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Math 103Statistics andProbability

Basic Concepts of Statistics

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Statistics

Statistics• Specific numbers that have been observed

• Observation, Presentation, Analysis and Interpretation of Chance Outcomes

Descriptive Statistics – methods concerned with collecting and describing data to yield meaningful

information.

Inferential Statistics – methods concerned with analysis

of a subset of data to predict or infer about the entire set

of data.

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Descriptive Statistics

Source : http://bsp.gov.ph

Dollar - Peso Rates

0.00

10.00

20.00

30.00

40.00

50.00

60.00

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Year

1$

=

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Observations

Independent Variables – data held constant to determine

the values of the dependent variables

Experiment – Any process that generates a set of data

Observations – The recorded information as a result

of an experiment

Dependent Variables – data as a result of an experiment

when independent variables are fixed

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Types of Data

Quantitative Data (Numerical)

– Discrete (countable)

ex. counts, salary, test

scores

– Continuous (no gaps)

ex. weight, time, force, distance, volume

Qualitative Data (Categorical)

– ex. Blood type,

gender, yes/no, car

model, profession

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4 Scales of Data

100 kilos is

twice as heavy

as 50 kilos

Prices; Weights;

50 kilos

70 kilos

80 kilos

Like interval, but with an

inherent starting point.

Ratios are meaningful

Ratio

90°F is not

twice as hot as

45°F.

Year; Seasonal

temperatures:

50°F75°F

100°F

Differences between

values can be found, but

there may be no inherent

starting point; ratios are

meaningless.

Interval

An order is

determined by

“compact, mid,

sport utility.”

Types of Autos:

5 compact

15 mid-size

20 sport –utility

Categories are ordered,

but differences cannot be

determined or are

meaningless

Ordinal

Categories or

names only.

Blood Types; Yes/No;

Baseball Players:

5 infielders

10 outfielders

15 pitchers

Categories only. Data

cannot be arranged in

ordered sequence

Nominal

ExplanationExampleSummaryLevel

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Where the data is from

Population

– Totality of all

observations we are concerned

– Parameter : a characteristic of a

population

– Census : collection of

data from every

element of population.

Sample

– Subset of Population

– Statistic – a

characteristic of a sample

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Samples

Why Sample?• The population may be too big to observe

• ex. all Filipino citizens

• Costs may be prohibitive • ex. Surveys may be expensive

• Experiments may be destructive• ex. Light bulb life

Biased sampling procedures consistently overestimates or underestimates some characteristic of the population.

Use inferential statistics to generalize

information about the population based on

information obtained from the sample.

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Example: A researcher wants to find out the average weight of 3,000 students in a college. How big must the

sample be to have a 5% margin of error ?

Slovin’s Formula

21 Ne

Nn

+

=

N = Population size

n = sample sizee = margin of error

3535.8

3000

)05.0(30001

30002

==

⋅+

=n

When used: When nothing is known about the population.Otherwise, more accurate formulas are available.

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Sampling Methods

• Simple Random Sample- Eliminates possibility of a bias

- choose sample so that every subset of n observations from the population has the same

chance of being selected.

Use random numbers – using mechanical devices, tables, or computers

• Systematic Sampling – selects every k-th element

with starting point chosen at random

• Stratified Random Sampling – partition population

and select proportional random samples from each subpopulaton

• Cluster Sampling – perform simple random sampling only on randomly selected subpopulations

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Simple Random Sample Example

Class of 270 students. Want a simple random sample of 10 students.

ROW 0 00157 37071 79553 31062 42411 79371 25506 69135

1 38354 03533 95514 03091 75324 40182 17302 64224

2 59785 46030 63753 53067 79710 52555 72307 10223

3 27475 10484 24616 13466 41618 08551 18314 57700

4 28966 35427 09495 11567 56534 60365 02736 32700

5 98879 34072 04189 31672 33357 53191 09807 85796

1. Number the units: Students numbered 001 to 270.

2. Choose a starting point: Row 3, 2nd column (10484…)

3. Read off consecutive numbers: (3-digit labels here)104, 842, 461, 613, 466, 416, 180, 855, 118, 314, 577, 002, 896, …

4. If number corresponds to a label, select that unit.

If not, skip it. Continue until desired sample size obtained.

Or use a computer to generate random numbers from 1 to 270.

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Systematic SamplingOrder the population of units in some way, select one of

the first k units at random and then every kth unit thereafter.

College survey: Order list of rooms starting at top floor of 1st undergrad dorm. Pick one of the first 11 rooms at

random => room 3, then pick every 11th room after that.

Note: often a

good alternative

to random

sampling but

can lead to a

biased sample.

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Stratified Random Sampling

Divide population of units into groups (called strata)

and take a simple random sample from each of the strata.

College survey: Two strata = undergrad & graduate dorms.

Take a simple

random sample

of 15 rooms from

each of the strata

for a total of 30

rooms.

Ideal: stratify

so little variability

in responses within

each of the strata.

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Stratified Proportional Allocation

Example :Suppose 38 students in a class were classified based

on place of birth. 20 are from NCR, 8 from Luzon (other than NCR), 6 from the Visayas, and 4 from Mindanao.

If a sample of 10 is to be made, how many from each classification should be selected?

Solution :

NCR: 20 * (10/38) = 5.26 ≈ 5Luzon: 8 * (10/38) = 2.10 ≈ 2

Visayas: 6 * (10/38) = 1.57 ≈ 2Mindanao: 4 * (10/38) = 1.05 ≈ 1

Total = 10

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Cluster Sampling

Divide population of units into groups (called clusters),

take a random sample of clusters and

measure only those items in these clusters.

College survey: Each floor of each dorm is a cluster.

Take a random sample

of 5 floors and all

rooms on those floors

are surveyed.

Advantage: need only

a list of the clusters

instead of a list of all

individuals.

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Summation

40 valuesdata x all sum ==∑ x

4054321

5

1

=++++=∑=

xxxxxxi

i

51040036169492

5

2

4

2

3

2

2

2

1

5

1

2=++++=++++=∑

=

xxxxxxi

i

Data: x1 = 7x2 = 3x3 = 4x4 = 6x5 = 20

y1 = 1y2 = 3y3 = 2y4 = -1y5 = 0

∑=

=++++=

5

1

1554321i

i

1106895

2

=+−+=∑=i

ii yx

∑∑ ∑= = =

=+++++=++=

2

1

3

1

2

1

60)693()14217(231i j i

iiijixxxyx

14400)120(3 2

25

1

==

∑

=i

ix

5

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Summation Theorems

∑∑∑===

+=+

n

i

i

n

i

i

n

i

ii yxyx111

)(

∑ ∑= =

=

n

i

n

i

ii xccx1 1

∑=

=

n

i

cnc1

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Multidimensional Data

25x24

41x23

31x22

42x21

2nd Floor

(i=2)

30

x14

45

x13

28

x12

40

x11

1st Floor

(i=1)

4th Room

(j=4)

3rd Room

(j=3)

2nd Room

(j=2)

1st Room

(j=1)

# of students

xij

282)25413142()30452840(

)()(

)(

2423222114131211

432

2

1

4

1

2

1

1

=+++++++=

+++++++=

+++=∑∑ ∑= = =

xxxxxxxx

xxxxx iii

i j i

iij

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Exercise

∑

∑

∑

=

=

=

−

=

=

5

1

2

5

1

2

5

1

3 evaluate

300 and

50 if

i

i

i

i

i

i

)(x

x

x

45)5(9)50(6300

96

)96(3

5

1

5

1

5

1

2

5

1

25

1

2

=+−=

+−=

+−=−

∑ ∑ ∑

∑∑

= = =

==

i i i

ii

i

ii

i

i

xx

xx)(x

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End