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CHAPTER 1 Introduction
Definition of Statistics
•

In its plural sense, statistics is a set of numerical data (e
...
, vital statistics in a beauty contest,
monthly sales of a company, daily P-$ exchange rate)
...


NATURE OF STATISTICS
General Uses of Statistics
a
...
Statistics summarizes data for public use
Examples on the Role of Statistics
• In the biological and medical sciences, it can help researchers discover
relationships worthy of further attention
...

• In the social sciences, it can guide and help researchers support theories and
models that cannot stand on rationale alone
...

• In business, a company can use statistics to forecast sales, design products, and
produce goods more efficiently
...
Results can
help the company decide whether to market the new formula or not
...


1

Example:
A quality controller can use Statistics to estimate the average
lifetime of the products produced by their current equipment
...
Statistical Methods of Applied Statistics - refer to procedures and techniques used in the
collection, presentation, analysis, and interpretation of data
...
Statistical Theory of Mathematical Statistics - deals with the development and exposition
of theories that serve as bases of statistical methods
...
Inferential Statistics
Descriptive

Inferential

•

A bowler wants to find his bowling •
average for the past 12 games
...


•

A housewife wants to determine the •
average weekly amount she spent on
groceries in the past 3 months
...


A housewife would like to predict
based on last year’s grocery bills, the
average weekly amount she will spend
on groceries for this year
...


2

POPULATION AND SAMPLE
Definition
...

A sample is a part or subset of the population from which the information is
collected
...
Toward this goal, 5,000 of his 200,000
customers are contacted and each is asked, “Are you satisfied with the performance
of the kerosene heater you purchased?” Identify the population and the sample for
this situation
...
A parameter is a numerical characteristic of the population
...

Example:

In order to estimate the true proportion of students at a certain college who smoke
cigarettes, the administration polled a sample of 200 students and determined that
the proportion of students from the sample who smoke cigarettes is 0
...
Identify
the parameter and the statistic
...

2
...

4
...


Define the problem
...

Collect the data
...

Interpret the results
...


A variable is a characteristic or attribute of persons or objects which can assume
different values or labels for different persons or objects under consideration
...


Measurement is the process of determining the value or label of a particular
variable for a particular experimental unit
...


An experimental unit is the individual or object on which a variable is measured
...
Discrete vs Continuous
Discrete variable

- a variable which can assume finite, or, at most, countably
infinite number of values; usually measured by counting or
enumeration
Continuous variable - a variable which can assume infinitely many values
corresponding to a line interval
2
...
g
...
g
...
of cars)
Levels of Measurement
1
...


4

Examples:
Sex
Marital status

M-Male
1-Single

F-Female
2-Married

3-Widowed

4-Separated

2
...

Examples:
Teaching ratings
Year level

1-poor
1-1st yr

2- fair
2 – 2nd yr

3-good
3 – 3rd yr

4-excellent
4 – 4th yr

3
...
An interval scale must have a common and constant unit of measurement
...

Examples:
IQ
Temperature (in Celsius)
4
...


...
Primary vs
...
Primary source

- data measured by the researcher/agency that published it

b
...


5

2
...
Internal
a
...
External data

- information that relates to some activity outside the organization
collecting the data

Example: The sales data of SM is internal data for SM but external data for any other
organization such as Robinson’s
...
Survey method

•

•
•
•

•

-

questions are asked to obtain information, either through
self-administered questionnaire or personal interview

Self-administered questionnaire
Obtained information is limited to
subjects’ written answers to pre-arranged
questions
Lower response rate
It can be administered to a large number
of people simultaneously
Respondents may feel freer to express
views and are less pressured to answer
immediately
It is more appropriate for obtaining
objective information

2
...
g
...
g
...


Experimental method - a method designed for collecting data under controlled
conditions
...

This is an excellent
method of collecting data for causation studies
...

4
...
g
...

• field sources – researchers who have done studies on the area of interest
are asked personally or directly for information needed
5
...
g
...

•
•

Census or complete enumeration is the process of gathering information from
every unit in the population
...


Survey sampling is the process of obtaining information from the units in the
selected sample
...
A sampling procedure that gives every element of the population a (known) nonzero
chance of being selected in the sample is called probability sampling
...

Whenever possible, probability sampling is used because there is no
objective way of assessing the reliability of inferences under non-probability
sampling
...
The target population is the population from which information is desired
...
The sampled population is the collection of elements from which the sample is
actually taken
...
The population frame is a listing of all the individual units in the population
...
purposive sampling

-

sets out to make a sample agree with the profile of the
population based on some pre-selected characteristic

2
...
convenience sampling-

selects sampling units that come to hand or are convenient
to get information from

4
...

2
...

4
...

6
...
The process of selecting the sample must give an equal chance of selection to any
one of the remaining elements in the population at any one of the n draws
...

In SRSWR, a chosen element is always replaced before the next selection is made, so that an
element may be chosen more than once
...

Step 2: Select n numbers from 1 to N using some random process, for example, the table of
random numbers
...

Step 3: The sample consists of the units corresponding to the selected random numbers
...

Inferential methods are simple and easy
...

A population frame is needed
...


•

Stratified Random Sampling
Description of the Design
In stratified random sampling, the population of N units is first divided into subpopulations
called strata
...

Sample Selection Procedure

9

Step 1: Divide the population into strata
...

Step 2: After the population has been stratified, a simple random sample is selected from
each stratum
...

It is administratively convenient
...

The stratification of the population may require additional prior information about the
population and its strata
...
Here k is called
the sampling interval; the reciprocal 1/k is the sampling fraction
...

Step 2: Determine k, the sampling interval using the formula k = N/n
...
The unit corresponding to r is the first
unit of the sample
...


Method B
Step 1: Number the units of the population consecutively from 1 to N
...


10

Step 3: Select the random start r, where 1 ≤ r ≤ N
...

Step 4: Consider the list of units of the population as a circular list, i
...
, the last unit in the
list is followed by the first
...
, r+ (n-1)k
...

It is possible to select a sample in the field without a sampling frame
...

Disadvantages

•

If periodic regularities are found in the list, a systematic sample may consist only of similar
types
...
)
Knowledge of the structure of the population is necessary for its most effective use
...

Similar to strata in stratified sampling, clusters are non-overlapping sub-populations which
together comprise the entire population
...
Unlike strata, however,
clusters are preferably formed with heterogeneous, rather than homogeneous elements so that
each cluster will be typical of the population
...
When all of the clusters are of the same size, the
number of elements in a cluster will be denoted by M while the number of clusters in the
population will be denoted by N
...

Step 2: Select n numbers from 1 to N at random
...

Step 3: Observe all the elements in the sample of clusters
...

Thus, listing cost is reduced
...

Disadvantages

•
•

The costs and problems of statistical analysis are greater
...


•

Multistage Sampling
Description of the Design
In multistage sampling, the population is divided into a hierarchy of sampling units
corresponding to the different sampling stages
...
In the second
stage of sampling, each selected PSU is subdivided into second-stage units (SSU) then a
sample of SSUs is drawn
...

Advantages

•
•

Listing cost is reduced
...

Disadvantages

•
•
•

Estimation procedure is difficult, especially when the primary stage units are not of the same
size
...

The sampling procedure entails much planning before selection is done
...
The far-flung Boeing fleet has now logged an estimated 1,803,704,000 miles
(22,855,948,000 kms
...

Passenger totals stand at upwards of 71
...

Advantages
•
•

This presentation gives emphasis to significant figures and comparisons
It is simplest and most appropriate approach when there are only a few numbers to be
presented

Disadvantages
•
•

When a large mass of quantitative data are included in a text or paragraph, the presentation
becomes almost incomprehensible
Paragraphs can be tiresome to read especially if the same words are repeated so many times

Tabular Presentation
•

the systematic organization of data in rows and columns

Advantages
•
•
•
•

more concise than textual presentation
easier to understand
facilitates comparisons and analysis of relationship among different categories
presents data in greater detail than a graph

Parts of a Formal Statistical Table
1
...
The title is a brief
statement of the nature, classification and time reference of the information presented
and the area to which the statistics refer
...

2
...

3
...
The stubhead describes the stub
listing as a whole in terms of the classification presented
...


4
...
Source note - an exact citation of the source of data presented in the table (should
always be placed when the figures are not original)
6
...
4 – CRIME VOLUME AND RATE BY TYPE: 1991 – 1993
(Rate per 100,000 population)
1991
Volume Crime
Rate
121,326
195

Type
Total

1992
Volume Crime
Rate
104,719
164

1993
Volume
Crime
Rate
96,686
148

Index Crimes
Murder
stub
Homicide
Physical Injury
Robbery
Theft
Rape

77,261
8,707
8,069
21,862
13,817
22,780
2,026

124
14
13
35
22
37
3

67,354
8,293
7,912
20,462
11,164
17,374
2,149

106
13
12
32
18
27
3

58,684
7,758
7,123
18,722
9,856
12,940
2,285

90
12
11
29
15
20
4

Nonindex Crimes

44,065

71

37,365

59

38,002

58

Source Note

boxhead

field

Source: Philippine National Police

Guidelines
•

The title should be concise, written in telegraphic style, not in complete sentence
...
Stress differences rather than similarities between adjacent
columns
...
This is frequently a signal that a spanner head is needed
...

Categories should not overlap
...

Show any relevant total, subtotals, percentages, etc
...

Tables should be self-explanatory, although they may be accompanied by a paragraph that
will provide an interpretation or direct attention to important figures
...
Accuracy

- A good chart should not be deceptive, distorted, misleading, or in any
way susceptible to wrong interpretations as a result of inaccurate or
careless construction
...

2
...
The graph
should focus on the message it is trying to communicate
...
The graph
must be able to aid the reader in the interpretation of facts
...
Simplicity - The basic design of a statistical chart should be simple, straightforward, not loaded with irrelevant, superfluous, or trivial symbols
and ornamentation
...

4
...
It must be artistic in that it embodies harmonious
composition, proportion, and balance
...
Line Chart - graphical presentation of data especially useful for showing trends over a
period of time
...
Pie Chart - a circular graph that is useful in showing how a total quantity is
distributed among a group of categories
...


Market Shares of Softdrinks in
Metro Manila
Sprite
Sarsi 5% Othe rs
7-up 5%
12%
8%

Pe ps i
30%

Coca-Cola
40%

16

3
...
If the bars are arranged vertically, the height of the bar represents the
quantity
...
Pictorial unit chart – a pictorial chart in which each symbol represents a definite and
uniform value
Growth Pattern of Philippine Population: 1960 - 2000

17

Chapter 4 The Organization of Data
Definition
...

Example: Final grades of Soc Sci 101 Students
82
83
79
72
71
84
59
77
50
87

82
82
69
88
80
76
62
78
69
73

83
63
74
84
72
75
79
87
75
84

79
75
53
80
60
82
82
75
70
68

72
50
73
68
81
76
72
86
77
85

71
85
71
50
89
53
81
82
87
62

84
76
50
74
94
91
60
74
86
87

59
79
76
84
80
69
84
73
77
92

77
68
57
71
84
60
68
72
75
69

50
69
81
73
81
89
66
84
96
52

87
62
62
68
50
79
94
51
66
65

Definition
...

Example: Final grades of Soc Sci 101 Students arranged in an array
50
50
50
50
50
50
51
52
53
53

57
59
59
60
60
60
62
62
62
62

63
65
66
66
68
68
68
68
68
69

69
69
69
69
70
71
71
71
71
72

72
72
72
72
73
73
73
73
74
74

74
75
75
75
75
75
76
76
76
76

77
77
77
77
78
79
79
79
79
79

80
80
80
81
81
81
81
82
82
82

82
82
82
83
83
84
84
84
84
84

84
84
85
85
86
86
87
87
87
87

87
87
88
89
89
91
92
94
94
96

Advantages:
•
•

easier to detect the smallest and largest value
easier to find the measures of position

18

Frequency Distribution Table
In the construction of a frequency distribution, the various items of a series are classified into
groups
...

Definition of terms
1
...

3
...


Class frequency
Class interval
Class limits
Class boundaries

5
...
Class mark (CM) 7
...

10
6
8
24
22
24
12
4

LCB
49
...
5
61
...
5
73
...
5
85
...
5

UCB
55
...
5
67
...
5
79
...
5
91
...
5

CM
52
...
5
64
...
5
76
...
5
88
...
5

Class
50 – 54
55 – 59
60 – 64
65 – 69
70 – 74
75 – 79
80 – 84
85 – 89
90 – 94
95 – 99

Freq
...
5
54
...
5
64
...
5
74
...
5
84
...
5
94
...
5
59
...
5
69
...
5
79
...
5
89
...
5
99
...

Determine the number of classes
...

There are no precise rules concerning the optimal number of classes but Sturges’ formula
can be used as a first approximation
...
322 log n
= approximate number of classes
n = number of observations
2
...
Whenever possible, all classes should be
of the same size
...

• Solve for the range, R = max – min
...

• Round-off C’ to a convenient number to work with, say C, and use C as the class
size
...

Determine the lowest class limit
...

4
...

5
...
Sum the frequencies and check against the
total number of observations
...


Relative Frequency (RF) Distribution and Relative Frequency Percentage (RFP)

RF = class frequency ÷ no
...

Cumulative Frequency Distribution (CFD)
- shows the accumulated frequencies of successive classes, beginning at either end of
the distribution
Greater than CFD – shows the no
...
of observations less than the UCB
Example:
Class

Freq
...
5
54
...
5
64
...
5
74
...
5
84
...
5

54
...
5
64
...
5
74
...
5
84
...
5
94
...
09

...
07

...
15

...
20

...
04

9
3
7
12
15
17
20
12
4

10
13
21
34
51
70
92
105
109

110
100
97
89
76
59
40
18
5

20

95 – 99

1

94
...
5


...

Frequency Histogram - a bar graph that displays the classes on the horizontal axis and
the frequencies of the classes on the vertical axis; the vertical lines of the bars are erected at
the class boundaries and the height of the bars correspond to the class frequency
25
20
15

No
...
5

3

4

54
...
5

5

6

7

8

64
...
5 74
...
5

9

84
...
5 94
...
5

Grades

2
...


0
...
2

Relativ
freq
...
15
0
...
05
0
1

2

3

4

5

6

7

8

9

10

11

12

49
...
5 59
...
5 69
...
5 79
...
5 89
...
5 99
...
Frequency Polygon – a line chart that is constructed by plotting the frequencies at the class
marks and connecting the plotted points by means of straight lines; the polygon is closed by
considering an additional class at each end and the ends of the lines are brought down to
the horizontal axis at the midpoints of the additional classes
...
of
students

15
10
5
0
47

57

62

67

72
77
Grades

82

87

92

97

102

Ogives - graphs of the cumulative frequency distribution
a
...
> ogive - the >CF is plotted against the LCB

< ogive

9

4


...
5
4
8


...
5
4
6

4


...
5

120
100
80
60
40
20
0
4

Cumulative Frequency

4
...
It
presents a histogram-like picture of the data, while allowing the experimenter to retain the
actual observed values of each data point
...

In creating a stem-and-leaf display, we divide each observation into two parts, the stem
and the leaf
...

Steps in Constructing the Stem-and-Leaf Display
1
...

2
...

3
...
Reorder the leaves from lowest to highest within each stem row
...

5
...
this subdivision can be
increased to five groups if necessary
...
Provide a key to your stem-and-leaf coding so that the reader can recreate the actual
measurements from your display
...

Example:
Unit = 0
...
2
1 | 2 represents 12
1 | 2 represents 120

24

Chapter 5 Measures of Central Tendency
and Measures of Location
Definition
...
It is often referred to as the average
...
easily understood
- not a distant mathematical abstraction
2
...
stable
- not affected materially by minor variations in the groups of items
4
...

The notation X1, X2,
...

Let the Greek letter Σ indicate the “summation of,” thus, we can write the sum of n
n

observations as

∑

i= 1

X i = X 1 + X 2 +
...


The numbers 1 and n are called the lower and the upper limits of summation,
respectively
...
The summation of the sum of variables is the sum of their summations
...
+ z i ) =

n

∑

i= 1

n

∑

i= 1

Yi

ai +

n

∑

i= 1

bi +
...
If c is a constant, then
n

∑

i= 1

n

cX i = c ∑ X i
i= 1

3
...


2
...


4

∑

i= 1

3
...


 Xi 


i = 1  Yi 

∑

n

4

4

X i ∑ Yi

6
...


N
n

The sample mean X (read as “X bar”) of n observations is computed as
X =

∑

i= 1

Xi


...


Examples:
1
...
Treating the
data as a population, find the mean number of employees for the 5 stores
...
Scores in the Statistics 101 first exam for a sample of 10 students are as follows: 60, 55,
30, 90, 88, 79, 45, 66, 93, and 80
...

3
...
Give the sample
mean
...


The weighted mean is a modification of the usual mean that assigns weights (or
measures of relative importance) to the observations to be averaged
...

Wi

27

Examples:
1
...
Jeffry obtains
marks of 83 for assignments, 72 for the project, 41 for the midterm exam, and 47 for the
final exam
...

2
...
0
1
...
0
3
...
0

Math 53 is a 5-unit course and all others are 3-unit courses
...

Characteristics of the Mean
1
...

2
...
In particular it is strongly influenced by
extreme values
...
Since the mean is a calculated number, it may not be an actual number in the data set
...
It possesses two mathematical properties that will prove to be important in subsequent
analyses
...

ii) The sum of the squared deviations is minimum when the deviations are taken from
the mean
...
a
...

b
...

Example:
Given 5 temperature readings measured in Fahrenheit: 98, 100, 107, 90, 92
...
If the assumption holds, the following equation may be used to
approximate the mean from a frequency distribution
...

(fi)
10
3
8
13
17
19
22
13
4
1
110

CM
(XI)
52
57
62
67
72
77
82
87
92
97

fiXi
520
171
496
871
1224
1463
1804
1131
368
97
8145

What is the mean grade?
Remarks:
1
...

2
...


29

THE MEDIAN
-

the positional middle of the arrayed data
in an array, one-half of the values precede the median and one-half follow it

The first step in calculating the median, denoted as Md, is to arrange the data in an array
...
, n
...
e
...
e
...
Given the following heights ( in inches ): 71, 72, 75, 75, and 67
...

2
...

3
...
Find the median
...
The median is a positional measure
...
The median is affected by the position of each item in the series but not by the value of
each item
...

Approximating the Median from the Frequency Distribution
possible only when the values of the observations falling in the median class can
be assumed to be evenly spaced throughout the class
...
)
Step 1
...

Step 2
...
This class is the median class
...


Approximate the median using the following formula:
 n / 2− < CFmd − 1 

Md = LCBmd + c
f md


where LCBmd
c
n
fmd

= the lower class boundary of the median class
= class size of the median class
= the total number of observations in the distribution
= less than cumulative freq
...

Class
50 – 54
55 – 59
60 – 64
65 – 69
70 – 74
Median
75 – 79
class
80 – 84
85 – 89
90 – 94
95 – 99

Freq
...
freq
...
5 + 5
 = 75
...

Examples: Find the mode of the following:
1
...
2, 5, 5, 2, 2, 5, 1, 3, 5, 4, 2, 5, 5, 2, 2, 5, 5, 2, 2, 1
2
...
Refer to the example on the final grades of 110 Soc Sci 101 students
...
It does not always exist; and if it does, it may not be unique
...

2
...

3
...

Approximating the Mode from the Frequency Distribution
Step 1:
Step 2:

Locate the modal class
...

Approximate the mode using the following formula:

f mo − f 1 

Mo = LCBmo + c
 2 f mo − f 1 − f 2 

where LCBmo
c
fmo
f1
f2

=
=
=
=
=

lower class boundary of the modal class
class size of the modal class
frequency of the modal class
frequency of the class preceding the modal class
frequency of the class following the modal class

32

Example :
Refer to the example on the final grades of 110 Soc Sci 101 students
...

10
3
8
13
17
19
22
13
4
1



22 − 19
 = 80
...
5 + 5
 2(22) − 19 − 13 

33

CHAPTER 6 MEASURES OF LOCATION
Definition
...

Definition
...
Thus,
P1, read as first percentile, is the value below which 1% of the values fall
...

•
•
•
P99, read as ninety-ninth percentile, is the value below which 99% of the values fall
...


34

Other Forms of Fractiles:
1
...
Thus,
D1, read as first decile, is the value below which 10% of the values fall
...

•
•
•
D9, read as ninth decile, is the value below which 90% of the values fall
...
Quartiles
- values that divide the array into 4 equal parts
...

Q2, read as second quartile, is the value below which 50% of the values fall
...

Examples: Use the data on Stat 101 final grades
a
...
P90 = X(90*[110+1]/100)
= X(99
...
9[X(100) - X(99)]
= 87 + 0
...
P90 = 84
...
2

2
...
Q2

2
...
Q2

35

Chapter 7 Measures of Dispersion and Measures of Skewness
Definition
...

Some Uses for Measuring Dispersion
to determine the extent of the scatter so that steps may be taken to control the
existing variation
•
used as a measure of reliability of the average value
•

General Classifications of Measures of Dispersion
1
...
Measures of Relative Dispersion

MEASURES OF ABSOLUTE DISPERSION
Measures of absolute dispersion are expressed in the units of the original
observations
...

•

The Range
Definition
...

Range = maximum - minimum
The range is approximated from a frequency distribution by getting the difference
between the upper class limit of the highest class interval and the lower class limit of the
lowest class interval
...
The IQ’s of 5 members of a certain family are 108, 112, 127, 116, and 113
...

2
...
Find the range
...
It uses only the extreme values
...

2
...

3
...

4
...

•

The Standard Deviation and the Variance
N

Definition
...
For a sample of size n, the sample variance is

s=

=

i

)2

n

and the sample standard deviation is

2

∑ (X

i

− X

i= 1

)

i

− X

)

2

n− 1

2

n− 1

Remarks:
1
...

2
...
It is not expressed in the same units
as the original observations
...
The following scores were given by 6 judges for a gymnast’s performance in the vault: 7,
5, 9, 7, 8, and 6
...

2
...
Find the standard deviation
...
Refer to the example on the final grades of 110 Soc Sci 101 students
...
11)

2

109

13798
...
25
109

=

Computational formula:
 n

n∑ X −  ∑ X i 
 i= 1 
s2 = i= 1
n(n − 1)
n

Example:

2

2
i

For the final grade of 110 Statistics 101 students,
s=

110(617936) − (8152) 2
=
110(109)

1517856
= 11
...

Class

Freq
...
It is affected by the value of every observation
...

2
...

3
...

4
...


MEASURES OF RELATIVE DISPERSION
Measures of relative dispersion are unitless and are used when one wishes to
compare the scatter of one distribution with another distribution
...
The coefficient of variation, CV, is the ratio of the standard deviation to the
mean and is usually expressed in percentage
...

1
...
In 1992 Bangko Sentral ng Pilipinas (BSP) put
the peso on a floating rate basis
...
Government intervenes through the BSP, only
when there are speculative elements in the market
...
Which of the two periods is more stable?
Mean

s
...


39

1989-1991
1992-1994

22
...
4

1
...
15

2
...
Individual pieces of cookies are
scanned by a spectrophotometer calibrated to reflect yellow-brown light
...
The cookies were also weighed in grams at this
stage
...


Color
Weight

Mean

s
...


41
...
7

10
3
...
The standard score measures how many standard deviations an observation is
above or below the mean
...
The standard score is not a measure of relative dispersion per se but is somewhat related
...
It is useful for comparing two values from different series specially when these two series
differ with respect to the mean or standard deviation or both are expressed in different
units
...
Robert got a grade of 75% in Stat 101 and a grade of 90% in Econ 11
...
Relative to the other students, where did he
perform better?
40

2
...
Different typing skills are required for secretaries depending on whether one is working
in a law office, an accounting firm, or for a mathematical research group at a major
university
...
A time penalty has been incorporated into the
scoring of each sample based on the number of typing errors
...

Sample
Law
Accounting
Scientific

Nancy’s Score
141 sec
7 min
33 min

180 sec
10 min
26 min

Mean

std
...

30 sec
2 min
5 min

Where do you think should Nancy be placed?

41

MEASURES OF SKEWNESS
Definition
...
It indicates not only the amount of skewness but also
the direction
...
Positively Skewed or Skewed to the Right
•
•
•
•

distribution tapers more to the right than to the left
longer tail to the right
more concentration of values below than above the mean
most skewed curves encountered in the social sciences are skewed to the right

Example: frequency distribution of income
2
...
Sk =

where

2
...
Since the mode is frequently only an approximation, formula 2 is preferred
...
Interpretation of the measure of skewness:
Sk > 0:
Sk < 0:
Sk = 0:

positively skewed since X > Md > Mo
negatively skewed since X < Md < Mo
symmetric since X = Md = Mo

Example: Refer to the final grade of 110 Soc Sci 101 students
...
1

Md = 75

Mo = 84

s = 11
...
1 − 84
= − 0
...
25

Using the second formula,
Sk =

3(74
...
24
11
...
The boxplot is a graph that is very useful for displaying the following features of
the data:
•
•
•
•
•

location
spread
symmetry
extremes
outliers

Steps in Constructing a Boxplot
1
...

2
...

3
...
5 IQR
FU = Q3 + 1
...
Locate the smallest value contained in the interval [F L , Q1]
...

5
...
Draw a line from this value to Q3
...
Values falling outside the fences are considered outliers and are usually denoted by “x”
...
The height of the rectangle is usually arbitrary and has no specific meaning
...

2
...
Such an observation is called a far outlier
...
Set A:

1
10
14

15
18
20

Q1 = 15
Q3 = 24
Md = 22

21
22
22

22
23
24

24
25
28

IQR = 9
FL = 1
...
5

44

Set B:

3
8
9

10
10
10

11
12
12

12
16
16

Q1 = 10
Q3 = 16
Md = 12

Set A

19
19
30

IQR = 6
FL = 1
FU = 25

x

Set B

x
0

2
...


50

55

60

65

70

75

80

85

90

95

100

p

45

Chapter 8 Probability
RANDOM EXPERIMENTS, SAMPLE SPACES AND EVENTS

Definition of Terms
1
...
Sample space
3
...
Event
5
...
Simple event
7
...
Mutually exclusive events

any process of generating a set of data or observations that can
be repeated under basically the same conditions, which lead to
well-defined outcomes
set of all possible outcomes of an experiment, usually denoted
by S
an element of the sample space, an outcome
any subset of the sample space, usually denoted by capital
letters
a subset of the sample space that contains no elements and
denoted by the symbol φ
...


•

The empty space can be viewed as an event that will never happen
...


•

The sample space S, as an event, always occurs, and is referred to as the certain or sure
event
...
A ∩ B

the intersection of events A and B is the event that both A and B occur

2
...
A1 or Ac

the complement of an event A with respect to S contains all elements of S that
are not in A and is the event that A does not occur

Some relationships between events can be illustrated by means of a Venn Diagram
...

Postulate 1
...

Postulate 3
...

P(S) = 1, where S is the sample space, and P(φ) = 0, where φ is the null space
...
A Priori or Classical Probability – probability is determined even before the experiment is
performed using the following rule:
If an experiment can result in any one of N different equally likely outcomes,
and if exactly n of these outcomes correspond to event A, then the probability of
event A is
P( A) =

no
...
of sample points in S N

2
...
of times event A occurred
no
...
Subjective Probability – probability is determined by the use of intuition, personal beliefs,
and other indirect information
...
Find the errors in each of the following statements:
a
...
40 and the probability that it will not rain
tomorrow is 0
...

b
...
19, 0
...
25, 0
...
29
...
The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given
day in February are, respectively, 0
...
38, 0
...
15
...
On a single draw from a deck of playing cards the probability of selecting a heart is 1/4,
the probability of selecting a black card is 1/2, and the probability of selecting both a
heart and a black card is 1/8
...
Answer the following:
a
...
In tossing a fair die, what is the probability of getting a 3? Of getting an even number?
Of getting a number greater than 6?
3
...
If the coin is tossed once,
what is the probability of getting a head?

Rules of Counting (Optional)
Theorem
...
n2 ways
...
How many sample points are there in the sample space when a pair of balanced dice is
thrown once?
2
...


(Multiplication Rule) If an operation can be performed in n 1 ways, if for each of
these a second operation can be performed in n2 ways, if for each of the first two a
third operation can be performed in n3 ways, and so on, then the sequence of k
operations can be performed in n1n2
...


Examples:
1
...
all digits are distinct?
b
...
Numbers must be even and repetition is not allowed?

48

2
...
A permutation is an arrangement or ordering of all or part of a set of objects
...


The number of permutations of n distinct objects is
n (n-1)(n-2)
...
0! = 1
...


The number of permutations of n distinct objects taken r at a time is
n!
n Pr =
(n − r )!
Examples:
1
...
Find the number of
sample points in the space S
...
In how many ways can the 5 starting positions on a basketball team be filled with 8 men
who can play any position?

Theorem
...
, nk of a kth kind is
n!
n1!n 2 !
...
Consider the word STATISTICS
...

2
...
A combination is a selection of r objects from n without regard to order
...


The number of combinations of n distinct objects taken r at a time is
n!
n Cr =
r!(n − r )!

49

Examples:
1
...
In how many
ways can he choose a set of 8 questions if he chooses arbitrarily?
2
...


(Additive Rule) If A and B are any two events, then
P(A∪B) = P(A) + P(B) - P(A∩B)

Corollary
...


If A1, A2,
...
∪ An) = P(A1) + P(A2) +
...


If A and Ac are complementary events, then
P(A) + P(Ac) = 1
...
Suppose A and B are two events for which P(A)=0
...
6
...
P(A∪B) b
...
A coin is tossed twice
...
A die is loaded in such a way that an even number is twice as likely to occur as an odd
number
...
a number less than 4 occurs
b
...
The probability that a student passes Stat 101 is 0
...
85
...
95, what
is the probability that he will pass both courses? fail both Stat 101 and Comm II?
5
...
Past evidence shows that the
probability that neither well produces oil is 0
...
18; and, the probability that both wells produce oil is 0
...
What is the
probability that at most one well produces oil? At least one?
6
...
In a poker hand consisting of 5 cards, find the probability of holding 2 aces and 3 jacks
...
A box contains 4 red and 2 white marbles
...
the two end marbles are red?
b
...

The probability of an event B occurring when it is known that some event A has
occurred is called a conditional probability
...

Examples:
1
...

Marital
Status
Never Married
Married
Widowed
Divorced
Totals (in millions)

Males
(in millions)
25
...
6
2
...
3
95
...
0
59
...
0
11
...
4

Totals
(in millions)
46
...
9
13
...
4
197
...
The probability that a student passes Stat 101 is 0
...
85, the probability that he passes both subjects is 0
...
If the student passes Stat
101, what is the probability that the student will pass Comm II?
3
...
Bigueras has two children
...
What is the probability that both children are boys given that the older child is a
boy?
4
...
If it is known that one die shows a 4, what is the probability that
a
...
the total of both dice is greater than 7?

Definition
...


51

Examples:
1
...
The probability that Robert will correctly answer the toughest question in an exam is 1/4
...
Find the probability
that both will answer the question correctly, assuming that they do not copy from each other
...
The probabilities that a student will get passing grades in Math 55, in Stat 105, or in both are
P(M)=0
...
80, and P(M∩E)=0
...
Are M and E independent events?

52

Chapter 9 Random Variable
Concept of a Random Variable
Definition
...

Remark
...


Examples:
1
...
1) An experiment consists of tossing a coin 3 times and observing the
result
...
(Experiment No
...
Let Y denote the
number of spots on the upper face
...


DISCRETE & CONTINUOUS PROBABILITY DISTRIBUTIONS
Definition
...

Definition
...


53

Definition
...

Definition
...


Discrete Probability Distributions
Definition
...

Remark
...


Examples:
1
...
1, the discrete probability distributions of the random variables X and Y
are
x
P(X=x)

0
1/8

1
3/8

2
3/8

3
1/8

y
P(Y=y)

-3
1/8

-1
3/8

1
3/8

3
1/8

2
...
2
...
The function with values f(x) is called a probability density function for the
continuous random variable X, if
the total area under its curve and above the horizontal axis is equal to 1; and
the area under the curve between any two ordinates x = a and x = b gives the
probability that X lies between a and b
...
A continuous random variable has a probability of zero of assuming exactly any of its values,
that is, if X is a continuous random variable, then P(X=x) = 0 for all real numbers x
...
The probability density function can not be represented in tabular form
...
5
f ( x) = 
otherwise
0
Find the following probabilities:
a
...

f(x)
½
0

1

2

x

area of shaded region = (length)(width) = P(1 < X < 2) = (1/2)(1) =1/2
b
...

d
...


P(X > 1
...
75)
P(X = 0
...
75)
...
Let X be a discrete random variable with probability distribution
x
P(X=x)

x1
f(x1)

x2
f(x2)


...


xn
f(xn)

The mean or expected value of X is

µ = E( X ) =

n

∑

i= 1

xi f ( x i )

Examples:
1
...
1
...
5
Y

-3

-1

1

3
55

P(Y=y)

1/8

3/8

3/8

1/8

E(Y) = (-3)(1/8) + (-1)(3/8) + (1)(3/8) + (3)(1/8) = 0
2
...
2
...
In a gambling game a man is paid P50 if he gets all heads or all tails when 3 coins are tossed,
and he pays out P30 if either 1 or 2 heads show
...


Let X be a discrete random variable with probability distribution
x
P(X=x)

x1
f(x1)

x2
f(x2)


...


xn
f(xn)

The mean or expected value of the random variable g(X) is
E ( g ( X )) =

n

∑

i= 1

Example:

Definition
...
4, one
car is 0
...
15, 3 cars is 0
...
08, five cars is 0
...
01
...
Find the salesman’s expected daily earnings
...


g ( xi ) f ( xi )

2

= Var ( X ) = E ( X − µ ) 2

Let X be a discrete random variable with probability distribution
X
P(X=x)

x1
f(x1)

x2
f(x2)


...


xn
f(xn)

The variance of X is

σ

2

= Var ( X ) = E ( X − µ ) 2 =

n

∑

i= 1

Theorem
...
1, find the variance of X
...
5
4

Var(X) =

∑

i= 1

( xi − 1
...
5)2(1/8) + (1-1
...
5)2(3/8) + (3-1
...
75
Using the computational formula of the Var(X),
56

Var(X) = E(X2) - [E(X)]2 = 3 – (1
...
75

Properties of the Mean and Variance
Let X and Y be random variables (discrete or continuous) and let a and b be constants
...
E(aX + b) = a E(X) + b
Special Cases:
a
...

b
...


a

2
...
E(XY) = E(X)E(Y) if X and Y are independent
...
E[ X - E(X) ] = 0
...


Var(aX + b) = a2Var(X)
...
if b = 0, then Var(aX) = a2Var(X)
...
if a = 0, then Var(b) = 0
...
If X and Y are independent then
Var(X + Y) = Var(X) + Var(Y)
Var(X - Y) = Var(X) + Var(Y)
Example :
If X and Y are independent random variables with E(X) = 3, E(Y) = 2, Var(X) = 2 and
Var(Y)=1, find
a
...

c
...


E(3X + 5)
Var(3X +5)
E(XY)
Var(3X - 2Y)

57

Chapter 10 The Normal Distribution
Definition
...
71828
and π ≈ 3
...

Notation:
Note:

If X follows the above distribution, we write X~ N(µ, σ2 )
...


The graph of the normal distribution is called the normal curve
...
The curve is bell-shaped and symmetric about a vertical axis through the mean µ
...
The normal curve approaches the horizontal axis asymptotically as we proceed in either
direction away from the mean
...
The total area under the curve and above the horizontal axis is equal to 1
...

The distribution of a normal random variable with mean zero and standard
deviation equal to 1 is called a standard normal distribution
...

z1 =

z2 =

x2 − µ
σ

Examples :
1
...
less than 45
b
...
more than 45
2
...
5, find
a
...
2578
b
...
1539
...
The achievement scores for a college entrance examination are normally distributed with
mean 75 and standard deviation equal to 10
...
A softdrink machine is regulated so that it dispenses an average of 200 ml
...
If the
amount of drink dispensed is normally distributed with a standard deviation equal to 15 ml
...

b
...

d
...
and 209 ml
...
cups are used for the next 1000 drinks ?
below what value do we get the smallest 25% of the drinks?

59

OTHER COMMON DISTRIBUTIONS
•

Binomial Distribution

Definition
...
The probability of a failure is equal to q=1-p
...

Definition
...

Note :

If X~Bi(n, p) then E(X) = np and Var(X) = npq, where q = 1-p
...
A multiple-choice quiz has 15 questions, each with 4 possible answers of which only 1 is the
correct answer
...
exactly 10 correct answers
b
...
to 12 correct answers
...
Suppose that airplane engines operate independently in flight and fail with probability 1/5
...
A civil service examination is prepared so that 80% of all persons with a high school diploma
can pass it
...

b
...

d
...
A hypergeometric experiment is one that possesses the following properties:
•

a sample of size n is taken without replacement from a population of size

N
•

k of the N are classified as “success” and (N-k) classified as “failure”
...

Definition
...


Note:

If X~H(N,n,k) then
nk
E( X ) =
N

Remark:

and

Var ( X ) =

( N − n) nk 
k
 1− 
( N − 1) N 
N

If n is small relative to N the probability of “success” for each draw will change
only slightly
...


Examples:
1
...
A lot of 20 personal computers was delivered to the Statistical Center
...
If at least 2 of these 10 are
defective, the entire lot of 20 computers will be returned
...
A production lot of 2000 units contains 50 units that do not meet the specifications
...
A poisson experiment is one that possesses the following properties:
the number of outcomes occurring in one time interval or specified region
is independent of the number that occur in any other disjoint time interval or
region of space
• the probability that a single outcome will occur during a very short time
interval or in a small region is proportional to the length of the time interval
• the probability that more than one outcome will occur in such a short time
interval or fall in such a small region is negligible
•

The random variable of interest X, the number of outcomes in a specified length of
time interval or region, is called a Poisson random variable
...
The probability distribution of the Poisson random variable is given by
e− µ µ x
, x = 0,1,2,…
...


Note:

If X~Poi(µ), then E(X) = µ and Var(X) = µ
...


Examples:
1
...
Suppose that
the number of accidents per month follows a Poisson distribution, what is the probability that
in any given month at this intersection,
a
...
less than 3 accidents will occur?
c
...
Suppose that an average of 3 cars arrive at a highway tollgate every min
...
exactly 5 cars will arrive in a 1-min period?
b
...
The probability that a person dies from a certain respiratory infection is 0
...
Find the
probability that fewer than 5 in a random sample of 2000 so infected will die
...


If X~Bi(n, p) with mean np and variance npq, then the distribution of
Z=

X − np
npq

as n approaches ∞ will approximate the standard normal distribution
...
The normal distribution gives a very good approximation of the Binomial distribution when n
is large and p is close to 1/2
...
Since a continuous distribution (in this case, the Normal) is used to approximate a discrete
distribution, then we must adjust for continuity
...

 (a − 0
...
5) − np 
P ( X = a ) ≈ P
< Z<


npq
npq


Example:
A certain pharmaceutical company knows that, on the average, 45% of a certain type of pill has
an ingredient that is below the minimum strength and thus unacceptable
...


The probability distribution function of a statistic is called its sampling
distribution
...
g
...


•

The sampling distribution of a statistic will depend on the size of the population, the size
of the sample, and the method of choosing the sample
...
It tells us the extent to which we expect the values of the statistic to vary from
different possible samples
...


Sampling Distribution of the Mean
Consider 4 observations making up the population values of a random variable X having the
probability distribution
f(x) = 1/ 4 , x = 0, 1, 2, 3
Note that µ = E(X) = 3/2 and σ2 = Var(X) = 5/4
...

1
2
3
4
5
6
7
8

Sample
0, 0
0, 1
0, 2
0, 3
1, 0
1, 1
1, 2
1, 3

X
0
...
5
1
...
5
0
...
0
1
...
0

No
...
0
1
...
0
2
...
5
2
...
5
3
...
5
1
...
5
2
...
5
3
...


Theorems:
1
...

2
...

E( X ) = µ and Var( X ) =
n  N − 1
 N − n
 in the formula of the variance of X is called the finite population
• The factor 
 N − 1
correction factor
...

3
...

Hence, the limiting form of the distribution of
X− µ
Z=
σ n
as n approaches infinity is the standard normal distribution
...


•

If n < 30, the approximation is good only if the population is not too different from the
normal
...


Example:
An electrical firm manufactures electric light bulbs that have a length of life which is
normally distributed with mean and standard deviation equal to 500 and 50 hours,
respectively
...

4
...

If X and S2 are the mean and variance, respectively, of a random sample of size n taken
from a population which is normally distributed with mean µ and variance σ2 , then
X− µ
T=
S n
is a random variable having the t - distribution with v = n-1 degrees of freedom
...

2
...

•

When the sample size is large, i
...
n ≥ 30, the t-distribution can be well
approximated by the standard normal distribution
...
That
is, if T~t(v) then tα is such that P(T> tα) = α
...
Find the following values on the t -table:
(a) t0
...

(b) t0
...

2
...
807) = 0
...
A manufacturing firm claims that the batteries used in their electronic games will
last an average of 30 hours
...
If the computed t-value falls between -t0
...
025, the firm is satisfied with
its claim
...
5 hours and standard deviation S = 5 hours? Assume the distribution of battery
lives to be approximately normal
...
Statistical inference refers to methods by which one uses sample
information to make inferences or generalizations about a population
...
Estimation
- point estimation
- interval estimation
2
...
An estimator is any statistic whose value is used to estimate an unknown
parameter
...

For example, the sample mean X , is an estimator of the population mean µ
...
An estimator is said to be unbiased if the average of the estimates it produces under
repeated sampling is equal to the true value of the parameter being estimated
...

Under random sampling with replacement, S2 is an unbiased estimator of
σ2, but S on the other hand is a biased estimator of σ with the bias
becoming insignificant for large samples
...
A parameter can have more than one unbiased estimator
...


Interval Estimation
Definition
...
This pair of numbers, (a,b), is called
an interval estimate or confidence interval
...


The running time (in minutes) of a sample of films produced by Star-Regal
Theater are as follows: 103 94 110 87 98
...
6, 109
...

The number 0
...

•

The endpoints 87
...
2 are called the lower and upper confidence
limits
...
In general, we construct a (1-α)100% confidence interval
...

2
...

3
...
Rather, the confidence coefficient is “the probability that the
interval estimator encloses the true value of the parameter
...
A good confidence interval is one that is as narrow as possible and has a large
confidence coefficient, near 1
...

However, for a fixed sample size, as the confidence coefficient increases, the length
of the interval also increases
...

(1-α) 100% Confidence Interval for µ
a
...

b
...

Remarks:
1
...

However, they provide good approximate (1-α)100% confidence intervals when the
distribution is not normal provided the sample size is large, i
...
n > 30
...
If σ2 is unknown and n > 30, use

 X − zα


S
/2

n

, X + zα

/2

S 

n

where zα/2 is the z-value leaving an area of α/2 to the right
...
An electrical firm manufactures light bulbs that have a length of life that is normally
distributed, with a standard deviation of 40 hours
...

2
...
M
...
D
...
In a random sample of 20 similar servings of AlphaBits, the mean sugar content was 11
...
45 grams
...

3
...
Construct a 99% confidence interval for the average
number of kilometers an automobile is driven annually in Virginia
...

Types of Sampling:
•
•

selecting two independent samples
paired sampling

Paired sampling is used to overcome the difficulty imposed by extraneous differences
between two groups when testing the difference between 2 means
...
Matching may be achieved by:
using the same subject in the 2 samples
pairing of subjects with respect to any extraneous variable which
might affect or influence the outcome
...
σ

2
1

and σ

2
2

known


 (X 1 − X 2 ) − z
α


b
...
σ 12 ≠ σ

/2

σ 12 σ 22
+
, ( X 1 − X 2 ) + zα
n1 n 2

v=

(S

(S

2
1

2
1

S12 S 22
+
, ( X 1 − X 2 ) + tα
n1 n2

/ 2(v)

n1 + S 22 n 2

)

2

(

)

/ 2( v )

S12 S 22 
+
n1 n 2 

2

)

n1
S2 n
+ 2 2
n1 − 1
n2 − 1

2

Remarks:
1
...
However, they provide good approximate (1-α)100% confidence
intervals when the distributions are not Normal provided both n 1 and n2 are
greater than 30
...
If σ 12 and σ

2
2

are unknown but n1 and n2 are greater than 30, use


 (X 1 − X 2 ) − z
α



/2

S12 S 22
+
, ( X 1 − X 2 ) + zα
n1 n2

/2

S12 S 22 
+
n1 n2 

3
...
Therefore, in a planned experiment, one should make every effort to
equalize the size of the samples
...
A statistics test was given to a random sample of 50 girls and another random sample
of 75 boys
...
Find a 95% confidence
interval for the difference µB - µG
...
Students may choose between a 3-unit course in Physics without lab and a 4-unit
course with lab
...
The
mean score of a random sample of 12 students in the section with lab is 84 with a
standard deviation of 4, and the mean score of another random sample of 18 students
in the section without lab is 77 with a standard deviation of 6
...
Assume the
populations to be approximately normally distributed with equal variances
...
The following data represent the running time of a random sample of films produced
by two motion picture companies:
Time (minutes)
Company 1
Company 2

103
97

94
82

110
123

87
92

98
175

88

118

Compute a 90% confidence interval for the difference between the mean running time
of films produced by the two companies
...


•

Based on Two Related/Paired Samples

 d − tα


Sd
/ 2(v )

n

, d + tα

/ 2(v )

Sd 

n

where di = xi - yi
n

d=

∑

i= 1

n

v = n-1

 n 
n∑ d −  ∑ d i 
i= 1
 i= 1 
n(n − 1)
n

di
Sd =

2

2
i

n = number
...
It is claimed that a new diet will reduce a person’s weight by 4
...
The weights of a random sample of 7 women who
followed this diet were recorded before and after a 2-week period:
Woman

Weight Before58
...
0

1

2

3

4

5

6

60
...
9

61
...
1

69
...
1

64
...
5

62
...
9

56
...
4

7

Compute a 95% confidence interval for the mean difference in the weight
...

2
...
One of each pair was selected at random and
assigned to a mathematics section using programmed materials only
...
At the
end of the semester each group was given the same examination and the following
results were recorded
...
Assume normality
...

If the unknown proportion is not expected to be too close to 0 or 1 and n is large, an
approximate (1-α)100% confidence interval for p is given by

 pˆ − zα



/2

pˆ qˆ
, pˆ + zα
n

/2

pˆ qˆ 

n 

74

Example:
In a random sample of 200 students who enrolled in Math 17, 138 passed on their
first take
...


ESTIMATING THE DIFFERENCE OF TWO PROPORTIONS
Given 2 independent random samples of size n1 and n2 , a point estimator of the
X Y
−
difference between the two proportions p1 and p2 is given by pˆ 1 − pˆ 2 =
,
n1 n2
where X is the number of successes in n1 trials (first sample) and Y is the number of
successes in n2 trials (second sample)
...
Construct a 95% confidence interval for p1- p2, where
p1 and p2 are the true proportions of females and males, respectively, who passed Math 17
on their first take
...
How large a sample is
needed if we wish to be 95% confident that the sample mean will be within 10 hours of
the true mean?

Sample Size for Estimating p
If pˆ will be used to estimate p, then we can be (1-α)100% confident that the error
will not exceed a specified amount, e, when the sample size is
z 2 pq
n = α / 22
e
When the value of p is unknown or cannot be approximated, then using p=0
...
25
...
05 of the true value
...
A statistical hypothesis is an assertion or conjecture concerning one or more populations
...
The null hypothesis (Ho) is the hypothesis that is being tested; it represents what the
experimenter doubts to be true
...
The alternative hypothesis (Ha) is the operational statement of the theory that the
experimenter believes to be true and wishes to prove
...

4
...

Examples:
a
...

c
...


Ho: µ = 14
Ho: µ = 14
Ho: µ1 - µ2 = 0
Ho: µ1 - µ2 = 0

vs
...

vs
...


Ha: µ > 14
Ha: µ < 14
Ha: µ1 - µ2 > 0
Ha: µ1 - µ2 < 0

A two-tailed test of hypothesis is a test where the alternative hypothesis does not specify a
directional difference for the parameter of interest
...
Ho: µ = 14
b
...

vs
...
A test statistic is a statistic whose value is calculated from sample measurements and on
which the statistical decision will be based
...
The critical region or rejection region is the set of values of the test statistic for which the
null hypothesis will be rejected
...
The acceptance and rejection
regions are separated by a critical value of the test statistic
...
The Type I error is the error made by rejecting the null hypothesis when it is true
...


77

The Type II error is the error made by accepting (not rejecting) the null hypothesis when it
is false
...

Null Hypothesis
True
False

Decision
Reject Ho

Type I error

Accept Ho Correct decision

Correct decision
Type II error

8
...


Steps in Hypothesis Testing
1
...

3
...

5
...

Choose the level of significance α
...

Collect the data and compute the value of the test statistic from the sample data
...
Reject Ho if the value of the test statistic belongs in the critical region
...


TESTING A HYPOTHESIS ON THE POPULATION MEAN
Ho
a
...
σ unknown
µ = µo

t=

Remarks:
1
...
However,
they provide good approximate α-level test when the distribution is not normal provided that
the sample size is large, i
...
n > 30
...
If σ is unknown and n > 30, use the test in (a) replacing the test statistic by
X− µo
Z=
S n
78

Examples:
1
...
Ha: µ≠50 if a random sample of 16 subjects had mean 48 and standard
deviation of 5
...
05 level of significance
...

2
...

To test this claim, a random sample of 100 automobile owners are asked to keep a record of
the kilometers they travel
...
01 level
of significance
...
According to Dietary Goals for the United States (1977), high sodium intake may be related
to ulcers, stomach cancer, and migraine headaches
...
A
random sample of 20 similar servings of Special K had mean sodium content of 244
milligrams of sodium and a standard deviation of 24
...
Is there sufficient
evidence to believe that the average sodium content for single servings of Special K exceeds
the human requirement for salt at α=0
...
05? at α = 0
...

The following remarks hold for any test:
1
...

Consequently, if Ho is rejected at α-level of significance then Ho will also be rejected at a
higher level of significance using the same data
...
05
then testing at α = 0
...
However, Ho will not necessarily
be rejected at α = 0
...

2
...
For a fixed sample size n, a decrease in the
probability of one will result in an increase in the probability of the other
...

3
...
The p-value is the
smallest value of α for which Ho will be rejected based on sample information
...
In particular, it enables each reader to choose their personal value of
α
...
Otherwise, Ho is not rejected
...
σ

2
1

• Based on 2 independent samples
Ho
Test Statistic
2
and σ 2 known

µ1 - µ2 = do
b
...
σ

Ha

The remarks made in Chapter 8
...


Examples:
1
...
The girls made an average of 80 with a
standard deviation of 4 and the boys had an average of 86 with a standard deviation of 6
...
05 level of significance that the average grades of girls and
boys differ?
2
...
Students were allowed to choose between a 3-unit
course without lab and a 4-unit course with lab
...
7, and in the section
without lab, a sample of 17 students had an average grade of 79 with a standard deviation of
80

6
...
Would you say that the laboratory course increases the average grade by more than 5
points? Use a 0
...

3
...
Use a 0
...

4
...
Twelve cars were driven twice over a prescribed test course,
each time using a different type of tires (radial and belted) in random order
...
2
4
...
6
7
...
7
4
...
7
6
...
4
4
...
1
5
...
1
4
...
2
6
...
8
4
...
7
5
...
9
4
...
0
4
...
025 level of significance, can we conclude that cars equipped with radial tires give
better fuel economy than those equipped with belted tires? Assume the populations to be
normally distributed
...

If the unknown proportion is not expected to be too close to 0 or 1 and n is large, a large
sample approximation is given by:
Ho
p = po

Test Statistic
x − npo
Z=
npo qo

Ha
p < po
p > po
p ≠ po

Critical region
z < - zα
z > zα
| z | > zα/2

Example:
A commonly prescribed drug on the market for relieving nervous tension is believed to
be only 60% effective
...
Is this sufficient evidence to conclude that the new drug is superior to the one
commonly prescribed? Use a 0
...


TESTING THE DIFFERENCE BETWEEN TWO PROPORTIONS
Consider a situation in which a researcher wishes to compare the proportions of an attribute
between two populations
...
Thus, the researcher is, in general, interested in testing the null hypothesis Ho:
p1 = p2, where p1 and p2 are the two population proportions of interest
...
The sample proportions p1 and p 2 are computed and the common
x1 + x2
(population) proportion p is given as the pooled estimate p =
where x1 and x2 are the
n1 + n2
observed number of units possessing the attribute of interest in the two samples
...
Will you agree that the proportion of
males who passed Math 17 on their first take is higher than the proportion of females who
passed the same course on their first take? Test at α=0
...


TEST FOR INDEPENDENCE
The test for independence is used to determine whether two variables are related or not
...
We then take a random sample and for each subject determine his music
preference and classify his IQ into different categories (high, medium, low)
...
The row
and column totals are called marginal frequencies
...

Procedure:
1
...

Ho: The two variables are independent
Ha: The two variables are not independent
...
Choose the level of significance
...
Compute the test statistic, given by
83

χ

=

2

r

c

∑∑

( Oij − Eij ) 2

Eij
where Oij= observed number of cases in the ith row of the jth column
Eij = expected number of cases under Ho
( column total) x( row total)
=
grand total
i= 1 j= 1

4
...


Remarks:
1
...

2
...
Generally, one should not pool
categories unless there is a natural way to combine them
...
For a 2x2 contingency table, a correction called Yates’ correction for continuity is applied
...
5)
ij
2
χ = ∑ ∑
Eij
i= 1 j= 1

Example:
Using the table above:
Ho: Music preference and intelligence are independent
Ha: Music preference and intelligence are not independent
Music
Preference
Classical
Pop
Rock
Total

IQ
High
40
47
83

(29
...
4)
(94
...
7)
(51
...
7)
189

Low
17
25
79

(20
...
0)
(67
...
05 level of significance that music preference and intelligence
are not independent?

84

Exercises

85



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Title: Notes on Basic statistics and Probability
Description: This provides descriptions and examples regarding topics in statistics. From simple topics such as measures of central to tendency to probability distributions and methods for hypothesis testing. Intended for highscool and college level students

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