Parameter and Statistic
Parameter
A numerical value that represents a
certain population characteristic
Statistic
A numerical value that represents a
certain sample characteristic
The average of weight of students from a
population of students in a university
The percentage of defective components in
a population of electrical components
manufactured in a day
Measurement
The average of weight for a sample of
female students selected from all students in
a university
The percentage of defective components in
a sample of 100 electrical components
Parameter
Mean (Average)
Variance
Statistic
x
2
s2
Standard deviation
s
Proportion
p
SZS2017
r
POINT ESTIMATE
INTERNAL ESTIMATE
VS
µ
I want to know the
Ex :
students Suppose
-
I
if
will be
µ
we
value to estimate
F-
g
I
:
took
✓ since
to
we can
estimate
from
154-6,
Eg :(
called
ESTIMATE
there's
number
get
only one
50/50 chance
the
value
the values
instead ,
*
POINT
120 students
sample of
a
a
range
right
Any
from
value
154-6 to
165
165
INTERNAL
-
in the
to
3)
n
possibility
Of
getting
the
right
value
INTERVAL
are
more
the
right
.
valves
chance
,
.
of
µ
interval higher
capture
of µ
3)
* higher
ESTIMATE
CONFIDENCE
since there
.
.
sample
from
of valves
161-8
is
This
(
=
0
of lump
-
estimated using
ONE
use
?
height
mean
.
*
=
value
( E-
E.
T
lover bound
=
b- a
E=b÷
É+E )
T
upper
found
Note that,
the
estimation
error E is
also called
as margin
of error
Ca
i
@nym17
b)
< b
a
two
,
-
sided CI
The term
is a critical value
obtained from the
statistical tables
A
•
•
P x
E
x
E
=
by
bT AT ( directly proportional )
(
of Ad
inversely proportional )
1
In general, for a specific value, let say
0.05 , 95% of the sample means will
fall within this error value on either side of the population mean,
bec
✗
T
2- ✗
t
so
F-
I@nym17
One-Sided Confidence Interval
"
In some cases, a researcher is interested to construct one-sided
confidence interval, namely, one-sided lower bound or one-sided upper only
bound for .
find upper
A one-sided confidence bound for results from replacing 2 by , bound
and
sign by either + or – sign in the confidence interval formula for . to
"&
•
"
"^n#
bound
f
la a)
T
,
the
.
highest
possible
valve
C-
,
T
also
could
be
depending
0
on
b)
situation
ego
(0
shore
,
b)
size
temperature
N
C-
@nym17
,
b)
④
is
0-10 gives
② What
population
is
the
in
the
sample
question ?
size ?
n
=
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?
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slider
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use Formula
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*
if
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t
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:
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form
@nym17
-
I
sided
in
the
shall be
with ✗
.
What
①
if the
the
choose
ask
question
formula from
For
example
then
choose
,
if
0
not
Formula
for
one
above
given
one
*
-
sided
and
bound
(
n
a
=
50
,
formula
the
one
-
sided upper bound
upper
,
a)
.
*mlycaku1ate_
.
✗
?
UB
:
ÉÉouer
CI
2
to ✗ from
LB
sided
.
LB
② change §
-
C-
•
eg
depending
variable
:
)
&
0
the
.
x
.
T
on
bound
temperature
no
.
of
:(
-
people :(
on
&
0
,
,
b)
b)
:
QUESTION 4 [8 MARKS] (Sem 21920)
A study was carried out to investigate atmospheric ammonia due to automobile traffic in
Kuantan, Pahang. The ammonia concentrations (µg/m3) were measured for 7 days at an urban
site in Kuantan. The data are collected and recorded as in Table 3.
Table 3: Ammonia reading for seven days
Day
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Ammonia
concentrations
3.1
3.0
3.7
3.4
3.6
3.2
4.2
3
(µg/m )
(i) Identify the variable involved. Then, determine its level of measurement.
Ammonia concentration
8-
grit can
-
✗
.
Ratio
confidence
level
←
level
.
(2 Marks, CO1PO1)
.
1-0.9=0-1
=
(ii) Construct a 90% confidence interval for the population mean of ammonia concentration.
-
use
your
→
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=
3
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calculator
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t
,
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,
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( 3.4571
=
(
3.
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1517,
(1-9432)
3-
0-4158
TF
)
76251 µg / m3
ur n
unit
t
I
0.05
,
6
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n
=
✗
?
=
1-0.98=0-02
(i) How large the sample size should be if if we wish to be at least 98% confident that the error
in estimating the mean is less than 3% given that the population
standard deviation is 0.4. ←
-
ETO-03
z
o.o ,
=
2.3263
round
QUESTION
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next whole
number
→
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•
962.0750
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963 samples
(4 Marks, CO2PO3)
=
The average heights of a random sample of 400 people from a city is 1.75 m. It is known that
=
the heights of the population are random variables that follow a normal distribution with a
-
variance of 0.16.
2=0.05
(i) Determine the interval of 95% confidence for the average heights of the population.
-2=0-1
population(ii)
With a confidence level of 90%, what would the minimum sample size need to be for the
~#
→ true
mean
)
i
n
mean of the heights to be less than 2 cm from the sample mean?
400
=
,
I
=
1-75,0
0
(
1-75 I
(1-96)
=
=
0-16
Or
¥4 )
4
=
,
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.
ozg
←
=
sentence means the difference
bet pop mean and the
estimated valve
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terror)
is
than
2-
(1-7108,1-7892)
less
m
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QUESTION
A simple random sample of 225 marketing experts were consulted on the level of use of some
quantitative techniques in their companies (on a scale from 0 to 5). The results obtained had a
sample mean of 3.35 and a sample standard deviation of 0.72.
(i)
Compute a one-sided upper bound confidence interval for the mean value of the use of
the techniques in all companies (the population), at a level of 99 %.
2=1-0 "
Which would be the confidence level corresponding to an interval from 3.25 to 3.45,
for these data?
-
'
(ii)
-
(iii)
If the values of the sample mean and standard deviation do not change, find the sample
sizes for which a confidence interval at the 99 % level would have a size smaller than
0.20.
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)
i
225
h=
,
I
3- 35
=
5=0-72
,
,
Zo -01
(
)
ii
0
3-35
+
2.3263
lowest scale
,
width
=
cannot
°¥÷g )
3-45-3-25
:
=
2-
§
0.2
÷
-
w
w
V
formula for
F-
F-
be
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0
=
sided
negative
)
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3.4617
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0
is
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2 F-
=
=
.
(
=
-
2-3263
=
bee
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g.
two
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✓ 225
2-
approximately
the same
%
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.
alternatively
OR
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2-
%
,
3.
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=3 -25
.
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3-25
=
J 225
Z
§
ZO.org
significance
→ ✗
level
:
CL
=
=
2-
=
=
a
0833
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2
0--038
(1-0-038)
96.2° /
o
✗
)
bound
-
In
§
Cuse love / upper
100%
iii.
size
=
width of CI
2E < 0.2
→
EL
F-
Ort
=
Zz ¥
.
.
2-0-005=2.5758
2- 5758
.
0.72
<
0.1
Tn
Tn
>
18.5458
n
>
343
n
I
344
-94672m¥
people
.
EXERCISE TOPIC 2.2
QUESTION 5 [9 MARKS]
Nitrogen-oxide emissions yield from car is one of the factors that contribute to air pollution. In
order to control the amount of nitrogen-oxide emissions in Malaysia, the Department of
Environment (Jabatan Alam Sekitar) conducted an on-road operation where fifty cars were
stopped randomly. Each car was tested for nitrogen-oxide emissions (in grams per mile (g/mi)).
The mean amount of nitrogen-oxide emissions for the cars was 0.12 g/mi, and the standard
deviation was 0.04 g/mi.
(a)
Construct a 98% confidence interval of the true mean amount of nitrogen-oxide
emissions yield from the cars.
(5 Marks, CO2/ PO3)
(b)
Suppose that we want the width of the confidence interval of the true mean amount of
nitrogen-oxide emissions to be 0.016 g/mi at 95% confidence. What sample size should
be used in this case?
(4 Marks, CO2/ PO3)
QUESTION
A biotechnology company produces a therapeutic drug with an active content which is
normally distributed with a mean of
mg/L and standard deviation of 0.9 mg/L. A researcher
from the company selects a sample of 50 from the therapeutic drug process and obtained the
active content data with a mean of 16.0745 mg/L and standard deviation of 0.9168 mg/L.
(a) Construct a one-sided 90% confidence interval that gives a lower bound for the true mean
of active contents of the therapeutic drug.
(4 Marks)
(b) How large the sample would be if the researcher wishes to be at least 95% confident that
the error in estimating the mean of active contents of the therapeutic drug lies within 5%?
(3 Marks)
QUESTION 5 [11 MARKS] TEST SEM11617
Suppose we are interested in finding a 95% confidence interval for the mean of Test 1 mark of
Applied Statistics course in the last semester. The total mark of Test 1 is 50. A random sample
of Test 1 mark is taken from Section I students and the data is shown as follows.
Student
Mark
A
40
B
33
C
47
D
32
E
40
F
33
G
19
H
30
I
31
J
32
K
18
(a) Calculate the estimation of error (margin of error) of the population mean?
(5 Marks, CO2/PO3)
(b) Use the answer in (a) to construct the 95% confidence interval for the population mean.
Interpret the answer.
(4 Marks, CO2/PO3)
(c) What would happen to the width of confidence interval if the sample size is increasing
while the confidence level is unchanged? Give a reason.
(2 Marks, CO2/PO3)
QUESTION 7 [11 MARKS]
A particular brand of margarine was analysed to determine the level of fatty acid (in
percentage). The company claims that the population mean of the fatty acid level of its
the
following table.
16.8
(i)
17.2
17.6
16.5
17.3
16.1
16.7
Construct a 98% confidence interval for the population mean of the fatty acid level of its
margerine.
(6 Marks, CO2/PO3)
(ii)
Give a comment on the parameter estimate.
(1 Mark, CO2/PO3)
(iii) Identify the estimation error of this confidence interval.
(1 Mark, CO2/PO3)
(iv) Does the confidence interval calculated in (i) contains the population mean?
(1 Mark, CO2/PO3)
(v)
The company wishes to have a wider confidence interval for the population mean of the
fatty acid level of its margerine. What value should be changed? Give a reason.
(2 Marks, CO2/PO3)