenly distributed about the mean, the distribution is said to be symmetrical. When the majority of the data values fall to the left

More information

Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions.

Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. Lecture 1 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a process consisting of 3 parts. Lecture

More information

z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

More information

Observed Value Mean Standard Deviation

Observed Value Mean Standard Deviation Handout 3: Normal Distribution Reading Assignment: Chapter 4 Towards Normality Recall that the Empirical Rule states for bell-shaped data about 8% of the values fall within one standard deviation of the

More information

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage 4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

More information

Measuring center and spread for density curves. Calculating probabilities using the standard Normal Table (CIS Chapter 8, p 105 mainly p114)

Measuring center and spread for density curves. Calculating probabilities using the standard Normal Table (CIS Chapter 8, p 105 mainly p114) Objectives 1.3 Density curves and Normal distributions Density curves Measuring center and spread for density curves Normal distributions The 68-95-99.7 (Empirical) rule Standardizing observations Calculating

More information

6 3 The Standard Normal Distribution

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

More information

Mean and Variance. Uniform Distribution. Uniform Distribution Example. Uniform Distribution. Uniform Distribution Example

Mean and Variance. Uniform Distribution. Uniform Distribution Example. Uniform Distribution. Uniform Distribution Example Uniform Distribution Mean and Variance Suppose that a bus always arrives at a particular stop between 8:00 and 8:0 A.M. and that the probability that the bus will arrive in any given subinterval of time

More information

But what if I ask you to find the percentage of the weights that are less than 2300 lbs???

But what if I ask you to find the percentage of the weights that are less than 2300 lbs??? 1 Chapter 6: Normal Probability Distributions 6 2 The Standard Normal Distributions In chapter 5, we considered only discrete probability distributions using tables, but in this chapter we will learn continuous

More information

Chapter 2 Summarizing and Graphing Data. Section 2-2 Frequency Distributions. Definition. Survey data: Pulse rate (15 sec)

Chapter 2 Summarizing and Graphing Data. Section 2-2 Frequency Distributions. Definition. Survey data: Pulse rate (15 sec) Chapter 2 Summarizing and Graphing Data Section 2-2 Frequency Distributions 2-1 Review and Preview 2-2 Frequency Distributions 2-3 Histograms 2-4 Graphs that Enlighten and Graphs that Deceive Definition

More information

CHAPTER Probability Density Functions

CHAPTER Probability Density Functions CHAPTER 8 1. Probability Density Functions For some types of data, we want to consider the distribution throughout the population. For example, age distribution in the US gives the percentage of the US

More information

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

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

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

MATH 10: Elementary Statistics and Probability Chapter 6: The Normal Distribution

MATH 10: Elementary Statistics and Probability Chapter 6: The Normal Distribution MATH 10: Elementary Statistics and Probability Chapter 6: The Normal Distribution Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

More information

Math 213: Applied Statistics, Gannon University MINITAB 15 Guide 1

Math 213: Applied Statistics, Gannon University MINITAB 15 Guide 1 Math 213: Applied Statistics, Gannon University MINITAB 15 Guide 1 November 15, 2007 This guide contains instructions for most of the MINITAB commands used in the course. More commands may be added to

More information

The Normal Distribution

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

More information

Last Name First Name Class Time Chapter 5-1. a. Define the random variable, X =

Last Name First Name Class Time Chapter 5-1. a. Define the random variable, X = Last Name First Name Class Time Chapter 5-1 Chapter 5: Continuous Random Variables -Uniform Distribution Example U1: A package delivery service divides their packages into weight classes. Suppose that

More information

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution Chapter 6: Random Variables and the Normal Distribution 6.1 Discrete Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.4 Standard

More information

Chapter 6: Continuous Random Variables

Chapter 6: Continuous Random Variables Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 6: Continuous Random Variables UE 2.2.2 Probability Density Function (Section

More information

Describing the data graphically: Frequency distributions, histograms, and other types of graphs

Describing the data graphically: Frequency distributions, histograms, and other types of graphs Lecture 2 Describing the data graphically: Frequency distributions, histograms, and other types of graphs. 2-1 2.1 Frequency Distributions and Histograms Frequency Distribution A summary of a set of data

More information

Continuous Random Variables

Continuous Random Variables Chapter 5 Continuous Random Variables A continuous random variable can take any numerical value in some interval. Assigning probabilities to individual values is not possible. Probabilities can be measured

More information

7.1: Normal Distributions

7.1: Normal Distributions In Chapter 7: Chapter 7: Normal Probability Distributions 7.1 Normal Distributions 7.2 Determining Normal Probabilities 7.3 Finding Values That Correspond to Normal Probabilities 7.4 Assessing Departures

More information

6.1 Discrete and Continuous Random Variables

6.1 Discrete and Continuous Random Variables 6.1 Discrete and Continuous Random Variables A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur. For example, suppose we toss a fair

More information

7.1 - Continuous Probability Distribution and The Normal Distribution

7.1 - Continuous Probability Distribution and The Normal Distribution 7.1 - Continuous Probability Distribution and The Normal Distribution Since a continuous random variable can assume an infinite number of uncountable values, we have to look at assuming a value within

More information

CHAPTER 6: Continuous Random Variables Essentials of Business Statistics, 4th Edition Page 1 of 83

CHAPTER 6: Continuous Random Variables Essentials of Business Statistics, 4th Edition Page 1 of 83 6 CHAPTER 6: Continuous Random Variables 228 Essentials of Business Statistics, 4th Edition Page 1 of 83 6.1 Learning Objectives After mastering the material in this chapter, you will be able to: _ Explain

More information

Statistics: Introduction:

Statistics: Introduction: Statistics: Introduction: STAT- 114 Notes Definitions Statistics Collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and

More information

Standard Normal Distribution

Standard Normal Distribution Standard Normal Distribution The standard normal distribution has three properties: 1. It s graph is bell-shaped. ( 0) 2. It s mean is equal to 0. 3. It s standard deviation is equal to 1 ( 1) Uniform

More information

Midterm 2 A. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Midterm 2 A. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Name: Class: Date: Midterm 2 A Multiple Choice Identify the choice that best completes the statement or answers the question. 1. An approach of assigning probabilities which assumes that all outcomes of

More information

Methods for Describing Sets of Data

Methods for Describing Sets of Data Chapter 2 Methods for Describing Sets of Data 2.1 Describing Qualitative Data Definition 2.1 Class: A class is one of the categories into which qualitative data can be classified. Definition 2.2 Class

More information

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage 4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the given frequency distribution to find the (a) class width. (b) class midpoints of

More information

Unit 16 Normal Distributions

Unit 16 Normal Distributions Unit 16 Normal Distributions Objectives: To obtain relative frequencies (probabilities) and percentiles with a population having a normal distribution While there are many different types of distributions

More information

Week in Review #6 ( , )

Week in Review #6 ( , ) Math 66 Week-in-Review - S. Nite 0/3/0 Page of 6 Week in Review #6 (.-.4, 3.-3.4) n( E) In general, the probability of an event is P ( E) =. n( S) The Multiplication Principle: Suppose there are m ways

More information

2. A sample is a subset of the population. 3. Construct a frequency distribution for the data of the grades of 25 students taking Math 11 last

2. A sample is a subset of the population. 3. Construct a frequency distribution for the data of the grades of 25 students taking Math 11 last Math 111 Chapter 12 Practice Test 1. If I wanted to survey 50 Cabrini College students about where they prefer to eat on campus, which would be the most appropriate way to conduct my survey? a. Find 50

More information

Exercise: Empirical Rule

Exercise: Empirical Rule Exercise: Empirical Rule Use the empirical rule to answer the following:! Monthly maintenance costs are distributed normally with a µ=$250 and σ=$50 " 1) What percent of months have maintenance costs in

More information

Exercise: Empirical Rule. Answer: " 1) STAT1010 Standard Scores. 5.2 Properties of the Normal Distribution

Exercise: Empirical Rule. Answer:  1) STAT1010 Standard Scores. 5.2 Properties of the Normal Distribution Exercise: Empirical Rule Use the empirical rule to answer the following:! Monthly maintenance costs are distributed normally with a µ=$250 and σ=$50 " 1) What percent of months have maintenance costs in

More information

Section 9.5 The Normal Distribution

Section 9.5 The Normal Distribution Section 9.5 The Normal Distribution The normal distribution is a very common continuous probability distribution model in which most of the data points naturally fall near the mean, with fewer data points

More information

Basic Statistics Review Part Two Page 1. Basic Statistics Review Part Two

Basic Statistics Review Part Two Page 1. Basic Statistics Review Part Two Basic Statistics Review Part Two Page 1 Basic Statistics Review Part Two Sampling Distribution of the Mean; Standard Error (See Zar 4 th ed. pages 65-76; or Zar 5 th ed. pages 66-72; 87-91) In our discussion

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

More information

Business Statistics: A First Course 5 th Edition

Business Statistics: A First Course 5 th Edition Business Statistics: A First Course 5 th Edition Chapter 6 The Normal Distribution Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc. Chap 6-1 Learning Objectives In this chapter, you learn:

More information

Chapter 2 - Graphical Summaries of Data

Chapter 2 - Graphical Summaries of Data Chapter 2 - Graphical Summaries of Data Data recorded in the sequence in which they are collected and before they are processed or ranked are called raw data. Raw data is often difficult to make sense

More information

You flip a fair coin four times, what is the probability that you obtain three heads.

You flip a fair coin four times, what is the probability that you obtain three heads. Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

More information

THE NORMAL DISTRIBUTION (Gaussian Distribution)

THE NORMAL DISTRIBUTION (Gaussian Distribution) THE NORMAL DISTRIBUTION (Gaussian Distribution) Marquis de Laplace (1749-1827) and Carl Friedrich Gauss (1777-1855) were jointly credited with the discovery of the normal distribution. However, in 1924,

More information

Continuous Random Variables

Continuous Random Variables Statistics 5 Lab Materials Continuous Random Variables In the previous chapter, we introduced the idea of a random variable. In this chapter we will continue the discussion of random variables. Our focus

More information

Curriculum Map - Prob and Statistics - Author: Susan Kelly

Curriculum Map - Prob and Statistics - Author: Susan Kelly Page 1 of 15 Map: Prob and Statistics Grade Level: 12 District: Island Trees Created: 11/10/2007 Last Updated: 11/10/2007 Essential Questions Content Skills Assessments Standards/PIs Re What is statistics?

More information

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability 6-2 The Standard Normal Distribution This section presents the standard normal distribution which has three properties: 1. Its graph is bell-shaped. 2. Its mean is equal to 0 (μ = 0). 3. Its standard deviation

More information

Normal Distribution Lab

Normal Distribution Lab Math 130 Name Solutions Normal Distribution Lab Goal: To gain experience with the normal distribution, both by hand and with the computer. Part 1 Sketching Normal Distributions A sample of daily temperatures

More information

Chapter 3: Probability Distributions and Statistics

Chapter 3: Probability Distributions and Statistics Chapter 3: Distributions and Statistics Section 3.4 3.4 Normal Distribution So far, we have focused on discrete random variables. Now we turn our attention to continuous random variables. Continuous random

More information

Standardization. The Normal Density. The Normal Distribution

Standardization. The Normal Density. The Normal Distribution The The is the most important distribution of continuous random variables. The normal density curve is the famous symmetric, bellshaped curve. The central limit theorem is the reason that the normal curve

More information

4B: Normal Probability Distributions

4B: Normal Probability Distributions Normal density curve 4B: Normal Probability Distributions The previous section used the binomial formula to calculate probabilities for binomial random variables. Outcomes were discrete, and probabilities

More information

Yesterday s lab. Mean, Median, Mode. Section 2.3. Measures of Central Tendency. What are the units?

Yesterday s lab. Mean, Median, Mode. Section 2.3. Measures of Central Tendency. What are the units? Yesterday s lab Section 2.3 What are the units? Measures of Central Tendency Measures of Central Tendency Mean, Median, Mode Mean: The sum of all data values divided by the number of values For a population:

More information

The Normal Distribution

The Normal Distribution CHAPTER 6 The Normal Distribution CHAPTER OUTLINE 6.1 The Standard Normal Distribution 6.2 Probability Calculations with the Normal Distribution 6.3 Applications of the Normal Distribution 6.4 Determining

More information

A continuous random variable can take on any value in a specified interval or range

A continuous random variable can take on any value in a specified interval or range Continuous Probability Distributions A continuous random variable can take on any value in a specified interval or range Example: Let X be a random variable indicating the blood level of serum triglycerides,

More information

FREQUENCY AND PERCENTILES

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

More information

Key Concept. Density Curve

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

More information

3.4 The Normal Distribution

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

More information

Z-Scores and What Exactly Does It Mean to be Normal Anyway? Discrete Math, Section 9.5

Z-Scores and What Exactly Does It Mean to be Normal Anyway? Discrete Math, Section 9.5 Z-Scores and What Exactly Does It Mean to be Normal Anyway? Discrete Math, Section 9.5 We first look at the concept of a z-score in isolation, then we tie it into the standard normal distribution. I. Z-Scores

More information

Describing Data with Tables and Graphs

Describing Data with Tables and Graphs IB Math Describing Data with Tables and Graphs Example A random sample of 20 management students (n=20) took a course to prepare them for a management aptitude test. The following table gives MAT their

More information

Chapter 5. Normal Probability Distribution

Chapter 5. Normal Probability Distribution Chapter 5 Normal Probability Distribution Lesson 5-1/5-2 The Standard Normal Distribution Review Random Variable A variable having a single numerical value, determine by chance, for each outcome of some

More information