Chapter 3. The Normal Distribution


 Anastasia Ross
 1 years ago
 Views:
Transcription
1 Chapter 3. The Normal Distribution Topics covered in this chapter: Zscores Normal Probabilities Normal Percentiles Zscores Example 3.6: The standard normal table The Problem: What proportion of observations on a standard Normal variable z take values less than 1.47? Find a zscore in SPSS. 1. Open SPSS. 2. Type the number 1.47 in the first cell of the Data Editor. 3. Go to the Transform menu. 4. Scroll to the Compute Variable option. The following window should open: 34
2 35 Chapter 3 5. Under Function Group, scroll down and select the CDF & Noncentral CDF option. 6. Under Functions and Special Variables, scroll down and doubleclick the Cdfnorm option. Now the window should appear like this:
3 The Normal Distribution Replace the question mark under Numeric Expression with the variable ZValue by highlighting the question mark, clicking on the variable ZValue to the left and then clicking the arrow to the left of the Numeric Expression box. 8. Under Target Variable type any variable name you like, for example Probability. 9. Click OK. Now the answer should be adjacent to the value of 1.47 in your Data Editor in a column entitled whatever you named the Target Variable as seen below. Normal Probabilities Example 3.8: Who qualifies for an athletic scholarship? The Problem: The NCAA considers a student a partial qualifier if the combined SAT score is at least 720. Partial qualifiers can receive athletic scholarships and practice with the team, but they can t compete during their first college year. What proportion of all students who take the SAT would be partial qualifiers, receiving a combined SAT score of between 720 and 820? SAT scores are distributed with a mean of 1026 and a standard deviation of Open a new window in SPSS. 2. Click on the Variable View tab and create a variable named SAT. 3. Click on the Data View tab and enter two data values: 720 and 820.
4 37 Chapter 3 4. Go to the Transform menu. 5. Scroll to the Compute Variable option. The following window should open: 6. Under Function Group, scroll down to the CDF & Noncentral CDF option.
5 The Normal Distribution Under Functions and Special Variables, scroll down to the Cdf.Normal option and doubleclick. Now the previous window should appear like this: 8. Replace the first question mark under Numeric Expression with the variable SAT by highlighting the first question mark, clicking on the variable Quant to the left and then clicking the arrow to the left of the Numeric Expression box. 9. Replace the second question mark under Numeric Expression with the mean of 1026 as given in the problem. 10. Replace the third question mark under Numeric Expression with the standard deviation of Under Target Variable type the variable name Probability.
6 39 Chapter Click OK. Now two probabilities may be viewed in the Data Editor, the probability that a student scores less than a 720 for their combined SAT score and the probability that a student scores less than an 820 for their combined SAT score. Since the question asked for the probability that a student scored between a 720 and an 820, the two probabilities should be subtracted, leaving a final probability of = 0.09 or 9 percent. Normal Percentiles Example 3.9: Find the top 10% using software The Problem: Scores on the SAT Verbal test in recent years follow approximately the N(504,111) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT? 1. Click on the Variable View tab. 2. Create three variables named Prob, Mean, and SD. Change the number of decimals for Mean and SD to 0.
7 The Normal Distribution Go to the Data View tab. 4. Type.90 under the Prob column in the first row. We want the location of the top 10% which has the same bordering point as the lower 90%, and the normal distribution uses only lower probabilities. 5. Type 504 under the Mean column. Type 111 under the SD column. 6. Go to the Transform menu. 7. Scroll to the Compute Variable option. The following window should open: 8. Under Function Group, scroll down to the Inverse DF option. 9. Under Functions and Special Variables, scroll down to the Idf.Normal option and doubleclick. 10. Replace the first question mark under Numeric Expression with the variable Prob by highlighting the first question mark, clicking on the variable Prob to the left and then clicking the arrow to the left of the Numeric Expression box. 11. Replace the second question mark under Numeric Expression with the variable Mean by highlighting the second question mark, clicking on the variable Mean to the left and then clicking the arrow to the left of the Numeric Expression box. 12. Replace the third question mark under Numeric Expression with the variable SD by highlighting the third question mark, clicking on the
8 41 Chapter 3 variable SD to the left and then clicking the arrow to the left of the Numeric Expression box. 13. Under Target Variable type a variable name you like, for example ANS. 14. Click OK. Now the answer should be adjacent to the three variables in your SPSS Data Editor in a column entitled whatever you typed in Target Variable.
9 The Normal Distribution 42 Chapter 3 Exercises 3.9 Men s and women s heights Monsoon rains Table A Standard normal drill Acid rain? 3.33 A milling machine In my Chevrolet The middle half What s your percentile? 3.41 Heights of men and women A surprising calculation Normal is only approximate: ACT scores Are the data normal? Fruit fly thorax lengths Are the data normal? Soil penetrability Where are the quartiles?
10 317 Chapter 3 SPSS Solutions 3.9 It s inconvenient to use Minitab for a computation such as this. Using a standard calculator, we can easily compute the zscores. To compute the zscores, we use the formula z = ( value μ)/ σ. Either do the subtraction first, or be sure to use parentheses. A woman six feet (72 ) tall is 2.96 standard deviations above the mean; the six foot tall man is standard deviations above the mean. The woman is much taller, relative to other women, than the man is, compared to other men To find the percent of years with less than 697 mm of rain, we use Transform, Compute Variable. Locate the CDF & Noncentral CDF Function group, then the CDF.Normal function in the lower box. Clicking on that will transfer the command shell into the Numeric Expression box. Notice that in the lower center of the box there is a description of the command and its parameters. Enter the parameters as shown, then OK computes the probability into the worksheet (as variable Drought, here). For more decimal places in your result (remember, the default is two), click on the Variable view tab and increase them. About 2.9% of all years will have less than 697 mm of rain. To find the percent of normal rainfall years (between 683 mm and 1022 mm), we ll find the cumulative probability for 1022 mm and subtract the cumulative probability of
11 We do this in one combination of CDF.Normal calculations as shown below. About 96.1% of all years will have normal rainfall Here, we are given a relative frequency under the standard Normal curve. We need to find the value of z. We ll again use Transform, Compute Variable. Locate the Inverse DF Function group, then the IDF.Normal function in the lower box. Clicking on that will transfer the command shell into the Numeric Expression box. Notice that in the lower center of the box there is a description of the command and its parameters. Enter the parameters as shown, then OK computes the probability into the worksheet (as variable Z here). The point z with 20% of the area below it is z = We repeat for part (b) using 0.6 as the area to the left of the point (since 40% of the observations are above it). This point is z = As with Exercise 3.13 above, use Transform, Compute Variable, we want the Inverse DF and IDF.Normal. As before, enter the area to the left of the desired point on the curve (0.8), the value of the mean (0) and standard deviation (1). This point is z =
12 319 Part (b) asks for the point with 35% of all observations above it; this means that 65% = 0.65 are below it. This point is z = To find the proportion of rainy days that meet the acid rain criteria, we use Transform, Compute Variable. Locate the CDF & Noncentral CDF Function group, then the CDF.Normal function in the lower box. Clicking on that will transfer the command shell into the Numeric Expression box. Notice that in the lower center of the box there is a description of the command and its parameters. Enter the parameters as shown, then OK computes the probability into the worksheet (as variable Acid, here). For more decimal places in your result (remember, the default is two), click on the Variable view tab and increase them. At this location 22.9% of days will qualify as acid rain days To find the proportion of slots that meet specifications, we ll use Transform, Compute Variable and find the cumulative probability for inch and subtract the cumulative probability of inch. We do this in one combination of CDF.Normal calculations as shown below. About 98.76% of slots will meet the specifications This problem refers to the information given about 2008 model vehicles. They had mean 18.7 mpg and standard deviation 4.3 mpg. We want to know the area to the left of the Chevy Malibu (with 25 mpg). Use Transform, Compute Variable and find the cumulative probability for the Malibu as below % of 2008 cars had worse mileage than the Chevy Malibu.
13 To find the quartiles, we want the points with (respectively) 25% and 75% of the area below them. We can find these values using Transform, Compute Variable. We want the Inverse DF and IDF.Normal. As before, enter the area to the left of the desired point on the curve (0.25, then 0.75), the value of the mean (18.7) and standard deviation (4.3). This point is z = We find that Q 1 (the 25 th percentile) is mpg and Q 3 (the 75 th percentile) is mpg The percentile corresponds to the area to the left of the value of interest. We find this using Transform, Compute Variable and find the cumulative probability for the Jacob as below. We see that Jacob is not quite at the 15 th percentile (his is 14.9) We want to know what proportion of women are taller than the average man (69.3 ). We ll use Transform, Compute Variable but subtract the percent of women shorter than 69.3 from 1 to find the proportion taller than 69.3 Be sure to use the values for the women s distribution: mean (64), and the standard deviation (2.7). We see that not quite 2.5% (2.48%) of women should be taller than the average man.
14 To find the proportion of students scoring at least 750, we ll use Transform, Compute Variable and subtract the proportion scoring less than 750 from 1 as we did in Exercise We see that 3.1% of men scored at least 750 while only 1.1% of women did this well To find the proportion scoring higher than 27, divide the given numbers; to find the proportion scoring 27 or more, add the number that scored 27 to the first. We find that 11.5% scored higher than 27, while 15.3% scored at least 27. To compare this with the Normal computation, use CDF.Normal to find the proportion scoring at least than 27 by subtracting the proportion scoring less than 27 from 1. We would expect 12.3% to score at least 27 if the scores were exactly Normal Open worksheet file ex We ll create a histogram of the lengths and compute summary statistics using Analyze, Descriptive Statistics, Explore. Click to enter variable Length in the Dependent List. Click Plots and be sure the Histogram box is checked. To find the quartiles of this distribution, click Statistics and ask for Percentiles. Weighted Average(Definition 1) Length Percentiles Percentiles Tukey's Hinges Length
15 322 Descriptives Length Statistic Std. Error Mean Median.8000 Variance.006 Std. Deviation Minimum.64 Maximum.94 Range.30 Interquartile Range.10 Skewness Kurtosis This distribution actually looks a bit skewed left (other windows also show this same general shape); there are no outliers. The mean ( x = 0.800) is the same (within rounding) as the median (Med = 0.8); the standard deviation is s = 0.078; the quartiles are Q 1 = 0.76 and Q 3 = The distances to the quartiles from the median (0.04 and 0.06) are roughly similar. These all suggest the distribution is rather symmetric.
16 323 In part (c), we want to find the percent of observations expected to be between the two quartiles (0.76 and 0.86) if the distribution is Normal. We ll use CDF.Normal to find the proportion by subtracting the proportion less than 0.76 from the proportion less than About 47.5% of all observations between 0.76 and To find what actual proportion lies between these values, sort the list using Data, Sort Cases. Enter the variable name Length in both the Sort by box. Click OK. Examining the worksheet after the sort, we find there are 11 values less than 0.76 and 12 values greater than 0.86; that means (49 23)/49 = 53.1% of the values are between the quartiles Open worksheet file ta We want stemplots of the data for both loose and intermediate compression. Use Analyze, Descriptive Statistics, Explore and enter Pent as the Dependent variable and Comp as the Factor. Pent StemandLeaf Plot for Comp= I Pent StemandLeaf Plot for Comp= L Frequency Stem & Leaf Frequency Stem & Leaf Extremes (>=4.3) Stem width: 1.00 Each leaf: 1 case(s) Extremes (>=4.89) Stem width:.10 Each leaf: 1 case(s)
17 324 We see below that both of these distributions are not Normal; they are skewed right with high outliers (indicated as Extremes) We ll find the zscores corresponding to the quartiles using Transform, Compute Variable, and ask for the IDF.Normal. We specify area to the left (0.25) of Q 1, the mean (0) and standard deviation (1). Since the Normal distribution is symmetric, we ll find only Q 1. (Q 3 will have the same value, but a positive number).
Using SPSS for Descriptive Statistics
Using SPSS for Descriptive Statistics This tutorial will show you how to use SPSS version 12.0 to perform exploratory data analysis and descriptive statistics. You will use SPSS to create histograms, frequency
More informationNormal distributions in SPSS
Normal distributions in SPSS Bro. David E. Brown, BYU Idaho Department of Mathematics February 2, 2012 1 Calculating probabilities and percents from measurements: The CDF.NORMAL command 1. Go to the Variable
More informationThe Normal Distributions
CHAPTER 3 3.1 3.2 3.3 The Normal Distributions Density Curves and Normal Distributions Common Errors Selected Exercise Solutions Introduction In this chapter, we use the TI calculators for calculations
More informationChapter 13. Binomial Distributions
Chapter 13. Binomial Distributions Topics Covered in this chapter: Binomial Probabilities Binomial Probabilities Example (Apply Your Knowledge 13.5): Proofreading The Problem: Typing errors in a text are
More information1.3 Density Curves and Normal Distributions
Looking at Data Distributions 1.3 Density Curves and Normal Distributions 2012 W.H. Freeman and Company Density curves A density curve is a mathematical model of a distribution. The total area under the
More information3.1. Sketches will vary. Use them to confirm that students understand the meaning of (a) symmetric and (b) skewed to the left.
Chapter Solutions.1. Sketches will vary. Use them to confirm that students understand the meaning of (a) symmetric and (b) skewed to the left..2. (a) It is on or above the horizontal axis everywhere, and
More informationChapter 14. Introduction to Inference
Chapter 14. Introduction to Inference Topics covered in this chapter: Estimating µ with a 95% Confidence Interval Estimating µ with a 99% Confidence Interval Significance Test with a Onesided Pvalue
More informationAnnotated Sample SPSS Output Descriptive Statistics
Annotated Sample SPSS Output Descriptive Statistics Source: http://www.ats.ucla.edu/stat/spss/output/descriptives.htm Case processing summary a. Valid  This refers to the nonmissing cases. In this column,
More informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 68 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationSPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav)
Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer
More informationUsing SPSS, Chapter 2: Descriptive Statistics
1 Using SPSS, Chapter 2: Descriptive Statistics Chapters 2.1 & 2.2 Descriptive Statistics 2 Mean, Standard Deviation, Variance, Range, Minimum, Maximum 2 Mean, Median, Mode, Standard Deviation, Variance,
More informationObtaining Summary Statistics with SPSS. Math 260
Obtaining Summary Statistics with SPSS Math 260 Open the New York Travel Times data from Exercise 2.2 File eg0203.sav. Your data should have n=20 rows Explore Procedure Select Analyze Descriptive Statistics
More informationMath 263 Section 005: Class 2 : Normal Distribution and zscores Deborah Hughes Hallett
Math 263 Section 005: Class 2 : Normal Distribution and zscores Deborah Hughes Hallett dhh@math.arizona.edu Course Policies and Information at http://math.arizona.edu/~dhh/26314.html. You need a WebAssign
More informationChapter 5: The standard deviation as a ruler and the normal model p131
Chapter 5: The standard deviation as a ruler and the normal model p131 Which is the better exam score? 67 on an exam with mean 50 and SD 10 62 on an exam with mean 40 and SD 12? Is it fair to say: 67 is
More informationContinuous Random Variables and the Normal Distribution
76 Chapter 6 Continuous Random Variables and the Normal Distribution Continuous Random Variables and the Normal Distribution Chapter 6 Section 6.5 Example 612, pg. 271 Finding Area under a Normal Curve
More information3In this chapter we cover...
CHAPTER 3In this chapter we cover... Density curves Describing density curves Normal distributions The 68 95 99.7 rule The standard Normal distribution Finding Normal proportions Using the standard Normal
More informationChapter 17. Inference About a Population Mean
Chapter 17. Inference About a Population Mean Topics covered in this chapter: Finding tcritical Values Confidence Interval One Sample ttest Matched Pairs t Procedure Finding tcritical Values Example
More informationVisual Display of Data in Stata
Lab 2 Visual Display of Data in Stata In this lab we will try to understand data not only through numerical summaries, but also through graphical summaries. The data set consists of a number of variables
More informationSTAT 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
More informationMinitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab StepByStep
Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different
More informationChapter 3. The Normal Distributions
Chapter 3. The Normal Distributions 1 Chapter 3. The Normal Distributions Density Curves Definition. A (probability) density curve is a curve that is always on or above the horizontal axis, and has area
More informationLearning goals for this chapter:
Chapter 1: Looking at DataDistributions Section 1.1: Introduction, Displaying Distributions with Graphs Section 1.2: Describing Distributions with Numbers Learning goals for this chapter: Identify categorical
More informationOrganization and Description of Data
10 Organization and Description of Data Organization and Description of Data Chapter 2 Section 2.3 Example 1, pg. 2526: Pie Charts A sample of 280 undergraduates was selected, and these students were
More informationGraphing Data and Descriptive Statistics
Chapter 2 Graphing Data and Descriptive Statistics 1 Graphing Data and Descriptive Statistics Chapter 2 Pie Charts A sample of 40 students at a university was randomly selected, and each was asked if they
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will get observations
More informationFirst 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
More informationTopics in SPSS 12 INDEX 1. Single Value Frequency Table Categorical Frequency Table Grouped ( Class Limits ) Table
Topics in SPSS 12 INDEX 1 1. OVERVIEW 2 2. DATA EDITOR 3 3. ORGANIZING DATA 4 Single Value Frequency Table Categorical Frequency Table Grouped ( Class Limits ) Table 4. DESCRIPTIVE MEASURES 7 Measures
More informationChapter 2: Exploring Data with Graphs and Numerical Summaries. Graphical Measures Graphs are used to describe the shape of a data set.
Page 1 of 16 Chapter 2: Exploring Data with Graphs and Numerical Summaries Graphical Measures Graphs are used to describe the shape of a data set. Section 1: Types of Variables In general, variable can
More informationMinitab 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 informationChapter 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)
More information3.1 Measures of Center
3.1 Measures of Center A value at the center or middle of a data set The (arithmetic) mean, often referred to as the average, is the sum of all values divided by the total number of values. The mean calculated
More information4. Descriptive Statistics: Measures of Variability and Central Tendency
4. Descriptive Statistics: Measures of Variability and Central Tendency Objectives Calculate descriptive for continuous and categorical data Edit output tables Although measures of central tendency and
More informationChapter 2. Objectives. Tabulate Qualitative Data. Frequency Table. Descriptive Statistics: Organizing, Displaying and Summarizing Data.
Objectives Chapter Descriptive Statistics: Organizing, Displaying and Summarizing Data Student should be able to Organize data Tabulate data into frequency/relative frequency tables Display data graphically
More informationChapter 2 Displaying and Summarizing Quantitative Data
Chapter 2 Displaying and Summarizing Quantitative Data The three primary rules of data analysis are: 1. Make a picture. 2. Make a picture. 3. Make a picture. TIcalculators can help with this, although
More informationMath 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 informationChapter 4 The Standard Deviation as a Ruler and the Normal Model
Chapter 4 The Standard Deviation as a Ruler and the Normal Model The standard deviation is the most common measure of variation; it plays a crucial role in how we look at data. Z scores measure standard
More informationBIOL Assessing normality
Assessing normality Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this
More informationChapter 6: RANDOM SAMPLING AND DATA DESCRIPTION. Part 1: Random Sampling Numerical Summaries StemnLeaf plots Histograms, and Box plots
Chapter 6: RANDOM SAMPLING AND DATA DESCRIPTION Part 1: Random Sampling Numerical Summaries StemnLeaf plots Histograms, and Box plots Sections 61 to 64 Random Sampling In statistics, we re usually
More informationProperties of the Normal Distributions. Normal Distributions. Normal Distributions. The Normal Distribution
The Normal Distribution Properties of the Normal Distribution Shapes of Normal Distributions Standard (Z) Scores The Standard Normal Distribution Calculate areas within the normal distribution Chapter
More informationSource: D. Morton, et al., Lead Absorption in Children of Employees in a LeadRelated Industry, American Journal of Epidemiology 155 (1982).
STAT E50  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 informationPSY 216: Elementary Statistics Exam 1
Name: PSY 216: Elementary Statistics Exam 1 This exam consists of 25 multiplechoice questions and 5 essay / problem questions. For each multiplechoice question, circle the one letter that corresponds
More informationUnit 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 bellshaped and completely characterized
More informationSection 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*, 1.146148. 1.109 Sketch some normal curves. (a) Sketch
More informationMINITAB COMMANDS TABLE OF CONTENTS:
MINITAB COMMANDS Minitab is one of many statistical software packages. This handout contains the commands that you will need throughout this course, and perhaps beyond, and it is meant as a reference for
More informationAnalyzing Data Finding centers of data set, describing variation of data set, and a shape of data set.
Chapter : Describing Distributions with umbers Descriptive Statistics Describe the important characteristics of a set of measurements. Analyzing Data Finding centers of data set, describing variation of
More informationProgram L5. Confidence intervals
Program L5 Confidence intervals Confidence interval around the mean, cont d Confidence intervals for small samples Find out if a variable is Normally distributed 1 Confidence interval Golf example A study
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More informationDescriptive Statistics. Frequency Distributions and Their Graphs 2.1. Frequency Distributions. Chapter 2
Chapter Descriptive Statistics.1 Frequency Distributions and Their Graphs Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number
More informationMeasuring 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 689599.7 (Empirical) rule Standardizing observations Calculating
More informationThe Standard Normal Distribution
Chapter 6 The Standard Normal Distribution Goal: To become familiar with how to use Excel 2007/2010 for the Normal Distribution. Instructions: There are four Stat Tools in Excel that deal with the Normal
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationPSY 307 Statistics for the Behavioral Sciences. Chapter 35 Mean, Variance, Standard Deviation and Zscores
PSY 307 Statistics for the Behavioral Sciences Chapter 35 Mean, Variance, Standard Deviation and Zscores Measures of Central Tendency (Representative Values) Quantitative data: Mode the most frequently
More informationChapter 3: Statistics for describing, exploring, and comparing data
Chapter 3: Statistics for describing, exploring, and comparing data Chapter Problem: A common belief is that women talk more than men. Is that belief founded in fact, or is it a myth? Data set 8 in Appendix
More informationProject Help
It is your responsibility to figure out the basis usage of Word, Excel and try out Minitab 12 tutorials (from Minitab website as explained below) and Minitab help (on the menu in Minitab). Please note
More informationSection 62 The Standard Normal Distribution
Section 62 The Standard Normal Distribution 6.11 Continuous Random Variables Continuous random variable A random variable X takes infinitely many values, and those values can be associated with measurements
More informationchapter 21 Descriptive Statistics Introduction
chapter 1 Descriptive Statistics Introduction Any text discussing measurement and evaluation will have chapters like this with Statistics in the title. The word statistics strikes fear in the hearts of
More informationChapter 6 The Standard Normal Distribution
The Standard Normal Distribution TOPIC SLIDE What is the Standard Normal Distribution? 2 What do zscores tell us? 3 The Empirical Rule 10 Steps for finding the area under the Normal Curve 13 Tutorial:
More informationExercise 1.12 (Pg. 2223)
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.
More informationDisplaying Quantitative Data
Displaying Quantitative Data Numerical data can be visualized with a histogram. Data are separated into equal intervals along a numerical scale, then the frequency of data in each of the intervals is tallied.
More informationLOOKING AT DATA DISTRIBUTIONS
CHAPTER 1 LOOKING AT DATA DISTRIBUTIONS SECTION 1.1 OVERVIEW Section 1.1 introduces several methods for exploring data. These methods should only be applied after clearly understanding the background of
More informationStatistic Lab 1. 0.a. Type data directly into SPSS (make sure you tell SPSS the level of measurement: nominal, ordinal, interval, ratio)
Statistics Lab 1 0. Data Acquisition 0.a. Type data directly into SPSS (make sure you tell SPSS the level of measurement: nominal, ordinal, interval, ratio) 0.b. Type data into Excel Copy and Paste into
More informationSection 4.4 ZScores and the Empirical Rule
Section 4.4 ZScores and the Empirical Rule 1 GPA Example A sample of GPAs of 40 freshman college students appear below (sorted in increasing order) 1.40 1.90 1.90 2.00 2.10 2.10 2.20 2.30 2.30 2.40 2.50
More informationChapter 6. The Standard Deviation as a Ruler and the Normal Model. Copyright 2012, 2008, 2005 Pearson Education, Inc.
Chapter 6 The Standard Deviation as a Ruler and the Normal Model Copyright 2012, 2008, 2005 Pearson Education, Inc. The Standard Deviation as a Ruler The trick in comparing very differentlooking values
More informationThe data analysis tools are located on the Data Tab in the Analysis Group on the Excel Ribbon, as shown below. Data Tab Data Analysis Tools
Getting to Know Your Data (Activity) Maryann Allen There are two components to this activity: Technical Component using Descriptive Statistics and Histogram Tools in Excel Quantitative Literacy Component
More informationUnivariate Statistics
Univariate Statistics Univariate analysis, looking at single variables, is typically the first procedure one does when examining data being used for the first time. There are a number of reasons why it
More informationGeoGebra Statistics and Probability
GeoGebra Statistics and Probability Project Maths Development Team 2013 www.projectmaths.ie Page 1 of 24 Index Activity Topic Page 1 Introduction GeoGebra Statistics 3 2 To calculate the Sum, Mean, Count,
More informationStats for Business MINITAB Intro Worksheet (Complete Week 4 With TA in Computer Lab Discussion)
Stats for Business MINITAB Intro Worksheet (Complete Week 4 With TA in Computer Lab Discussion) DIRECTIONS: Spend 2025 minutes working this MINITAB Intro with your TA and classmates. (Finish later if
More information13.2 Measures of Central Tendency
13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers
More informationCHAPTER 6 NORMAL DISTIBUTIONS
CHAPTER 6 NORMAL DISTIBUTIONS GRAPHS OF NORMAL DISTRIBUTIONS (SECTION 6.1 OF UNDERSTANDABLE STATISTICS) The normal distribution is a continuous probability distribution determined by the value of µ and
More informationNumerical Measures of Central Tendency
Numerical Measures of Central Tendency Often, it is useful to have special numbers which summarize characteristics of a data set These numbers are called descriptive statistics or summary statistics. A
More informationDescribing, Exploring, and Comparing Data
24 Chapter 2. Describing, Exploring, and Comparing Data Chapter 2. Describing, Exploring, and Comparing Data There are many tools used in Statistics to visualize, summarize, and describe data. This chapter
More informationSelfPaced Calculator Practice for Statistics
SelfPaced Calculator Practice for Statistics Clear and Reset Your Calculator Clear your calculator before you begin working with it and at the end of your assignments. (Especially if you are borrowing
More informationLab #5. Normal Distribution, Central Limit Theorem
36220 Lab #5 Normal Distribution, Central Limit Theorem Please write your name below, tear off this front page and give it to a teaching assistant as you leave the lab. It will be a record of your participation
More informationSection 1.4 Range, IQR and Finding Outliers
Section 1.4 Range, IQR and Finding Outliers In earlier sections we discussed measures of center (mean and median) and measures of spread (variance and standard deviation). In this section, we will introduce
More informationIBM SPSS Statistics for Beginners for Windows
ISS, NEWCASTLE UNIVERSITY IBM SPSS Statistics for Beginners for Windows A Training Manual for Beginners Dr. S. T. Kometa A Training Manual for Beginners Contents 1 Aims and Objectives... 3 1.1 Learning
More informationStatistics: 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 informationIntroduction to the Practice of Statistics Sixth Edition Moore, McCabe
Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 1.3 Homework Answers 1.99 Test scores. Many states have programs for assessing the skills of students in various grades. The
More informationUSING THE TINspire CAS AND TINspire HANDHELDS
Using the Function Table Feature A function can be displayed in a table of values. 1. Enter the function into the data entry line of the Graphs and Geometry application. For example, to enter the function
More informationTest 2A AP Statistics Name:
Test 2A AP Statistics Name: Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. The heights of American men aged 18 to 24 are approximately Normally distributed with a mean
More informationChapter 23. Inferences for Regression
Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily
More informationDensity Curves and Normal Distributions
Density Curves and Normal Distributions Recall: For data on a quantitative variable, the histogram gives a graphical picture of the distribution. Histogram will show us approximate shape, center, spread,
More informationGETTING STARTED WITH MINITAB INTRODUCTION TO MINITAB STATISTICAL SOFTWARE
Six Sigma Quality Concepts & Cases Volume I STATISTICAL TOOLS IN SIX SIGMA DMAIC PROCESS WITH MINITAB APPLICATIONS CHAPTER 2 GETTING STARTED WITH MINITAB INTRODUCTION TO MINITAB STATISTICAL SOFTWARE 201012
More informationFinal Exam Review. Topics
Final Exam Review The final exam is comprehensive, but focuses more on the material from chapters 58. The final will take place during the last class session Thursday 7/18. That is the only thing we will
More informationChapter 2. The Normal Distribution
Chapter 2 The Normal Distribution Lesson 21 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 informationName: School: University of Houston High School Contest Spring 2006 Statistics Test
University of Houston High School Contest Spring 2006 Statistics Test Part I Multiple Choice. Each problem is worth 5 points. 1. Fifty cars were selected to test the effect of a gasoline additive on exhaust
More informationUsing the normal tables from A1, look up a standard score of The corresponding area from to is percent.
Review Exercises Normal Approximation to Data Chapter 5, FPP, p. 9396 Dr. McGahagan Problem 1. Test scores and the normal approximation. Given: Mean = 50, SD = 10 A 1.25 SD interval around the mean =
More informationThe 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 information7. The following histogram pictures the number of students who visited the Career Center each week during the school year.
Chapter 2 Review MULTIPLE CHOICE.. The following list is ordered from smallest to largest: 25, 26, 25, 30, y, y, y, 33, 50. Which of the following statements is (are) true? I. The mean is greater than
More informationExploratory data analysis (Chapter 2) Fall 2011
Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,
More information6. The amount of spread in the data is a measure of what characteristic of a data set? a. Center b. Variability c. Shape
CHAPTER 7 1. Name the four kinds of useful information that you can get about a set of measurement data once it has been organized and summarized. ANSWER: 1) THE CENTER; 2) THE VARIABILITY; 3) THE SHAPE;
More informationIntroduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data
A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine jutts@uci.edu 1. Statistical Methods in Water Resources by D.R. Helsel
More informationInstructions for carrying out statistical procedures and tests using SPSS
Instructions for carrying out statistical procedures and tests using SPSS These instructions are closely linked to the author s book: Essential Statistics for the Pharmaceutical Sciences John Wiley & Sons
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationThe Normal Distribution & Descriptive Statistics. Kin 304W Week 2: May 14, 2013
The Normal Distribution & Descriptive Statistics Kin 304W Week 2: May 14, 2013 1 Writer s Corner Grammar Girl Quick and Dirty Tips For Better Writing http://grammar.quickanddirtytips.com/ 4 Writer s Corner
More informationLecture Numerical Measures  1  http://wiki.stat.ucla.edu/socr/index.php/socr_courses_008_thomson_econ61 DESCRIPTIVE STATISTICS PART II DESCRIBING YOUR DATA USING NUMERICAL MEASURES Grace S. Thomson Lecture
More informationWhen the conditions are met, the standardized sample difference between the means of two independent groups, t= SE(y  y )
STAT E50  Introduction to Statistics Comparing Means; Paired Samples The Sampling Distribution for the Difference between Two Means When the conditions are met, the standardized sample difference between
More informationProbability & Statistics Chapter 2b Pretest
Probability & Statistics Name Chapter 2b Pretest Date For the given data, construct a frequency distribution and frequency histogram of the data using five classes. Describe the shape of the histogram
More informationThe Normal Distributions
P1: FBQ PB286A03 PB286MooreV5.cls April 16, 2003 21:44 CHAPTER (Mike Powell/Allsport Concepts/Getty Images) 3 In this chapter we cover... Density curves The median and mean of a density curve Normal
More informationCalculating the Standard Deviation
Calculating the Standard Deviation If x is a number, then the difference "x  mean" is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations
More informationCh. 56 Review Exercises
Ch. 56 Review Exercises 1. Prenatal care. Results of a 1996 American Medical Association report about the infant mortality rate for twins carried for the full term of a normal pregnancy are shown below,
More information