Math 10 - Elementary Statistics

Report

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
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 Surrounded

by examples
crime, sports, politics
 Interpret
data to make decisions
 Analyze information

Survey 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
Text: Collaborative Statistics by
Susan Dean and Barbara Illowsky
Available online as a free download.
3
Sampling and Data
4
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
 Statistics

collection, analysis, interpretation
and presentation of data


descriptive statistics
inferential statistics
 Probability

mathematical tool used to study
randomness


theoretical
empirical
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 Population

entire collection of persons, things
or objects under study
 Sample

a portion of the larger population
 Parameter

number that is a property of the
population

Average, standard deviation,
proportion (µ, σ, p)
 Statistic

number that is a property of the
sample

Average, standard deviation,
proportion (x-bar, s, p’)
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 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
 discrete
 continuous
An ‘in context’ example
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 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
Population
Sample
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 Simple

random sampling
each member of a population
initially has an equal chance of
being selected for the sample

Random number generator
 With replacement
 Without replacement
 Stratified

divide population into groups and
then take a sample from each
group
 Cluster

sample
sample
divide population into groups and
then randomly select some of the
groups and sample all members of
those groups
10
 Systematic

sample
select a starting point and take every nth
piece of data from a listing of the
population
 Convenience

sample
using results that are readily available –
just happen to be there

Why a problem?
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





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.
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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.
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 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
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 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
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 Data
value
 Frequency

how many times the data value
occurs
 Relative

Frequency
frequency/(total number of data
values)
 Cumulative

Relative Frequency
summation of previous relative
frequencies
An example – How many siblings do
you have?
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A


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.
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.
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Descriptive Statistics: Displaying and
Measuring Data
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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.

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 Mean

or average
Use calculator
 Median




x
n
- the middle data value
50% of data below, 50% above
Data MUST be ordered from lowest
to highest
Use calculator
 Mode
- the most frequent data
value

Have to count (or put in a
frequency table)
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


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)
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 Range

Difference between high value and
low value
 Standard
deviation
‘distance’ from the mean
 Sample versus population
2
2
 ( x  x)

(
x


)
s

n 1
N

 Variance


Sample
s2
Population 2
Using calcuator
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 ‘Charts’



Stem and Leaf Graphs – example
Line Graphs – not using
Bar Graphs – not using
 Boxplots

–
need min, median, first and second
quartile, max
 Histograms




–
sort data into bars or intervals
5 to 15 bars
horizontal axes is what the data
represents
vertical axes labeled “frequency”
or “relative frequency”
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Probability Topics
Chapter Objectives
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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.
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#
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) = ____
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








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
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
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
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 Experiment

Toss 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)}
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 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)}
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 P(A)
= ___ P(B) = ___ P(D) = ___
 P(D
and A) = ____
 P(A
and B) = ____
 P(A
or D) = ____
 P(D|A)
= ____
 P(A|D)
= ____
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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) =
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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
Performance on Job
Below
Average Above
Average
Average
Poor
23
60
29
Average
28
79
60
Very Good 9
49
63
TOTAL
60
188
152
TOTAL
112
167
121
400
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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
 Experiment
- cup with 8 black
and 3 yellow beads. Draw 2
beads , one at a time, with
replacement. Record bead color.
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 What’s



Chapter 1
Chapter 2
Chapter 3
 21

multiple choice questions
The last 3 quarters exams
 What

fair game
to bring with you
Scantron (#2052), pencil, eraser,
calculator, 1 sheet of notes (8.5x11
inches, both sides)
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