CHAPTER 9
Estimation from Sample Data
to accompany
Introduction to Business Statistics
fourth edition,by Ronald M,Weiers
Presentation by Priscilla Chaffe-Stengel
Donald N,Stengel
? 2002 The Wadsworth Group
Chapter 9 - Learning Objectives
? Explain the difference between a point and
an interval estimate.
? Construct and interpret confidence
intervals:
– with a z for the population mean or proportion.
– with a t for the population mean.
? Determine appropriate sample size to
achieve specified levels of accuracy and
confidence.
? 2002 The Wadsworth Group
Chapter 9 - Key Terms
? Unbiased estimator
? Point estimates
? Interval estimates
? Interval limits
? Confidence
coefficient
? Confidence level
? Accuracy
? Degrees of
freedom (df)
? Maximum likely
sampling error
? 2002 The Wadsworth Group
Unbiased Point Estimates
Population Sample
Parameter Statistic Formula
? Mean,μ
? Variance,s2
? Proportion,p
x x = xi?n
1–
2)–(22
n
xixss ?=
p p = x successesn trials
? 2002 The Wadsworth Group
Confidence Interval,μ,s Known
where = sample mean ASSUMPTION:
s = population standard infinite population
deviation
n = sample size
z = standard normal score
for area in tail = a/2
a ? 2 a ? 2? ? a
nzxxnzxx
zzz ss
×+×
+
–:
0–:
x
? 2002 The Wadsworth Group
where = sample mean ASSUMPTION:
s = sample standard Population
deviation approximately
n = sample size normal and
t = t-score for area infinite
in tail = a/2
df = n – 1
a ? 2 a ? 2? ? a
n
stxx
n
stxx ttt ×+× +–,0–:
x
Confidence Interval,μ,s Unknown
? 2002 The Wadsworth Group
Confidence Interval on p
where p = sample proportion ASSUMPTION:
n = sample size n?p ? 5,
z = standard normal score n?(1–p) ? 5,
for area in tail = a/2 and population
infinite
a ? 2 a ? 2? ? a
nn
zzz 0–:
ppzppppzpp )–1()–1(–,×+×
+
? 2002 The Wadsworth Group
Converting Confidence Intervals to
Accommodate a Finite Population
? Mean:
or
? Proportion:
?
?
?
?
?
?
?
?
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???
???
1–
–
2
1–
–
2
N
nN
n
stx
N
nN
n
zx
a
s
a
??
?
?
?
?
?
??
?
?
?
?
?
??? 1––)–1(
2
N nNn ppzp a
? 2002 The Wadsworth Group
Interpretation of
Confidence Intervals
? Repeated samples of size n taken from the
same population will generate (1–a)% of
the time a sample statistic that falls within
the stated confidence interval.
OR
? We can be (1–a)% confident that the
population parameter falls within the
stated confidence interval.
? 2002 The Wadsworth Group
Sample Size Determination for μ
from an Infinite Population
? Mean,Note s is known and e,the bound
within which you want to estimate μ,is given.
– The interval half-width is e,also called the
maximum likely error:
– Solving for n,we find,2 22
e
zn
nze
s
s
×=
×=
? 2002 The Wadsworth Group
Sample Size Determination for μ
from a Finite Population
? Mean,Note s is known and e,the bound
within which you want to estimate μ,is given.
where n = required sample size
N = population size
z = z-score for (1–a)% confidence
n = s 2
e 2
z 2
+ s 2N
? 2002 The Wadsworth Group
Sample Size Determination for p
from an Infinite Population
? Proportion,Note e,the bound within which
you want to estimate p,is given.
– The interval half-width is e,also called the
maximum likely error:
– Solving for n,we find:
2
)–1(2
)–1(
e
ppzn
n ppze
=
×=
? 2002 The Wadsworth Group
Sample Size Determination for p
from a Finite Population
? Mean,Note e,the bound within which you
want to estimate μ,is given.
where n = required sample size
N = population size
z = z-score for (1–a)% confidence
p = sample estimator of p
n = p(1– p)e2
z2 +
p(1– p)
N
? 2002 The Wadsworth Group
An Example,Confidence Intervals
? Problem,An automobile rental agency has the
following mileages for a simple random sample of
20 cars that were rented last year,Given this
information,and assuming the data are from a
population that is approximately normally
distributed,construct and interpret the 90%
confidence interval for the population mean.
55 35 65 64 69 37 88
39 61 54 50 74 92 59
38 59 29 60 80 50
? 2002 The Wadsworth Group
A Confidence Interval Example,cont.
? Since s is not known but the population is approximately
normally distributed,we will use the t-distribution to
construct the 90% confidence interval on the mean.
a ?2 a ?2? ? a
n
stxx
n
stxx ttt ×+× +–,0–:
)621.64,179.51(721.69.57
20
384.17729.19.57
729.1So,
05.02/,191–20
384.17,9.57
?±
×±?×±
=
===
==
n
stx
t
df
sx
a
? 2002 The Wadsworth Group
A Confidence Interval Example,cont.
?Interpretation:
–90% of the time that samples of
20 cars are randomly selected
from this agency’s rental cars,
the average mileage will fall
between 51.179 miles and
64.621 miles.
? 2002 The Wadsworth Group
An Example,Sample Size
? Problem,A national political candidate
has commissioned a study to determine
the percentage of registered voters who
intend to vote for him in the upcoming
election,In order to have 95% confidence
that the sample percentage will be within
3 percentage points of the actual
population percentage,how large a simple
random sample is required?
? 2002 The Wadsworth Group
A Sample Size Example,cont.
? From the problem we learn:
– (1 – a) = 0.95,so a = 0.05 and a /2 = 0.025
– e = 0.03
? Since no estimate for p is given,we will use 0.5
because that creates the largest standard error.
To preserve the minimum confidence,the candidate
should sample n = 1,068 voters.
1.067,12
)03.0(
)5.0)(5.0(296.1
2
)–1)((2 ===
e
ppzn
? 2002 The Wadsworth Group