1
5.1
DATA SUMMARY AND DISPLAY
Statistics ???
Meaning :
Numerical facts
Field or discipline of study
Collection of methods for planning experiments, obtaining data and
organizing, analyzing, interpreting and drawing the conclusions or
making a decision.
2
BASIC TERMS IN STATISTICS
Population
-
Entire collection of individuals which are characteristic being
studied.
Sample
-
Subset of population.
Population
Sample
3
Census
-
Survey includes every member of population.
Sample survey
-
Collecting information from a portion of population (techniques)
Element
-
Specific subject or object about which information collected.
Variable
-
Characteristics which make different values.
4
Observation
-
Value of variable for an element.
Data Set
-
A collection of observation on one or more variables.
Table 1: Student’s Score for Business Statistic
Element
Name
Score
Mohd Amirul bin Hamdi
90
Hashimah
78
Variable
Observation/
Measurement
5
TYPES OF VARIABLES
Variable
Quantitative
Discrete
(e.g, number of
houses, cars
accidents
Qualitative
Continuous (e.g.,
length, age, height,
weight, time)
e.g., gender,
marital status
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QUANTITATIVE AND QUALITATIVE
VARIABLE
1) Quantitative variable
A variable that can be measured numerically.
Data collected on a quantitative variable are called quantitative data.
There are two types of quantitative variables:i. Discrete Variable
A variable whose values are countable, can assume only
certain values with no intermediate values.
ii. Continuous Variable
A variable that can assume any numerical value over a
certain interval or intervals.
2) Qualitative variable
A variable that cannot assume a numerical value but can be classified into
two or more nonnumeric categories.
Data collected on such a variable are called qualitative data.
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STATISTICS
DESCRIPTIVE
STATISTICS
Using tables, graphs &
summary
measures
INFERENTIAL
STATISTICS
Using sample result in
making decision or
predict about a
population.
Also called inductive
reasoning
or inductive statistics.
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Descriptive Statistics
Consists of methods for organizing, displaying and describing data
by using tables, graphs and summary measures.
In general divided by two categories :- Data presentation (display)
- Statistics
9
Inferential Statistics
Consists of methods that use sample results to help make decisions
or predictions about a population.
Area statistics which are deal with decision making procedures.
Example :- In order to find the salary of a college graduate, we may
select 2000 recent college graduates, find the starting salaries
and make decision based on the information.
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DATA PRESENTATION
A data with a lot of observations usually looks non
informative
-
We cannot get much information with the raw data
We have to summarize or organize in such a way so that
we can get some information about the data.
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DATA PRESENTATION OF QUALITATIVE DATA
Tabular presentation for qualitative data is usually in the
form of frequency table
Frequency table- table represent the number of times the
observation occurs in data
A graphic display can reveal at a glance the main
characteristics of a data set.
Three types of graphs used to display qualitative data:- bar graph
- pie chart
- line chart
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Example 5.1
Table 5.1 shows that the data of 50 UNIMAP students with
their data and background.
Code used :
• For gender: 1 is male and 2 is female
• For ethnic group: 1 is Malay, 2 is
Chinese, 3 is Indian and 4 is others
• Not much information can be
obtained from the data 1 in the raw
form. It has to be summarized so that
we can get more informations.
13
If data from table 5.1 summarized into gender and ethnic
group, then the frequency tables can get as below :
Observation
Frequency
Male
28
Female
22
Total
50
Table 5.2: Frequency Table for the Gender
Observation
Frequency
Malay
33
Chinese
9
Indian
6
Others
2
Total
50
Table 5.3: Frequency Table for the Ethnic Group
14
Bar Chart
 Bar chart is used to display the frequency distribution in the
graphical form. It consists of two orthogonal axes and one of
the axes represent the observations while the other one
represents the frequency of the observations. The frequency of
the observations is represented by a bar.
*Bar chart is for data from Table 5.3.
Figure 1: Bar Chart of the Ethnic Group
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2.1.2 Pie Chart
Pie Chart is used to display the frequency distribution. It
displays the ratio of the observations. It is a circle consists
of a few sectors. The sectors represent the observations
while the area of the sectors represent the proportion of
the frequencies of that observations.
*Pie chart is for data from Table 5.2.
Figure 2: The Pie Chart for the Gender
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2.1.3 Line Chart
 Line chart is used to display the trend of observations. It consists of
two orthoganal axes and one of the axes represent the observations
while the other one represents the frequency of the observations. The
frequency of the observations are joint by lines.
Example :
Table 2.4 below shows the number of sandpipers recorded between
January 1989 till December 1989.
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
10
7
5
10
39
7
260
316
142
11
4
9
Table 2.4 : The number of sandpipers
Figure 3: The line Chart for the numbers of common Sandpipers
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DATA PRESENTATION OF QUANTITATIVE DATA
Tabular presentation of quantitative data is usually in the form of frequency
distribution
Frequency distribution – table that represents the frequency of the observation that
fall inside some spesific classes (intervals).
The are a few graph available for graphical presentation of the quantitative data. The
most popular are:
-
Histogram
-
Frequency polygon
-
Ogive
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FREQUENCY DISTRIBUTION
When summarizing large quantities of raw data, it is often useful to distribute the
data into classes. In determining the classes, there is no spesific rules but
statistician suggest the number of classes are between 5 to 20
Sturges’s Rule
Number of classes , c=1+3.3 log n
Where n is the numbers of observations in the data set.
Class width:
Largest value-smallest value
Number of classes
Range
i
c
i
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Example 5.2
CGPA (Class)
2.50 - 2.75
2.75 - 3.00
3.00 - 3.25
3.25 - 3.50
3.50 - 3.75
3.75 - 4.00
Total
Frequency
2
10
15
13
7
3
50
Table 5.5: The Fequency Distribution of
the Students’ CGPA
20
Cumulative Frequency Distributions
A cumulative frequency distribution gives the total number of values that fall
below the upper boundary of each class.
In cumulative frequency distribution table, each class has the same lower limit
but a different upper limit.
Table5.7 : Class Limit, Class Boundaries, Class Width , Cumulative Frequency
Weekly Earnings
(dollars)
(Class Limit)
Number of
Employees, f
Class Boundaries
Class Width
Cumulative Frequency
801-1000
9
800.5 – 1000.5
200
9
1001-1200
22
1000.5 – 1200.5
200
9 + 22 = 31
1201-1400
39
1200.5 – 1400.5
200
31 + 39 = 70
1401-1600
15
1400.5 – 1600.5
200
70 + 15 = 85
1601-1800
9
1600.5 – 1800.5
200
85 + 9 = 94
1801-2000
6
1800.5 – 2000.5
200
94 + 6 = 100
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 Histogram
The histogram looks like the bar chart except that the
horizontal axis represent the data which is quantitative in
nature. There is no gap between the bars.
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 Frequency Polygon
The frequency polygon looks like the line chart except that the
horizontal axis represent the class mark of the data which is
quantitative in nature.
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 Ogive
Ogive is a line graph with the horizontal axis represent the
upper limit of the class interval while the vertical axis represent
the cummulative frequencies.
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DATA SUMMARY
What is statistic?
Statistis is a number that describe the sample such as sample
mean which describe the sample average.
Type of statistic
i.
Measure of central tendency
ii. Measure of dispersion
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MEASURE OF CENTRAL TENDENCY
There are 3 popular central tendency measures, mean, median &
mode.
1) Mean
The mean of a sample is the sum of the measurements divided by
the number of measurements in the set. Mean is denoted by ( )
Mean = Sum of all values / Number of values
Mean can be obtained as below :-
- For raw data, mean is defined by,
_
x
x1  x2  .......  xn

x
, for n  1,2,..., n or x 
n
n
_
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For tabular/group data, mean is defined by:
n

x 

i 1
n
f i xi

i 1
or
fi
 fx
f
Where f = class frequency;
x = class mark (mid point)
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Example
The mean sample for students CGPA (raw) is
x
x
n

160.98
 3.22
50
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Example :
The mean sample for Table 5.8
Frequency, f
Class Mark
(Midpoint),
x
2.50 - 2.75
2
2.625
5.250
2.75 - 3.00
10
2.875
28.750
CGPA (Class)
fx
n

x
3.00 - 3.25
15
3.125
46.875
3.25 - 3.50
13
3.375
43.875
3.50 - 3.75
7
3.625
25.375
3.75 - 4.00
3
3.875
11.625
Total
50
f
i
xi
i 1
n
f

161 .75
 3.235
50
i
i 1
161.750
Table 5.8
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2) Median
 Median is the middle value of a set of observations arranged in
order of magnitude and normally is devoted by ~
x
i) The median for ungrouped data.
- The median depends on the number of observations in the
data, .
- If n
is odd, then the median is the(
ordered
n observations.
n 1
)
2
th observation of the
- If n
is even, then the median is the arithmetic mean of the
n
n
th observation and the (  1) th observation.
2
2
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ii) The median of grouped data / frequency of distribution.
The median of frequency distribution is defined by:
f F 
j 1

x  L  c 2
fj




~
where,
• L = the lower class boundary of the median class;
• c = the size of the median class interval;
• Fj 1= the sum of frequencies of all classes lower than the median
class
•
fj =
the frequency of the median class.
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Example for ungrouped data :The median of this data 4, 6, 3, 1, 2, 5, 7, 3 is 3.5.
Proof :- Rearrange the data in order of magnitude
becomes 1,2,3,3,4,5,6,7. As n=8 (even), the
median is the mean of the 4th and 5th
observations that is 3.5.
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Example for grouped data :Find median for frequency distribution below
Cum.
CGPA (Class)
Frequency, f
frequency
2.50 - 2.75
2
2
2.75 - 3.00
10
12
3.00 - 3.25
15
27
f F 
j 1

x  L  c 2
fj




~
 25  12 
Median , x  3.00  0.25
 3.217

 15 
~
3.25 - 3.50
13
40
3.50 - 3.75
7
47
3.75 - 4.00
3
50
Total
50
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3) Mode
• The mode of a set of observations is the observation with the
highest frequency and is usually denoted by ( 
). Sometimes
x
mode can also be used to describe the qualitative
data.
i)
Mode of ungrouped data :- Defined as the value which occurs most frequent.
- The mode has the advantage in that it is easy to calculate and
eliminates the effect of extreme values.
- However, the mode may not exist and even if it does exit, it
may not be unique.
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*Note:
If a set of data has 2 measurements with higher frequency,
therefore the measurements are assumed as data mode and
known as bimodal data.
If a set of data has more than 2 measurements with higher
frequency so the data can be assumed as no mode.
ii) The mode for grouped data/frequency distribution data.
- When data has been grouped in classes and a frequency curve
is drawn to fit the data, the mode is the value of corresponding
to the maximum point on the curve.
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ii) The mode for grouped data/ frequency distribution data
 1 
x  L  c

 1   2 

where
L = the lower class boundary of the modal class;
c = the size of the modal class interval;
1 = the difference between the modal class frequency and the class
before it;
and
 2 = the difference between the modal class frequency and the class
after it.
*Note:
- The class which has the highest frequency is called the modal
class.
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Example for ungrouped data :
The mode for the observations 4,6,3,1,2,5,7,3 is 3.
Example for grouped data based on table :
CGPA (Class)
Modal
Class
Frequency
2.50 - 2.75
2
2.75 - 3.00
10
3.00 - 3.25
15
3.25 - 3.50
13
3.50 - 3.75
7
3.75 - 4.00
3
Total
50
 1 
x  L  c
  3.179
 1   2 

 1 
5
x  L  c

3
.
00

0
.
25
(
)  3.179

52
 1   2 

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Measure of Dispersion
The measure of dispersion/spread is the degree to which a set of
data tends to spread around the average value.
It shows whether data will set is focused around the mean or
scattered.
The common measures of dispersion are:
1) range
2) variance
3) standard deviation
The standard deviation actually is the square root of the variance.
The sample variance is denoted by s2 and the sample standard
deviation is denoted by s.
38
39
Variance
i) Variance for ungrouped data
 The variance of a sample (also known as mean square) for the
raw (ungrouped) data is denoted by s2 and defined by:

2
(
x

x
)
S2  
n 1
ii) Variance for grouped data

The variance for the frequency distribution is defined by:

fx

 fx  n
2
S2 
2
f
(
x

x
)

 fx  1
2

n 1
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Example for ungrouped data :
given income for 5 workers are : RM 1000, RM 2500, RM 2000, RM 4000, RM 3500. Find
variance of this data.
Solution:
1000  2500  2000  4000  3500
5
 2600
Mean, x 

Variance, S
2
 ( x  x)

2
n 1
1000  2600    2500  2600    2000  2600    4000  2600   3500  2600 

2
2
2
2
2
5 1
5700, 000
4
 142500

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Example for grouped data :
The variance for frequency distribution in Table is:
Class
boundaries
Frequency, f
Class Mark,
x
2.50 - 2.75
2.75 - 3.00
3.00 - 3.25
3.25 - 3.50
3.50 - 3.75
3.75 - 4.00
2
10
15
13
7
3
2.625
2.875
3.125
3.375
3.625
3.875
Total
50
S2 
 fx
2
f x



n 1
n
2
fx
fx2
5.250
28.750
46.875
43.875
25.375
11.625
13.781
82.656
146.484
148.078
91.984
45.047
161.750
528.031
(161.75) 2
528.031 
50

 0.0973
49
42
ESTIMATION
Introduction
The field of statistical inference consist of those
methods used to make decisions or to draw
conclusions about a population. These methods
utilize the information contained in a sample
from the population in drawing conclusions
43
ESTIMATOR VS ESTIMATE
Estimator
• In statistics, the method used
Estimate
• The value that obtained from a sample
I have a sample of 5 numbers and I take the average. The
estimator is taking the average of the sample.
The estimator of the mean.
Let say, the average = 4
the estimate.
44
CONFIDENCE INTERVAL ESTIMATES
Definition : An Interval Estimate
In interval estimation, an interval is constructed around
the point estimate and it is stated that this interval is
likely to contain the corresponding population
parameter.
Definition : Confidence Level and Confidence Interval
Each interval is constructed with regard to a given confidence
level and is called a confidence interval. The confidence level
associated with a confidence interval states how much confidence
we have that this interval contains the true population parameter. The
confidence level is denoted by
45
CONFIDENCE INTERVAL ESTIMATES FOR
POPULATION MEAN
The (1 - a )100% Confidence Interval of Population Mean, m
(i) x ± za
s
2
n
if s is known and normally distributed population
æ
s
s ö
÷
or çç x - za
< m < x + za
÷
2
2
n
nø
è
s
if s is unknown, n large (n ³ 30)
2
n
æ
s
s ö
÷
or çç x - za
< m < x + za
÷
2
2
n
n
è
ø
(ii) x ± za
46
s
(iii) x ± tn - 1,a
if s is unknown, normally distributed population
2
n
and small sample size ( n < 30)
æ
s
s ö
÷÷
or çç x - tn - 1,a
< m < x + tn - 1,a
2
2
n
nø
è
47
EXAMPLE
If a random sample of size n = 20 from a normal population
with the variance s 2 = 225 has the mean x = 64.3, construct
a 95% confidence interval for the population mean, m.
48
SOLUTION
It is known that, n = 20, m = x = 64.3 and s = 15
For 95% CI,
95% = 100(1 – a )%
1 –a = 0.95
a = 0.05
a
= 0.025
2
za = z0.025 = 1.96
2
49
æs ö
÷
Hence, 95% CI = x ± za çç
÷
n
ø
2 è
æ 15 ö
÷
= 64.3 ± 1.96 çç
÷
è 20 ø
= 64.3 ± 6.57
= [57.73, 70.87]
@
57.73 < m < 70.87
Thus, we are 95% confident that the mean of random variable
is between 57.73 and 70.87
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Example :
A publishing company has just published a new textbook.
Before the company decides the price at which to sell this
textbook, it wants to know the average price of all such
textbooks in the market. The research department at the
company took a sample of 36 comparable textbooks and
collected the information on their prices. This information
produced a mean price RM 70.50 for this sample. It is known
that the standard deviation of the prices of all such textbooks
is RM4.50. Construct a 90% confidence interval for the mean
price of all such college textbooks.
51
solution
It is known that, n = 36, m = x = RM70.50 and s = RM 4.50
For 90% CI,
90% = 100(1 – a )%
1 –a = 0.90
a = 0.1
a
= 0.05
2
za = z0.05 = 1.65
2
52
æs ö
÷
Hence, 90% CI = x ± za çç
÷
n
ø
2 è
æ4.50 ö
÷
= 70.50 ± 1.65 çç
÷
36
è
ø
= 70.50 ± 1.24
= [ RM 69.26, RM 71.74]
Thus, we are 90% confident that the mean price of all such
college textbooks is between RM69.26 and RM71.74
53
EXAMPLE
Consider a survey on male students height in a certain IPTA: a
random sample of 100 male students are taken. The height of the
male students is normally distributed with mean 178.2 cm and
variance 17.75 cm2.
i)
Construct a 95% CI for the mean of male students height
ii) If mean of the female students height is 170.2 cm height, at 98%
CI, verify whether if this can proof that the male are taller than
the female students.
54
SOLUTION
It is known that
n  100
x  178.2
 2  17.75
For 95 % CI
95%  1   100%
1    0.95
  0.95

 0.025
2
z  z0.05  1.96
2
55
Hence 95% CI;
  
 x  z 

n
2 
 17.75 
 178.2  1.96 

 100 
 178.2  0.83
 177.37,179.03
56
ii)
It is known that   x  178.2 and  2  17.75
For 98 % CI 98%  1   100%
1    0.98
  0.02 thus

2
 0.01
z  2.33
2
Hence, 98% CI
  
 x  z 

n
2 
 17.75 
 178.2  2.33 

100


 178.2  0.98  177.22,179.18
We can see that mean of female students does not lies in the interval 177.22,179.18
hence this indicate that the male students are taller than female students.
57
CONFIDENCE INTERVAL ESTIMATES FOR POPULATION
PROPORTION
The (1 - a )100% Confidence Interval for p for Large Samples (n ³ 30)
pˆ ± za
2
pˆ (1 - pˆ )
n
or
pˆ - za
2
pˆ (1 - pˆ )
< p < pˆ + za
2
n
pˆ (1 - pˆ )
n
58
Example
According to the analysis of Women Magazine in June
2005, “Stress has become a common part of everyday life
among working women in Malaysia. The demands of work,
family and home place an increasing burden on average
Malaysian women”. According to this poll, 40% of working
women included in the survey indicated that they had a
little amount of time to relax. The poll was based on a
randomly selected of 1502 working women aged 30 and
above. Construct a 95% confidence interval for the
corresponding population proportion.
59
Solution
Let p be the proportion of all working women age 30 and above,
who have a limited amount of time to relax, and let pˆ be the
corresponding sample proportion. From the given information,
n = 1502 , pˆ = 0.40, qˆ = 1 - pˆ = 1 – 0.40 = 0.60
æ pq
ö
ˆˆ÷
ç
Hence, 95% CI = pˆ ± za ç
n ÷
2 è
ø
æ 0.40(0.60) ö
÷
= 0.40 ± 1.96 çç
1502 ÷
è
ø
= 0.40 ± 0.02478
= [0.375, 0.425] or 37.5% to 42.5%
Thus, we can state with 95% confidence that the proportion of all
working women aged 30 and above who have a limited amount of
time to relax is between 37.5% and 42.5%.
60
EXERCISE
61
HYPOTHESIS TESTS
Hypothesis and Test Procedures
A statistical test of hypothesis consist of :
1. The Null hypothesis, H 0
2. The Alternative hypothesis,
H1
3. The test statistic and its p-value
4. The rejection region
5. The conclusion
62
Definition
Hypothesis testing can be used to determine whether a statement
about the value of a population parameter should or should not
be rejected.
Null hypothesis, H0 : A null hypothesis is a claim (or statement)
about a population parameter that is assumed to be true.
(the null hypothesis is either rejected or fails to be rejected.)
Alternative hypothesis, H1 : An alternative hypothesis is a claim
about a population parameter that will be true if the null
hypothesis is false.
63
Test Statistic is a function of the sample data on
which the decision is to be based.
p-value is the probability calculated using the
test statistic. The smaller the p-value, the more
contradictory is the data to H 0 .
DEVELOPING NULL AND ALTERNATIVE
HYPOTHESIS
 It is not always obvious how the null and alternative hypothesis should be
formulated.
 When formulating the null and alternative hypothesis, the
nature or purpose of the test must also be taken into
account. We will examine:
1) The claim or assertion leading to the test.
2) The null hypothesis to be evaluated.
3) The alternative hypothesis.
4) Whether the test will be two-tail or one-tail.
5) A visual representation of the test itself.
 In some cases it is easier to identify the alternative hypothesis first. In
other cases the null is easier.
9.1.1 Alternative Hypothesis as a Research Hypothesis
•
Many applications of hypothesis testing involve
an attempt to gather evidence in support of a
research hypothesis.
•
In such cases, it is often best to begin with the
alternative hypothesis and make it the conclusion
that the researcher hopes to support.
•
The conclusion that the research hypothesis is true
is made if the sample data provide sufficient
evidence to show that the null hypothesis can be
rejected.
Example 9.1: A new drug is developed with the goal
of lowering blood pressure more than the existing drug.
•
•
Alternative Hypothesis:
The new drug lowers blood pressure more than
the existing drug.
Null Hypothesis:
The new drug does not lower blood pressure more
than the existing drug.
9.1.2 Null Hypothesis as an Assumption to be Challenged
•
We might begin with a belief or assumption that
a statement about the value of a population
parameter is true.
•
We then using a hypothesis test to challenge the
assumption and determine if there is statistical
evidence to conclude that the assumption is
incorrect.
•
In these situations, it is helpful to develop the null
hypothesis first.
Example 9.2 : The label on a soft drink bottle states
that it contains at least 67.6 fluid ounces.
•
•
Null Hypothesis:
The label is correct. µ > 67.6 ounces.
Alternative Hypothesis:
The label is incorrect. µ < 67.6 ounces.
Example 9.3: Average tire life is 35000 miles.
•
Null Hypothesis: µ = 35000 miles
•
Alternative Hypothesis: µ ≠ 35000 miles
9.1 DEVELOPING NULL AND ALTERNATIVE
HYPOTHESIS
 It is not always obvious how the null and alternative hypothesis should
be formulated.
 When formulating the null and alternative hypothesis, the
nature or purpose of the test must also be taken into
account. We will examine:
1) The claim or assertion leading to the test.
2) The null hypothesis to be evaluated.
3) The alternative hypothesis.
4) Whether the test will be two-tail or one-tail.
5) A visual representation of the test itself.
 In some cases it is easier to identify the alternative hypothesis first. In
other cases the null is easier.
9.1.1 Alternative Hypothesis as a Research Hypothesis
•
Many applications of hypothesis testing involve
an attempt to gather evidence in support of a
research hypothesis.
•
In such cases, it is often best to begin with the
alternative hypothesis and make it the conclusion
that the researcher hopes to support.
•
The conclusion that the research hypothesis is true
is made if the sample data provide sufficient
evidence to show that the null hypothesis can be
rejected.
Example 9.1: A new drug is developed with the goal
of lowering blood pressure more than the existing drug.
•
•
Alternative Hypothesis:
The new drug lowers blood pressure more than
the existing drug.
Null Hypothesis:
The new drug does not lower blood pressure more
than the existing drug.
9.1.2 Null Hypothesis as an Assumption to be Challenged
•
We might begin with a belief or assumption that
a statement about the value of a population
parameter is true.
•
We then using a hypothesis test to challenge the
assumption and determine if there is statistical
evidence to conclude that the assumption is
incorrect.
•
In these situations, it is helpful to develop the null
hypothesis first.
Example 9.2 : The label on a soft drink bottle states
that it contains at least 67.6 fluid ounces.
•
•
Null Hypothesis:
The label is correct. µ > 67.6 ounces.
Alternative Hypothesis:
The label is incorrect. µ < 67.6 ounces.
Example 9.3: Average tire life is 35000 miles.
•
Null Hypothesis: µ = 35000 miles
•
Alternative Hypothesis: µ ≠ 35000 miles
How to decide whether to reject or accept H 0 ?
The entire set of values that the test statistic may assume is
divided into two regions. One set, consisting of values that
support the H1 and lead to reject H 0 , is called the rejection
region. The other, consisting of values that support the H 0 is
called the acceptance region. H0 always gets “=“.
Tails of a Test
Sign in H
0
Sign in H1
Rejection Region
Two-Tailed
Test
=
¹
In both tail
Left-Tailed Right-Tailed
Test
Test
= or ³
= or £
<
>
In the left tail In the right tail
77
Population Mean,  , ( known and unknown  )
Null Hypothesis : H 0 : m = m0
Test Statistic :
x- m
Z=
s
n
• any population,  is known and n is large
or
• normal population,  is known and n is small
Z = x -s m
• any population,  is unknown and n is large
n
t = x -s m
n
v =n- 1
• normal population,  is unknown and n is
small
78
Alternative hypothesis
Rejection Region
H1 : m¹ m0
Z < - za 2 or Z > za 2
H1 : m > m0
Z > za
H1 : m < m0
Z< - za
79
Definition: p-value
The p-value is the smallest significance level at which the null
hypothesis is rejected.
Using the p - value approach, we reject the null hypothesis, H 0 if
p - value < a for one - tailed test
p - value a
<
for two - tailed test
2
2
and we do not reject the null hypothesis, H 0 if
p - value ³ a for one - tailed test
p - value a
³
for two - tailed test
2
2
80
Example
The average monthly earnings for women in managerial and
professional positions is RM 2400. Do men in the same positions
have average monthly earnings that are higher than those for women ?
A random sample of n = 40 men in managerial and professional
positions showed x = RM 3600 and s = RM 400. Test the appropriate
hypothesis using a = 0.01
81
Solution
1.The hypothesis to be tested are,
H 0 : m £ 2400
H1 : m > 2400
2.We use normal distribution n > 30
3. Rejection Region : Z > za ; za = z0.01 = 2.33
4. Test Statistic
Z=
x - m 3600 - 2400
=
= 18.97
s
400
n
40
Since 18.97 > 2.33, falls in the rejection region, we reject H 0
and conclude that average monthly earnings for men in managerial
and professional positions are significantly higher than those for women.
82
POPULATION PROPORTION, P
Null Hypothesis :
Test Statistic
:
H 0 : p = p0
pˆ - p0
Z=
p0 q0
n
Alternative hypothesis
Rejection Region
H1 : p ¹
Z < - za
p0
2
or Z > za
H1 : p > p0
Z > za
H1 : p < p0
Z< - za
2
83
Example
When working properly, a machine that is used to make chips for calculators
does not produce more than 4% defective chips. Whenever the machine
produces more than 4% defective chips it needs an adjustment. To check if
the machine is working properly, the quality control department at the
company often takes sample of chips and inspects them to determine if the
chips are good or defective. One such random sample of 200 chips taken
recently from the production line contained 14 defective chips. Test at the 5%
significance level whether or not the machine needs an adjustment.
84
Solution
The hypothesis to be tested are ,
H 0 : p £ 0.04
H1 : p > 0.04
Test statistic is
pˆ - p0
0.07 - 0.04
Z=
=
= 2.17
p0 q0
0.04(0.96)
200
n
Rejection Region : Z > za
; za = z0.05 = 1.65
Since 2.17 > 1.65, falls in the rejection region, we can reject H 0
and conclude that the machine needs an adjustment.
85
REGRESSION AND CORRELATION
Regression – is a statistical procedure for establishing the
relationship between 2 or more variables.
This is done by fitting a linear equation to the observed data.
The regression line is then used by the researcher to see the
trend and make prediction of values for the data.
There are 2 types of relationship:
 Simple ( 2 variables)
 Multiple (more than 2 variables)
THE SIMPLE LINEAR REGRESSION MODEL
 is an equation that describes a dependent variable (Y) in
terms of an independent variable (X) plus random error 
Y = b 0 + b1 X + e
where,
0
1
= intercept of the line with the Y-axis

= random error
= slope of the line
 Random error,  is the difference of data point from the
deterministic value.
 This regression line is estimated from the data collected by
fitting a straight line to the data set and getting the
equation of the
straight
line,
Ù
Ù
Ù
Y = b 0+ b1 X
LEAST SQUARES METHOD
• The least squares method is commonly used to
determine values for  0 and 1 that ensure a
best fit for the estimated regression line to the
sample data points
• The straight line fitted to the data set is the line:
Ù
Ù
Ù
y = b 0+ b1 x
y is the estimated or predicted value of y for a given value of x. In
other words, the predicted value of the dependent variable y for a
given independent variable x can simply be obtain by substituting
the given value of x.
We can find the least squares estimators  0 and
formula
Ù
b1 =
Ù
1 by using the
Sxy
Sxx
Ù
b 0 = y - b1 x
where
1 n
x   xi
n i 1
1 n
y   yi
n i 1
æ n öæ n ö
çå xi ÷çå yi ÷
n
S xy = å xi yi - è i =1 øè i =1 ø
n
i =1
æ n ö2
çå xi ÷
n
2
S xx = å xi - è i =1 ø
n
i =1
n
S yy = å
i =1
æ n ö2
çå yi ÷
2
yi - è i =1 ø
n
89
EXAMPLE
Suppose we take a sample of seven household from a low moderate
income neighborhood and collect information on their income and food
expenditures for the past month. The information obtained (in hundreds of
Ringgit Malaysia) is given below
Income
Food expenditures
35
9
49
15
21
7
39
11
15
5
28
8
25
9
Find the least squares regression line of food expenditure (Y) on income (X)
90
SOLUTION
Income
Food Expenditure
x
35
49
21
39
15
28
25
y
9
15
7
11
5
8
9
xy
315
735
147
429
75
224
225
x2
1225
2401
441
1521
225
784
625
y2
81
225
49
121
25
64
81
∑ x = 212
∑ y = 64
∑ xy = 2150
∑ x2 = 7222
∑ y2 = 646
Compute
 x,  y, x and y
91
 x  212
x
 y  64
 x  212 =30.2857
n
7
compute  xy and
 xy  2150
x
x
2
y
 y  64  9.1429
n
7
2
 7222
compute S xy and S xx
 n  n 
  xi   yi 
n
 212  64   211.7143
S xy   xi yi   i 1  i 1   2150 
n
7
i 1
2
 n 
2
  xi 
n
212


S xx   xi 2   i 1   7222 
 801.4286
n
7
i 1


compute  0 and 1

1 

S xy
S xx

211.7143
 0.2642
801.4286

 0  y  1 x  9.1429   0.2642  30.2587   1.1414
Thus our regression model is y  1.1414  0.2642x
92
CORRELATION (R)
Correlation measures the strength of a linear relationship
between the two variables. Also known as Pearson’s
product moment coefficient of correlation.
The symbol for the sample coefficient of correlation is r
Formula :

r =
S xy
S xx .S yy
Properties of r :
- 1£ r £ 1
Values of r close to 1 implies there is a strong
positive linear relationship between x and y.
Values of r close to -1 implies there is a strong
negative linear relationship between x and y.
Values of r close to O implies little or no linear
relationship between x and y.
EXAMPLE
Refer example before.
Calculate the value of r and interpret its meaning.
Solution
From example before we know that
S xy  211.7143
and
S xx  801.4286
compute
2
 n 
2
  yi 
n
64


S yy   yi 2   i 1   646 
 60.8571
n
7
i 1
S xy
211.7143
r

 0.9587
S xx S yy
801.4286  60.8571
Since the r value close to 1, implies that there is strong positive linear
relationship between income (x) and food expenditure (y).
95