WELCOME TO ELEMENTARY STATISTICS

Statistics is a ‘do’ field. In order to learn, it you must ‘do’ it.

I can recite the rules, I can explain with examples, but whether you learn the material or not is up to you.

We depend on the TI-83/84 to eliminate the drudgery of calculations.

This is a collaborative class Hints for success in this class

Work on class topics every day Form a study group Don’t get discouraged As you solve problems, ask yourself

“Does this answer make sense?’ Get help as soon as you need it

From me Tutorial Center

Don’t get behind

1

WHAT’S THE POINT?

Surrounded by examplescrime, sports, politics

Interpret data to make decisions

Analyze informationSurvey results and your critical

eye Do samples represent population Is sample big enough? How was sample chosen? What

‘type’ of people/things selected? Are survey questions loaded? Are graphs properly displayed,

data complete, context stated? Was there anything ‘confounding’

the results?

2

MATH 10 - ELEMENTARY

STATISTICSText: Collaborative Statistics by Susan Dean and Barbara

IllowskyAvailable online as a free download.

3

CHAPTER 1

Sampling and Data

4

CHAPTER 1 OBJECTIVES

The Student will be able to Define, in context, key

statistical terms. Define, in context, and

identify different sampling techniques.

Understand the variability of data.

Create and interpret Relative Frequency Tables.

5

SOME DEFINITIONS Statistics

collection, analysis, interpretation and presentation of data descriptive statistics inferential statistics

Probabilitymathematical tool used to

study randomness theoretical empirical

6

KEY TERMS

Populationentire collection of persons,

things or objects under study Sample

a portion of the larger population

Parameternumber that is a property of the

population Average, standard deviation,

proportion (µ, σ, p)

Statisticnumber that is a property of the

sample Average, standard deviation,

proportion (x-bar, s, p’)

7

KEY TERMS Variable

the characteristic of interest for each person or thing in a population numerical categorical

Data - data type example the actual values of the variable

qualitative quantitative

discretecontinuous

8

An ‘in context’ example

SAMPLING Taking a portion of the total

population Need for random sample

Represent the population (has the same characteristics as population)

each element of the population should have an equal chance of being chosen

9

Population

Sample

SAMPLING METHODS

Simple random samplingeach member of a population

initially has an equal chance of being selected for the sample Random number generator

With replacementWithout replacement

Stratified sampledivide population into groups

and then take a sample from each group

Cluster sampledivide population into groups

and then randomly select some of the groups and sample all members of those groups

10

SAMPLING METHODS

Systematic sampleselect a starting point and take

every nth piece of data from a listing of the population

Convenience sampleusing results that are readily

available – just happen to be there Why a problem?

11

DETERMINE THE TYPE OF SAMPLING USED (SIMPLE RANDOM, STRATIFIED, SYSTEMATIC, CLUSTER, OR CONVENIENCE).

A soccer coach selects 6 players from a group of boys aged 8 to 10, 7 players from a group of boys aged 11 to 12, and 3 players from a group of boys aged 13 to 14 to form a recreational soccer team.

A pollster interviews all human resource personnel in five different high tech companies.

An engineering researcher interviews 50 women engineers and 50 men engineers.

A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.

A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.

A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.

12

EXAMPLE 1.6.1 – DETERMINE WHETHER OR NOT THE FOLLOWING SAMPLES ARE REPRESENTATIVE. IF THEY ARE NOT, DISCUSS THE REASONS.

1. To find the average GPA of all students in a university, use all honor students at the university as the sample. 2. To find out the most popular cereal among young people under the age of 10, stand outside a large supermarket for three hours and speak to every 20th child under the age of 10 who enters the supermarket. 3. To find the average annual income of all adults in the U.S., sample U.S. congresspersons. Create a cluster sample by considering each state as a stratum (group). By using a simple random sampling, select states to be part of the cluster. Then survey every U.S. congressperson in the cluster. 4. To determine the proportion of people taking public transportation to work, survey 20 people in NYC. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you. 5. To determine the average cost of a two day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.

13

VARIATION In data (within the sample) In samples (between samples)

The larger the sample the better it represents the population – Law of Large numbers – and sample statistics get closer to population parameters

14

CRITICAL EVALUATION

Problems with samples Self-selected samples Sample size issues Undue influence Non-response or refusal of

subject to participate Causality Self-funded or Self-interest

studies Misleading Use of Data Confounding

15

FREQUENCY TABLE Data value Frequency

how many times the data value occurs

Relative Frequency frequency/(total number of data

values) Cumulative Relative

Frequencysummation of previous relative

frequencies

An example – How many siblings do you have?

16

AND, OH, BTW

A word on fractions You DO NOT have to reduce

fractions in this course. In fact, I INSIST that you don’t.

If you convert to decimal, take answer to 4 decimal places.

A word on rounding answers Don’t round until the final answer In general, the final answer

should have one more decimal place than the data used to get the answer HOWEVER, the rule of thumb for this

course will be probabilities (relative frequencies) to 4 decimal places, everything else to 2, unless you are told otherwise.

17

CHAPTER 2Descriptive Statistics: Displaying and

Measuring Data

18

CHAPTER 2 OBJECTIVES

The Student will be able to Display data graphically and

interpret graphs: stemplots, histograms and boxplots.

Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.

Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.

Recognize, describe, and calculate the measures of the spread of data: variance, standard deviation, and range.

19

MEASURES OF THE “CENTER” OF THE

DATA Mean or average

Use calculator

Median - the middle data value50% of data below, 50% aboveData MUST be ordered from

lowest to highestUse calculator

Mode - the most frequent data valueHave to count (or put in a

frequency table)

20

nx

MEASURES OF LOCATION OF DATA Relative to other data values

Quartiles Splits data into 4 equal groups

that contain the same percentage of data

Data must be put in numerical order

Use calculator Percentiles

Splits data into 100 equal groups Data must be put in numerical

order Relative to the mean

x = x-bar + zs z < 0, data value is below the

mean z > 0, data value is above the

mean IQR – interquartile range

IQR = Q3 – Q1 Middle 50% of data Determine potential outliers

Data value < Q1 – 1.5(IQR) Data value > Q3 + 1.5(IQR)

21

MEASURES OF THE “SPREAD” OF DATA

RangeDifference between high value

and low value Standard deviation

‘distance’ from the meanSample versus population

VarianceSample s2

Population 2

22

Nx

2)( 1)( 2

nxx

s

Using calcuator

PICTURES WORTH A 1000 WORDS

‘Charts’Stem and Leaf Graphs –

exampleLine Graphs – not usingBar Graphs – not using

Boxplots – need min, median, first and

second quartile, max Histograms –

sort data into bars or intervals5 to 15 barshorizontal axes is what the data

representsvertical axes labeled

“frequency” or “relative frequency”

23

CHAPTER 3

Probability TopicsChapter Objectives

24

CHAPTER 3 OBJECTIVES

The student will be able to Understand and use the

terminology of probability. Calculate probabilities by

listing event sample spaces and counting.

Determine whether two events are mutually exclusive or independent.

Calculate probabilities using the Addition Rules and Multiplication Rules.

Construct and interpret Contingency Tables.

Construct and interpret Tree Diagrams.

25

ACTIVITY # of students in class ____

# with change in pocket or purse ____

# who have a sister ____ # who have change and a sister

____

P(change) = ____ P(sister) = ____ P(change and sister) = ____ P(change|sister) = ____

26

DEFINITIONS Experiment - planned operations

carried out under controlled conditions

Chance experiment - results not predetermined

Outcome - result of an experiment Sample space - set of all possible

outcomes Event - any combination of outcomes Probability - long-term relative

frequency of an outcome, I.E. it is a fraction - a number between 0 and 1, inclusive

OR - as in A OR B - outcome is in A or is in B

AND - outcome is in both A and B at the same time

Complement - denoted A’ (read “A prime”) - all outcomes that are not in A

27

MORE DEFINITIONS Conditional Probability of A

given B - probability of A is calculated knowing B has already occurred P(A|B) = P(A and B) ÷ P(B)

Independent events - the chance of event A occurring does not affect the chance of event B occurring and vice versa must prove one of the following

P(A|B) = P(A) P(B|A) = P(B) P(A and B) = P(A)P(B)

Mutually Exclusive - event A and event B cannot occur at the same time, they don’t share outcomes P(A and B) = 0

28

THE TALE OF TWO DIE

ExperimentToss two die, record value

showing on each die Sample space (S)

{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

29

THE TALE OF TWO DIE

Let A = the event the sum of the faces of the die is odd A = {(1,2), (1,4), (1,6), (2,1), (2,3),

(2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5)}

Let B = event of getting a double B = {(1,1), (2,2), (3,3), (4,4), (5,5),

(6,6)}

Let D = event that at least one face is a 2 D = {(1,2), (2,1), (2,2), (2,3), (2,4),

(2,5), (2,6), (3,2), (4,2), (5,2), (6,2)}

30

THE TALE OF TWO DIE

P(A) = ___ P(B) = ___ P(D) = ___

P(D and A) = ____

P(A and B) = ____

P(A or D) = ____

P(D|A) = ____

P(A|D) = ____

31

WHAT IF THE SAMPLE SPACE IS

NOT LISTED?Need formulas:

Addition Rule: P(A OR B) = P(A) + P(B) – P(A AND B)

Multiplication Rule: P(A AND B) = P(B)*P(A|B)

P(A AND B) = P(A)*P(B|A)

Example: P(C) = 0.4, P(D) = 0.5, P(C|D) = 0.6

P(C and D) = _____

Are C and D mutually exclusive?

Are C and D independent?

P(C or D) =

P(D|C) =

32

CONTINGENCY TABLES

A table that displays sample values in relation to two different variables that may be contingent on one another.

Example - Performance on Job vs. performance in training

33

Performance on JobBelowAverage

Average AboveAverage

TOTAL

Poor 23 60 29 112Average 28 79 60 167Very Good 9 49 63 121TOTAL 60 188 152 400

TREE DIAGRAM A “graph” used to determine

outcomes of an experiment Consists of “branches” that

are labeled with either frequencies or probabilities

Once probability (frequency) entered on branches, probability (frequency) can be “read” by multiplying down branches and/or adding across branches

34

TREE DIAGRAM Experiment - cup with 8 black

and 3 yellow beads. Draw 2 beads , one at a time, with replacement. Record bead color.

35

REVIEW FOR EXAM 1

What’s fair gameChapter 1Chapter 2Chapter 3

21 multiple choice questionsThe last 3 quarters exams

What to bring with youScantron (#2052), pencil,

eraser, calculator, 1 sheet of notes (8.5x11 inches, both sides)

36