MATERI TEKNIK STATISTIKA MATERI BAHAN AJAR UNIVERSITAS

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1. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 1

2. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 2
Define a point estimator, a point estimate, and desirable
properties of a point estimator such as
unbiasedness, efficiency, and consistency.
Define an interval estimator and an interval estimate
Define a confidence interval, confidence level, margin of
error, and a confidence interval estimate
Construct a confidence interval for the population mean
when the population standard deviation is known
When you have completed this chapter, you will be
able to:

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9 - 3
Construct a confidence interval for the population mean
when the population is normally distributed and the
population standard deviation is unknown
Construct a confidence interval for a population proportion
Determine the sample size for attribute and
variable sampling
Construct a confidence interval for the population variance
when the population is normally distributed

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9 - 4
Terminology
Point Estimate…is a single value (statistic) used to
estimate a population value (parameter)
Confidence Interval…is a range of values within which
the population parameter
is expected to occur
Interval Estimate …states the range within which a
population parameter probably lies

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Desirable properties of a point estimator
• unbiased
… possible values are concentrated
close to the value of the parameter
…unbiased when the expected value equals the value
of the population parameter being estimated.
Otherwise, it is biased!
…values are distributed evenly on
both sides of the value of the
parameter
• efficient
• consistent

6. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 6
Standard error of the sample mean
…is the standard deviation
of the sampling distribution of the
sample means
It is computed by
…is the symbol for the standard error of the
sample mean
…is the standard deviation of the population
n…is the size of the sample
Terminology

x

n



x

7. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 7Standard Error
of the Means
If  is not known and n > 30,
the standard deviation of the
sample(s) is used
to approximate the population standard
deviation
n

sx s
Computed by…

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1.The sample size, n
2.The variability in the population,
usually estimated by s
3.The desired level of confidence
…that determine the width of a confidence interval
are:

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9 - 9
IN GENERAL,
A confidence interval for a mean is computed
by:
Constructing
Confidence Intervals
Interpreting…
z
x
n
s
α/2

10. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 10Interpreting
Confidence Intervals
The Globe
Suppose that you read that
“…the average selling price
of a family home in
York Region is
$200 000 +/- $15000
at 95% confidence!”
This means…what?

11. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 11Interpreting
Confidence Intervals
In statistical terms, this means:
…that we are 95% sure that the
interval estimate obtained
contains the value of the
population mean.
Lower confidence limit is
$185 000
Upper confidence limit is
$215 000
The Globe
“…the average
selling price of a
family home in
York Region is
$200 000 +/-
$15 000 at 95%
confidence!”
Also…
($200 000 - $15 000)
($200 000 + $15 000)

12. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 12Interpreting
Confidence Intervals
The Globe
“…the mean
time to sell a
family home
in York
Region
is 40 days.
Your newspaper also reports
that…
You select a random
sample of 36 homes sold
during the past year,
and determine a
90% confidence
interval estimate
for
the population mean to
be (31-39) days.Do your sample results support the paper’s claim?

13. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 13Interpreting
Confidence Intervals
You select a
random sample
of 36 homes
sold during
the past year,
and
determine a
90%
confidence interval
estimate

for the population
mean to be
(31-39) days.
There is a 10% chance (100%-90%)
that the interval estimate
does not contain the value
of the population mean!
Lower confidence limit
is 31 days
Upper confidence limit
is 39 days
Our evidence does not support the
statement made by the newspaper,
i.e., the population mean is not 40 days,
when using a 90% interval estimate

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9 - 14
31 39
i.e.  = 0.10
.05 .05
90%
… 10% chance of falling outside this interval
Interpreting
Confidence Intervals
90% Confidence Interval
…or, focus on
tail areas …
 is the probability of a value falling
outside the confidence interval

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9 - 15
Find the appropriate value of z:
n
z
X
n
z
XP



(
0
1.75
Locate Area on
the normal curve
1
This is a 92%
confidence interval
2
Look up a= 0.46 in Table
to get the corresponding
z-score
Search in the centre of the
table for the area of 0.46
Z = +/- 1.75
-1.75
0.92
92.

16. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 16Constructing
Confidence Intervals
n
s
X96.1
95% C.I. for
the mean:
Common
Confidence
Intervals
99% C.I. for
the mean:
X
s
n
258.
About 95% of
the constructed
intervals will
contain the
parameter being
estimated.
Also, 95% of the
sample means for
a specified sample
size will lie within
1.96 standard
deviations of the
hypothesized
population mean.
z
x
n
s
α/2

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9 - 17
Interval Estimates
If the population
standard deviation is
known or n > 30
If the population
standard deviation is
unknown and n<30
Use the z table…
z
x
n
s
α/2
Use the t-table…
t
x
n
s
α/2
More
on this
later…

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9 - 18
Our best estimate is 24 hours.
The Dean of the Business School wants to
estimate the mean number of hours
worked per week by students.
A sample of 49 students
showed a mean of 24 hours
with a standard deviation of 4 hours.
What is the population
mean?
This is a point estimate.

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9 - 19
Find the 95 percent confidence
interval for the population mean.
95% Confidence Z = +/- 1.96
Substitute
values:
24 = 24 +/- 1.12
The Confidence Limits range from 22.88 to 25.12
Commonly denoted as 1-
95 percent confidence
+ 1.96
4
Mean = 24 SD = 4 N = 49
49
z
x
n
s
α/2

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9 - 20
90% confidence level
1- = 0.9
or  = 0.10
1- = 0.99
or  = 0.010
99% confidence level
Interval Estimates

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9 - 21
Student’s t-distribution
….used for small sample sizes
Characteristics
…like z, the t-distribution is continuous
…takes values between –4 and +4
…it is bell-shaped and symmetric about zero
…it is more spread out and flatter at the centre
than the z-distribution
…for larger and larger values of degrees of
freedom, the t-distribution becomes closer and
closer to the standard normal distribution

22. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 22
Chart 9-1
Comparison of The Standard Normal Distribution
and the Student’s t Distribution
Z distribution
The t distribution should be flatter and more spread out
than the z distribution
t distribution
Student’s t-distribution

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9 - 23
Student’s t-distribution
…with df = 9 and 0.10 area in the upper tail…
t = 1.383
t
0.10
T -table

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9 - 24
Student’s t-distribution
Confidence Intervals
80% 90% 95% 98% 99%
Level of Significance for One-Tailed Test
0.100 0.050 0.025 0.010 0.005
Level of Significance for Two-Tailed Test
0.20 0.10 0.05 0.02 0.01
0.10
1.3839

25. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 25
When?
…to use the z Distribution or the t Distribution
NO
Population standard
deviation known?
Use a
nonparametric
test
(see
Ch16)
Use the z
distribution
Use the t
distribution
YES
Population Normal?
NO YES
n 30 or more?
NO YES
Use the z
distribution

26. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 26
Student’s t-distribution
The Dean of the Business School wants to
estimate the mean number of hours worked
per week by students.
A sample of only 12 students
showed a mean of 24 hours with a standard
deviation of 4 hours.
Find the 95 percent confidence interval
for the population mean.
so use the t - Distribution
n is small

27. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 27…sample of only 12 students
…a mean of 24 hours
…a standard deviation of 4 hours

Data

X
= 24 n = 12 s = 4 df = 12-1 = 11  = 1 – 95% = .05
Looking up 5% level of significance for
a two-tailed test with 11df, we find…
Formula t
x
n
s
α/2

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9 - 28
Student’s t-distribution
Confidence Intervals
80% 90% 95% 98% 99%
Level of Significance for One-Tailed Test
0.100 0.050 0.025 0.010 0.005
Level of Significance for Two-Tailed Test
0.20 0.10 0.05 0.02 0.010.05
2.20111

29. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 29…sample of only 12 students
…a mean of 24 hours
…a standard deviation of 4 hours

Data

X
= 24 n = 12 s = 4 df = 12-1 = 11  = 1 – 95% = .05
Looking up 5% level of significance for
a two-tailed test with 11df, we find…
t
0.025
= 2.201
Compare these with earlier limits of 22.88 to 25.12
= 24 +/- 2.54 24 2.201
12

4
The confidence limits range from 21.46 to 26.54
Formula t
x
n
s
α/2

30. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 30
The manager of the college cafeteria wants to
estimate the mean amount spent per customer
per purchase. A sample of 10 customers
revealed the following amounts spent:
$4.45 $4.05 $4.95 $3.25 $4.68
$5.75 $6.01 $3.99 $5.25 $2.95
Determine the 99% confidence interval
for the mean amount spent.
Student’s t-distribution

31. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 31
= $4.53 +/- $1.03
$4.45 $4.05 $4.95 $3.25 $4.68
$5.75 $6.01 $3.99 $5.25 $2.95
Determine the sample mean and standard deviation.Step 1
n = 10 – 1 = 9 1-99% = .01
Enter the key data into the appropriate formula.Step 2
= 4.53 3.25
We are 99% confident that the mean amount spent
per customer is between $3.50 and
$5.56
Student’s t-distribution
= $4.53 s = $1.00
X
 =df =10

1.00
10
Formulat
x
n
s
α/2

32. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 32Constructing Confidence Intervals
for Population Proportions
A confidence interval for a population
proportion is estimated by:
…is the symbol for the sample proportionp
n
pp
zp
)1(
Formula

33. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 33
A sample of 500 executives who own their
own home revealed 175 planned to sell their
homes and retire to Victoria.
Develop a 98% confidence
interval for the proportion of executives that
plan to sell and move to Victoria.
Constructing Confidence Intervals
for Population Proportions

34. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 34
A sample of 500
executives who own
their own home
revealed 175
planned to sell their
homes and retire to
Victoria.
Develop a 98%
confidence interval
for the proportion
of executives…
n = p = z =500 175/500 = .35

2.33
Constructing Confidence Intervals
for Population Proportions
1
/2
p p
p z
n
( )
ˆ


Formula
98% CL =
500
)35.1(35.
33.235.


0497.35. 

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9 - 35
Finite-Population
Correction Factor
Used when n/N is 0.05 or more
The attendance at the college hockey game last night
was 2700. A random sample of 250 of those in
attendance revealed that the average number of
drinks consumed per person was 1.8
with a standard deviation of 0.40.
Formula

x
n
N -n
N -

1
Develop a 90% confidence interval estimate for the mean
number of drinks consumed per person.
Correction
Factor

36. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 36
N = n = x =
s = /2 =
Finite-Population
Correction Factor
The attendance at the
college hockey game
last night was 2700.
A sample of 250 of
those in attendance
revealed that the
average number of
drinks consumed per
person was 1.8 with a
standard deviation of
0.40.
Develop a 90%
confidence interval
estimate.…
Formula
N -n
N - 1
2700 250 1.8
0.40 0.05
Since 250/2700 >.05, use the correction factor
)
12700
2502700
)(
250
4.
(645. 18.1



04.08.190% CL =
n
s
Z
α/2
X


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9 - 37
Selecting the
Sample
Size

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1.The degree of confidence selected
2.The maximum allowable error
3.The variation in the population
…that determine the sample size
are:
Factors

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9 - 39
E … is the allowable error
Z …is the z-score for the chosen level of confidence
S …is the sample deviation of the pilot survey
Selecting the

Sample Size

2





E
sz
α/2n =
Formula

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9 - 40
A consumer group would like to estimate the
mean monthly electricity charge for a single
family house in July (within $5) using a
99 percent level of confidence.
Based on similar studies the
standard deviation is
estimated to be $20.00.
Selecting the

Sample Size
How large a sample is required?

41. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 41Selecting the

Sample Size
A consumer group
would like to estimate the
mean monthly electricity
charge for a single family
house in July (within $5)
using a 99 percent level
of confidence.
Based on
similar studies the
standard deviation is
estimated to be $20.00.

Formula
= (10.32)
2
= 106.5
A minimum of
107 homes
must be sampled.
90% CL =
2
5.00
202.58


 





2





E
sz
α/2

42. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 42Selecting the

Sample Size
The Kennel Club wants to estimate the proportion of
children that have a dog as a pet.
Assume a 95% level of confidence and that the club
estimates that 30% of the children have a dog as a pet.
If the club wants the estimate to be
within 3% of the population proportion,
how many children would
they need to contact?

43. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 43
The Kennel Club
wants to estimate the
proportion of children
that have a dog as a
pet.
Assume a
95% level of
confidence and that
the club estimates that
30% of the children
have a dog
as a pet.
Selecting the

Sample Size
New
Formula
np p
Z
E
 





( )1
2
2
03.
96.1
)3.1(3. 






 233.65)21(.
n = 896.4
A minimum of 897 children
must be sampled.

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9 - 44
Test your learning…
www.mcgrawhill.ca/college/lindClick on…
Online Learning Centre
for quizzes
extra content
data sets
searchable glossary
access to Statistics Canada’s E-Stat data
…and much more!

45. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
9 - 45
This completes Chapter 9