Post on 12-Jan-2016
Introduction to thePractice of Statistics
Fourth Edition
Chapter 1:Exploring Data
RESOURCES/SUPPLIES:
Our textbook is The Practice of Statistics (Starnes, Yates, Moore, 4th ed.). Pay careful attention to the examples, the calculator procedures, and the AP Exam tips that are located in the page margins.
• I will refer to this book as TPS4e throughout the year. This textbook is well-aligned to the AP Statistics curriculum and the sample problems and activities will prepare you well for the AP Statistics exam.
• Companion Web Site: www.whfreeman.com/tps4e
RESOURCES/SUPPLIES:
• You will need a TI-84+ graphing calculator. (I have a class set to use in the classroom). I will be demonstrating problems using the TI-84 all year and tips on how to use this calculator are provided throughout the TPS4e textbook. The textbook also explains how to use the TI-89 as well as the TI Inspire.
• You will receive a packet with instructions for the TI-84 graphing calculator . Keep it on your binder since you will refer to it throughout the year.
• I recommend a large 2 ½” binder since I will provide a large number of AP Practice problems, handouts , and additional documents that you will find at my website www.hialeahhigh.org and that you may find helpful to print.
RESOURCES/SUPPLIES:
• Vocab Flash Cards
• Free Study Resources for AP Tests
• Textbook Website
• Free Response Questions
• Online Writing Lab, Quick Writing Reference
• Matching types of inference
• Khan Academy
I will be preparing you for the Advanced Placement Statistics Exam taking place on Thursday May 12, 2016 at 12 m.
This Exam is made of 2 Sections for a total of 3 hours. Section I: 40 MC, 90 minutes, 50 % of the exam score. No
penalty for guessing. Section II: 6 Free Response (FR), 90 minutes, 50 % of the
exam score. Questions 1-5 take about 13 minutes each and count for 75% of the Section II. The last question is an “Investigative Task” should take about 25 min and is worth 25% of the Section II score.
THE AP EXAM
THE AP EXAM
You CAN be successful on this exam IF you put forth the effort ALL YEAR LONG.
I will provide you with LOTS of preparation materials as well as insight from the grading of the exam.
I need you to provide the effort...
TOPIC OUTLINE:
THE TOPICS FOR THE AP STATISTICS ARE DIVIDED INTO 4 MAJOR THEMES:1. EXPLORATORY ANALYSIS( 20-30 %)2. PLANNING AND CONDUCTING A STUDY (10-15%)3. PROBABILITY ( 20-30%)4. STATISTICAL INFERENCE (30-40%)
WHAT IS STATISTICS?
The Science of Learning from Data The Collection and Analysis of Data
Experimental DesignChapter 4
Descriptive Statistics(Data Exploration)
Chapters 1, 2, 3
Inferential StatisticsChapters 8-12
ProbabilityChapter 5, 6, 7
BRANCHES OF STATISTICS:
THE PRACTICE OF STATISTICS, 4TH EDITION - FOR AP*
STARNES, YATES, MOORE
Chapter 1: Exploring DataIntroductionData Analysis: Making Sense of Data
CHAPTER 1EXPLORING DATA
Introduction: Data Analysis: Making Sense of Data
1.1 Analyzing Categorical Data
1.2 Displaying Quantitative Data with Graphs
1.3 Describing Quantitative Data with Numbers
INTRODUCTIONDATA ANALYSIS: MAKING
SENSE OF DATA
After this section, you should be able to…
DEFINE “Individuals” and “Variables”
DISTINGUISH between “Categorical” and “Quantitative” variables
DEFINE “Distribution”
DESCRIBE the idea behind “Inference”
LEARNING OBJECTIVES
DA
TA
AN
ALY
SIS
Statistics is the science of data. Data Analysis is the process of organizing,
displaying, summarizing, and asking questions about data.
Definitions:
Individuals – objects (people, animals, things) described by a set of data
Variable - any characteristic of an individual
Categorical Variable– places an individual into one of several groups or categories.
Quantitative Variable – takes numerical values for which it makes sense to find an average.
DA
TA
AN
ALY
SIS
A variable generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value.
Definition:
Distribution – tells us what values a variable takes and how often it takes those values
2009 Fuel Economy Guide
MODEL MPG
1
2
3
4
5
6
7
8
9
Acura RL 22
Audi A6 Quattro 23
Bentley Arnage 14
BMW 5281 28
Buick Lacrosse 28
Cadillac CTS 25
Chevrolet Malibu 33
Chrysler Sebring 30
Dodge Avenger 30
2009 Fuel Economy Guide
MODEL MPG <new>
9
10
11
12
13
14
15
16
17
Dodge Avenger 30
Hyundai Elantra 33
Jaguar XF 25
Kia Optima 32
Lexus GS 350 26
Lincolon MKZ 28
Mazda 6 29
Mercedes-Benz E350 24
Mercury Milan 29
2009 Fuel Economy Guide
MODEL MPG <new>
16
17
18
19
20
21
22
23
24
Mercedes-Benz E350 24
Mercury Milan 29
Mitsubishi Galant 27
Nissan Maxima 26
Rolls Royce Phantom 18
Saturn Aura 33
Toyota Camry 31
Volkswagen Passat 29
Volvo S80 25
MPG14 16 18 20 22 24 26 28 30 32 34
2009 Fuel Economy Guide Dot Plot
Variable of Interest:MPG
Variable of Interest:MPG
Dotplot of MPG Distribution
Dotplot of MPG Distribution
ExampleExample
MPG14 16 18 20 22 24 26 28 30 32 34
2009 Fuel Economy Guide Dot Plot
2009 Fuel Economy Guide
MODEL MPG <new>
9
10
11
12
13
14
15
16
17
Dodge Avenger 30
Hyundai Elantra 33
Jaguar XF 25
Kia Optima 32
Lexus GS 350 26
Lincolon MKZ 28
Mazda 6 29
Mercedes-Benz E350 24
Mercury Milan 29
2009 Fuel Economy Guide
MODEL MPG <new>
16
17
18
19
20
21
22
23
24
Mercedes-Benz E350 24
Mercury Milan 29
Mitsubishi Galant 27
Nissan Maxima 26
Rolls Royce Phantom 18
Saturn Aura 33
Toyota Camry 31
Volkswagen Passat 29
Volvo S80 25
2009 Fuel Economy Guide
MODEL MPG
1
2
3
4
5
6
7
8
9
Acura RL 22
Audi A6 Quattro 23
Bentley Arnage 14
BMW 5281 28
Buick Lacrosse 28
Cadillac CTS 25
Chevrolet Malibu 33
Chrysler Sebring 30
Dodge Avenger 30
Add numerical summaries
DA
TA
AN
ALY
SIS
Examine each variable by itself.
Then study relationships among
the variables.
Start with a graph or graphs
How to Explore DataHow to Explore Data
DA
TA
AN
ALY
SIS
• A population is the collection of all outcomes, responses, measurements, or counts that are of interest. A sample is a subset of a population
PopulationPopulation
SampleSample
Collect data from a representative Sample...
Perform Data Analysis, keeping probability in mind…
Make an Inference about the Population.
ACTIVITY: HIRING DISCRIMINATIONFollow the directions on Page 5
Perform 5 repetitions of your simulation.
Turn in your results to your teacher.
Teacher: Right-click (control-click) on the graph to edit the counts.
Data
Analy
sis
INTRODUCTIONDATA ANALYSIS: MAKING SENSE OF DATA
In this section, we learned that…
A dataset contains information on individuals.
For each individual, data give values for one or more variables.
Variables can be categorical or quantitative.
The distribution of a variable describes what values it takes and how often it takes them.
Inference is the process of making a conclusion about a population based on a sample set of data.
SUMMARY
LOOKING AHEAD…
We’ll learn how to analyze categorical data.Bar GraphsPie ChartsTwo-Way TablesConditional Distributions
We’ll also learn how to organize a statistical problem.
In the next Section…
CW # 1. PG. 7 EXC. 2, 4, 6HW # 2. PG. 7 EXC. 1,3,5,7,8
Practice:
RECALL OUR EARLIER QUESTION 1
1. What percent of the 60 randomly chosen fifth grade students have an IQ score of at least 120?
Numerically?
How to Represent
Graphically?18.3%+15%+3.3%=36.6%
(11+9+2)/60=.367 or 36.7%
Grey Shaded Region corresponds to this 36.6% of data
What is Different Fromthe Histogram we Generated
In Class??
Let’s Look at the Distribution we Just Created:•Overall Pattern:
Shape (modes, tails (skewness), symmetry) Center (mean, median)Spread (range, IQR, standard deviation)
•Deviations:Outliers
Descriptors we will be interested
in for data and population
distributions.
•Overall Pattern:Shape, Center, Spread?
•Deviations:Outliers?
Example 1.9 page 18-19
Data Analysis – An Interesting Example (Example 1.10, p. 9-10)
80 Calls
•Overall Pattern:Shape, Center, Spread?
•Deviations:Outliers?
Time Plots – For Data Collected Over Time…
Example: Mississippi River Discharge p.19 (data p. 21)
Example – Dealing with Seasonal Variation
EXTRA SLIDES FROM HOMEWORK
Problem 1.19
Problem 1.20
Problem 1.21
Problem 1.31
Problem 1.36
Problem 1.37-1.38
Problem 1.19, page 30
Problem 1.20, page 31
Problem 1.21, page 31
Problem 1.31, page 36
Problem 1.36, page 38
Problems 1.37 – 1.39
Section 1.2Describing
Distributions with Numbers
TYPES OF MEASURES
Measures of Center:Mean, Median, Mode
Measures of Spread:Range (Max-Min), Standard Deviation, Quartiles, IQR
MEANS AND MEDIANS
Consider the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
What is the Average (or Mean) Test Score?
What is the Median Test Score?
Consider the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Draw a Stem and Leaf Plot (Shape, Center, Spread?)Find the Mean and the MedianLet’s Use our TI-83 Calculators! Enter data into a list via Stat|EditStat|Calc|1-Var StatsWhat happens to the Mean and Median if the lowest
score was 20 instead of 65?What happens to the Mean and Median if a low score
of 20 is added to the data set (so we would now have 11 data points?)
What can we say about the Mean versus the Median?
Quartiles: Measures of Position
A Graphical Representation of Position of Data(It really gives us an indication of how the data is spread
among its values!)
Using Measures of Position to Get Measures of Spread
And what was the range again???
5 NUMBER SUMMARY, IQR, BOX PLOT, AND WHERE OUTLIERS WOULD BE FOR TEST SCORE DATA:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
What do we notice about symmetry?
HISTOGRAMS OF FLOWER LENGTHSPROBLEM 1.58GENERATED VIA MINITAB
length
Perc
ent
514845423936
48
36
24
12
0
514845423936
48
36
24
12
0
bihai red
yellow
Panel variable: variety
Histogram of Flower Length
Box Plots for Flower Lengths
30
35
40
45
50
55
Bihai Red Yellow
Flower Color
Len
gth
s (i
n m
m)
Bihai Red Yellow
Median 47.12 39.16 36.11
Q1 46.71 38.07 35.45
Min or In Fence 46.34 37.4 34.57
Max or In Fence 50.26 43.09 38.13
Q348.24
5 41.69 36.82
BOX PLOT AND 5-NUMBER SUMMARY FOR FLOWER LENGTH DATAGENERATED VIA BOX PLOT MACRO FOR EXCEL
Outliers?
Remember this histogram from the Service Call Length Data on page 9? How do you expect the Mean and Median to compare for this data?
Mean 196.6, Median 103.5
Box Plot for Call Length Data
MORE ON MEASURES OF SPREAD
Data Range (Max – Min)IQR (75% Quartile minus 25% Quartile
2, range of middle 50% of data)Standard Deviation (Variance)Measures how the data deviates from the mean….hmm…how can we do this?
Recall the Sample Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Recall the Sample Mean (X bar) was 78.8…
COMPUTING VARIANCE AND STD. DEV. BY HAND AND VIA THE TI83:Recall the Sample Test Score Data:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Recall the Sample Mean (X bar) was 78.8
x
65 70 75 80 9085 95
65 83
78.8
-13.8 4.2
What does the number 4.2 measure? How
about -13.8?
Consider (again!) the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94What happens to the standard deviation and the location of the 1st and 3rd quartiles if the lowest score was 20 instead of 65?
What happens to the standard deviation and the location of the 1st and 3rd quartiles if a low score of 20 is added to the data set (so we would now have 11 data points?)
What can we say about the effect of outliers on the standard deviation and the quartiles of a data set?
Effects of Outliers on the Standard Deviation
Example 1.18:Stemplots of Annual Returns forStocks (a) and Treasury bills (b)On page 53 of text. What are the
stem and leaf units????
Consider (again!) the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94Xbar=78.8 s=10.2 (rounded)
Suppose we “curve” the grades by adding 5 points to every test score (i.e. Xnew=Xold+5). What will be new mean and standard deviation?
Suppose we “curve” the grades by multiplying every test score times 1.5 (i.e. Xnew=1.5*Xold). What will be the new mean and standard deviation?
Suppose we “curve” the grades by multiplying every test score times 1.5 and adding 5 points (i.e. Xnew=1.5*Xold+5). What will be the new mean and standard deviation?
Effects of Linear Transformations on the MeanAnd Standard Deviation
Box Plots for Problems 1.62-1.64
Section 1.3Density Curves and Normal Distributions
BASIC IDEAS
One way to think of a density curve is as a smooth approximation to the irregular bars of a histogram.
It is an idealization that pictures the overall pattern of the data but ignores minor irregularities.
Oftentimes we will use density curves to describe the distribution of a single quantitative continuous variable for a population (sometimes our curves will be based on a histogram generated via a sample from the population).
Heights of American WomenSAT Scores
The bell-shaped normal curve will be our focus!
Shape?Center?Spread?
Density Curve
Page 64
Sample Size =105
Shape?Center?Spread?
Density Curve
Page 65
Sample Size=72 Guinea pigs
1. What proportion (or percent) of seventh graders from Gary,Indiana scored below 6?
2. What is the probability (i.e. how likely is it?)that a randomly chosenseventh grader from Gary, Indiana will have a test score less than 6?
Two Different butRelated Questions!
Example 1.22Page 66
Sample Size = 947
Relative “area under the curve”
VERSUSRelative “proportion of
data” in histogrambars.
Page 67 of text
Shape?Center?Spread?
The classic “bell shaped” Density curve.
A “skewed” density curve.Median separates area under curve into two equal areas
(i.e. each has area ½)
What is the geometric interpretationof the mean?
The mean as “center of mass” or “balance point” of the density curve
The normal density curve!Shape? Center? Spread?
Area Under Curve?
How does the magnitude of the standard deviation affect a density curve?
How does the standard deviation affect the shape of the normal density curve?
Assume Same Scale onHorizontal and Vertical
(not drawn) Axes.
The distribution of heights of young women (X) aged 18 to 24 is approximately normal with mean mu=64.5 inches and standard deviation sigma=2.5 inches (i.e. X~N(64.5,2.5)). Lets draw the density curve for X and observe the empirical rule!
(aka the “Empirical Rule”)
Example 1.23, page 72How many standard deviations from the mean height is the height of a woman who is 68 inches? Who is 58 inches?
The Standard Normal Distribution
(mu=0 and sigma=1)
Horizontal axis in units of z-score!
Notation:Z~N(0,1)
Let’s find some proportions (probabilities) using normal distributions!
Example 1.25 (page 75)Example 1.26 (page 76)(slides follow)
Let’s draw the distributions by hand
first!
Example 1.25, page 75
TI-83 Calculator Command: Distr|normalcdfSyntax: normalcdf(left, right, mu, sigma) = area under curve from left to right
mu defaults to 0, sigma defaults to 1Infinity is 1E99 (use the EE key), Minus Infinity is -1E99
Example 1.26, page 76
Let’s find the same probabilities using z-scores!
On the TI-83: normalcdf(720,820,1026,209)
THE INVERSE PROBLEM:GIVEN A NORMAL DENSITY PROPORTION OR PROBABILITY, FIND THE CORRESPONDING Z-SCORE!
What is the z-score such that 90% of the data has a z-score less than that z-score?
(1) Draw picture!(2) Understand what you are solving for!(3) Solve approximately! (we will also use
the invNorm key on the next slide)
Now try working Example 1.30 page 79!(slide follows)
TI-83: Use Distr|invNorm
Syntax:invNorm(area,mu,sigma) gives value of x with area to left of x under normal curve with mean mu and standard deviation sigma.
invNorm(0.9,505,110)=?invNorm(0.9)=?
Page 79
How can we use our TI-83s to solve this??
How can we tell if our data is “approximately normal?”
Box plots and histograms should show essentially symmetric, unimodal data. Normal Quantile plots are also used!
Histogram and Normal Quantile Plot for Breaking Strengths (in pounds) of Semiconductor Wires
(Pages 19 and 81 of text)
Histogram and Normal Quantile Plot for Survival Time of Guinea Pigs (in days) in a Medical Experiment
(Pages 38 (data table), 65 and 82 of text)
USING EXCEL TO GENERATE PLOTS
Example Problem 1.30 page 35Generate Histogram via MegastatGet Numerical Summary of Data via Megastat or Data Analysis AddinGenerate Normal Quantile Plot via Macro (plot on next slide)
Normal Quantile plot for Problem 1.30 page 35
EXTRA SLIDES FROM HOMEWORKProblem 1.80
Problem 1.82
Problem 1.119
Problem 1.120
Problem 1.121
Problem 1.222
Problem 1.129
Problem 1.135
Problem 1.80 page 84
Problem 1.83 page 85
Problem 1.119 page 90
Problem 1.120 page 90
Problem 1.121 page 92
Problem 1.122 page 92
Problem 1.129 page 94
Problem 1.135 page 95-96