Download t - YSU

Chapter 8
Interval Estimation
1
Chapter Outline
 Population Mean:  Known
 Population Mean:  Unknown
 Population Proportion
2
Introduction
 The sampling distribution introduced last
chapter connects sample statistics to
population parameters.
 In reality, we probably don’t know any of
the population parameters. However, a
study on sampling distributions can provide
reasonable references to the population
parameters.
3
Margin of Error and the Interval Estimate
 A point estimator cannot be expected to provide
the exact value of the population parameter. For
instance, the probability of any particular sample
mean equal population mean is zero, i.e. p( x = m) =
0.
 An interval estimate can be computed by adding
and subtracting a margin of error to the point
estimate.
 The purpose of an interval estimate is to provide a
reasonable value range of the population
parameters.
4
Margin of Error and the Interval Estimate
 The general form of an interval estimates of
a population mean m is
x  Margin of Error
5
Interval Estimate of A Population Mean:
 Known
 In the first scenario, we assume  to be known.
Although  is rarely known in reality, a good
estimate can be obtained based on historical data
or other information.
 Let’s use the example of Checking Accounts from
last chapter as an illustration. Here, we assume
that the population standard deviation is known
(=66). Our goal is to come up with an interval
estimate of population mean m based on the
sample mean x =280.
6
Summary of Point Estimates of A Simple
Random Sample of 121 Checking Accounts
Population
Parameter
Parameter
Value
Point
Estimator
Point
Estimate
m = Population mean
$310
x = Sample mean
$306
s = Sample std.
deviation for
account balance
$61
account balance
 = Population std.
$66
deviation for
account balance
p = Population proportion of account
balance no less than
$500
.3
account balance
p = Sample pro-
.27
portion of account
balance no less than
$500
7
Interval Estimate of A Population Mean:
 Known
 The sample mean distribution of 121 checking account
balances can be approximated by a normal distribution
with E( x ) = m. Let’s first figure out the values of x that
provide the middle area about m of 95%.
x =
66
121
=6
95%
a
E x ) = m
b
x
8
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
 Given the middle area of 95%, we can find z values first and then
convert z values to the corresponding values of x .
95%
a
m
95%
2.5
%
b
x
-z0.025
0
2.5
%
z0.025
z
9
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
 Convert z values to the corresponding values of x .
z0.025 =
 z0.025 =
bm
x
am
x
b = m  z0.025   x
a = m  z0.025   x
10
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
 We set the margin of error as z0.025   x. So, the interval estimate of
population mean is x  z0.025   x.
As long as x falls between a
and b, the interval x  z0.025   x
will include the population
mean m .
95%
a z  m z  b
0.025
x
0.025
x
[---------
x  z0.025   x -----------]
[---------
[---------
x  z0.025   x -----------]
x
x  z0.025   x -----------]
11
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
The rationale behind the interval estimate –
 For any particular sample mean x , we cannot compare it with the
population mean m since m is unknown. But, what we are certain is
that as long as x falls between a and b. The interval x  z0.025   x
will include the true value of m.
 In the example, x = 306. So, the interval estimate of population
account balance is 306  z0.025   x . Because z0.025=1.96 and  x = 6
, the interval estimate is calculated as
3061.96·6 = 306 11.76
Margin of Error
or
$294.24 to $317.76
 We are 95% confident that x will fall between a and b. So, the chance is
95% that the true value of m is no less than $294.24 and no more than
$317.76.
 On the other hand, there is a 5% chance that we make a mistake and the
above interval estimate doesn’t include m .
12
Interval Estimate of A Population Mean:
 Known
 Interval Estimate of m
x  z /2
where:

n
x is the sample mean
1 - is the confidence level
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
 is the population standard deviation
n is the sample size
13
Interval Estimate of A Population Mean:
 Known
 Values of z/2 for the Most Commonly Used
Confidence Levels
Confidence
Level
90%
95%
99%

/2
Area to the
left of z/2
.10
.05
.01
.05
.025
.005
1- /2= .9500
1- /2= .9750
1- /2= .9950
z/2
1.645
1.960
2.576
14
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
Confidence
Level
Margin
of Error
Interval Estimate
90%
95%
99%
9.87
11.76
15.46
296.13 to 315.87
294.24 to 317.76
290.54 to 321.46
The higher the confidence level, the wider the
Interval estimate.
15
Interval Estimate of A Population Mean:
 Unknown
 When  is unknown, we will have to use the sample
standard deviation s to estimate  .
 In this case, the interval estimate for m is based on the t
distribution. (See Table 2 of Appendix B in the textbook)
• A specific t distribution depends on a parameter known as the
degrees of freedom.
• Degrees of freedom refer to the number of independent pieces of
information that go into the computation of s.
• As the degrees of freedom increases, t distribution is approaching
closer to the Standard Normal Distribution.
16
t Distribution
t distribution
(20 degrees
of freedom)
Standard
normal
distribution
t distribution
(10 degrees
of freedom)
z, t
0
17
t Distribution
 For more than 100 degrees of freedom, the standard
normal z value provides a good approximation to the t
value.
 The standard normal z values can be found in the infinite
degrees (  ) row of the t distribution table.
Degrees
Area in Upper Tail
of Freedom
.20
.10
.05
.025
.01
.005
.
.
.
.
.
.
.
50
.849
1.299
1.676
2.009
2.403
2.678
60
.848
1.296
1.671
2.000
2.390
2.660
80
.846
1.292
1.664
1.990
2.374
2.639
100
.845
1.290
1.660
1.984
2.364
2.626
.842
1.282
1.645
1.960
2.326
2.576

Standard normal
z values
18
Interval Estimate of A Population Mean:
 Unknown
 Interval Estimate
x  t / 2
s
n
where: 1 - = the confidence level
t/2 = the t value providing an area of /2
in the upper tail of a t distribution
with n - 1 degrees of freedom
s = the sample standard deviation
n = sample size
19
Interval Estimate of A Population Mean:
 Unknown
 Example: Consumer Age
The makers of a soft drink want to identify the
average age of its consumers. A sample of 20 consumers
was taken. The average age in the sample was 21 years
with a standard deviation of 4 years.
.
Construct a 95% confidence interval for the true
average age of the consumers.
20
Interval Estimate of A Population Mean:
 Unknown
 Example: Consumer Age
At 95% confidence,  = .05, and  /2 = .025.
t.025 is based on n - 1 = 20 - 1 = 19 degrees of freedom.
In the t distribution table we see that t.025 = 2.093.
Degrees
Area in Upper Tail
of Freedom
.20
.100
.050
.025
.010
.005
15
.866
1.341
1.753
2.131
2.602
2.947
16
.865
1.337
1.746
2.120
2.583
2.921
17
.863
1.333
1.740
2.110
2.567
2.898
18
.862
1.330
1.734
2.101
2.520
2.878
19
.861
1.328
1.729
2.093
2.539
2.861
.
.
.
.
.
.
.
21
Interval Estimate of A Population Mean:
 Unknown
 Example: Consumer Age
x  t.025
s
n
4
21  2.093 
= 21  1.87
20
Margin of
Error
We are 95% confident that the average age of the
soft drink consumers is between 19.13 and 22.87.
22
Summary of Interval Estimation Procedures
for a Population Mean
Is the
population standard
deviation 
known ?
Yes
Use the sample
standard deviation
s to estimate s
 Known
Case
Use
x  z /2

n
No
 Unknown
Case
Use
x  t /2
s
n
23
Interval Estimate of A Population
Proportion
The general form of an interval estimate of a
population proportion is
p  Margin of Error
24
Interval Estimate of A Population
Proportion
 Just as the sampling distribution of x is key in
estimating population mean, the sampling
distribution of p is crucial in estimating population
proportion.
 The sampling distribution of p can be
approximated by a normal distribution whenever
np  5 and n(1-p)  5.
25
Interval Estimate of A Population
Proportion

Normal Approximation of Sampling Distribution of
Sampling
distribution
of p
/2
p
p(1  p)
p =
n
/2
1-
p
z /2 p
p
z /2 p
26
Interval Estimate of A Population
Proportion

Interval Estimate of p
p  z / 2
where:
p1  p )
n
1 - is the confidence level
z/2 is the z value providing an area of
/2 in the upper tail of the standard
normal probability distribution
p is the sample proportion
27
Interval Estimate of A Population Mean:
 Known
 Example: Checking Accounts
 Refer to our previous example of Checking Accounts. Out of the
simple random sample of 121 accounts, the sample proportion of
account balance no less than $500 is .27. Develop a 95%
confidence interval estimate of the population proportion.
p  z / 2
p1  p )
n
where: n = 121, p = .27, z/2 = 1.96
.271  .27)
.27  1.96
= .27  .04
121
Margin of Error
We are 95% confident that the proportion of all checking accounts
with a balance no less than $500 is between .23 and .31, which
correctly includes the population proportion .30.
28