Transcript Chap. 5: Inferences Based on a Single Sample: Estimation

Statistics for Business and
Economics
Chapter 5
Inferences Based on a Single Sample:
Estimation with Confidence Intervals
Learning Objectives
1. State What Is Estimated
2. Distinguish Point & Interval Estimates
3. Explain Interval Estimates
4. Compute Confidence Interval Estimates for
Population Mean & Proportion
5. Compute Sample Size
6. Discuss Finite Population Correction Factor
Thinking Challenge
Suppose you’re
interested in the
average amount of
money that students in
this class (the
population) have on
them. How would you
find out?
Introduction
to Estimation
Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
Hypothesis
Testing
Estimation Process
Population


Mean, , is
unknown




Sample





Random Sample
Mean


X
=
50

I am 95%
confident that 
is between 40 &
60.
Unknown Population
Parameters Are Estimated
Estimate Population
Parameter...
Mean

Proportion
p
Variance

Differences
with Sample
Statistic
x
p^
2
1 -  2
s
2
x1 -x2
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
Point Estimation
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
Point Estimation
1. Provides a single value
• Based on observations from one sample
2. Gives no information about how close the
value is to the unknown population parameter
3. Example: Sample mean x = 3 is point
estimate of unknown population mean
Interval Estimation
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
Interval Estimation
1. Provides a range of values
•
Based on observations from one sample
2. Gives information about closeness to unknown
population parameter
•
Stated in terms of probability
– Knowing exact closeness requires knowing unknown
population parameter
3. Example: Unknown population mean lies between 50
and 70 with 95% confidence
Key Elements of
Interval Estimation
Sample statistic
Confidence interval
(point estimate)
Confidence limit
(lower)
Confidence limit
(upper)
A probability that the population parameter falls
somewhere within the interval.
Confidence Limits
for Population Mean
Parameter =
Statistic ± Error
© 1984-1994
T/Maker Co.
(1)
  X  Error
(2)
Error  X   or X  
X 
(3)
Z
(4)
Error  Z x
(5)
  X  Z x
x

Error
x
Many Samples Have Same
Interval
X =  ± Zx
x_
-1.65
-2.58x
x
-1.96x

+1.65x
+2.58x
+1.96x
90% Samples
95% Samples
99% Samples
X
Confidence Level
1. Probability that the unknown population
parameter falls within interval
2. Denoted (1 – 
•  is probability that parameter is not within
interval
3. Typical values are 99%, 95%, 90%
Intervals & Confidence Level
Sampling Distribution of Sample Mean
/2
_
1- 
x
/2
 x = 
_
X
(1 – α)% of
intervals
contain μ
Intervals
extend from
X – ZσX to
X + ZσX
α% do not
Large number of intervals
Factors Affecting
Interval Width
1. Data dispersion
• Measured by 
Intervals extend from
X – ZX toX + ZX
2. Sample size

X 
n
3. Level of confidence
(1 – )
•
Affects Z
© 1984-1994 T/Maker Co.
Confidence Interval Estimates
Confidence
Intervals
Mean
σ Known
σ
Unknown
Proportion
Confidence Interval
Estimate Mean ( Known)
Confidence Interval Estimates
Confidence
Intervals
Mean
σ Known
σ
Unknown
Proportion
Confidence Interval
Mean ( Known)
1. Assumptions
•
•
•
Population standard deviation is known
Population is normally distributed
If not normal, can be approximated by normal
distribution (n  30)
2. Confidence interval estimate


X  Z / 2 
   X  Z / 2 
n
n
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence interval
estimate for  if  = 10.


X  Z / 2 
   X  Z / 2 
n
n
10
10
50  1.96 
   50  1.96 
25
25
46.08    53.92
Thinking Challenge
You’re a Q/C inspector for Gallo.
The  for 2-liter bottles is .05
liters. A random sample of 100
bottles showed x = 1.99 liters.
What is the 90% confidence
interval estimate of the true
mean amount in 2-liter bottles?
22 liter
liter
© 1984-1994 T/Maker Co.
Confidence Interval
Solution*
X  Z / 2 

n
   X  Z / 2 

n
.05
.05
1.99  1.645 
   1.99  1.645 
100
100
1.982    1.998
Confidence Interval
Estimate Mean ( Unknown)
Confidence Interval Estimates
Confidence
Intervals
Mean
σ Known
σ
Unknown
Proportion
Confidence Interval
Mean ( Unknown)
1. Assumptions
•
•
Population standard deviation is unknown
Population must be normally distributed
2. Use Student’s t–distribution
Student’s t Distribution
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
Z
t
Degrees of Freedom (df)
1. Number of observations that are free to vary
after sample statistic has been calculated
2. Example
–
Sum of 3 numbers is 6
X1 = 1 (or any number)
X2 = 2 (or any number)
X3 = 3 (cannot vary)
Sum = 6
degrees of freedom
=n-1
=3-1
=2
Student’s t Table
/2
v
t .10
t .05
t .025
1 3.078 6.314 12.706
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
2 1.886 2.920 4.303
/2
3 1.638 2.353 3.182
t values
0 2.920
t
Confidence Interval
Mean ( Unknown)
S
S
X  t / 2 
   X  t / 2 
n
n
df  n  1
Estimation Example
Mean ( Unknown)
A random sample of n = 25 has x = 50 and s = 8.
Set up a 95% confidence interval estimate for .
S
S
X  t / 2 
   X  t / 2 
n
n
8
8
50  2.064 
   50  2.064 
25
25
46.69    53.30
Thinking Challenge
You’re a time study analyst in
manufacturing. You’ve
recorded the following task
times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence
interval estimate of the
population mean task time?
Confidence Interval Solution*
• x = 3.7
• s = 3.8987
• n = 6, df = n - 1 = 6 - 1 = 5
• t.05 = 2.015
3.8987
3.8987
3.7  2.015 
   3.7  2.015 
6
6
.492    6.908
Confidence Interval
Estimate of Proportion
Confidence Interval Estimates
Confidence
Intervals
Mean
σ Known
σ
Unknown
Proportion
Confidence Interval
Proportion
1. Assumptions
•
•
Random sample selected
Normal approximation can be used if
npˆ  15 and nqˆ  15
2. Confidence interval estimate
ˆˆ
ˆˆ
pq
pq
pˆ  z 2 
 p  pˆ  z 2 
n
n
Estimation Example
Proportion
A random sample of 400 graduates showed 32
went to graduate school. Set up a 95%
confidence interval estimate for p.
ˆˆ
ˆˆ
pq
pq
pˆ  Z / 2 
 p  pˆ  Z / 2 
n
n
.08  .92
.08  .92
.08  1.96 
 p  .08  1.96 
400
400
.053  p  .107
Thinking Challenge
You’re a production manager
for a newspaper. You want to
find the % defective. Of 200
newspapers, 35 had defects.
What is the 90% confidence
interval estimate of the
population proportion
defective?
Confidence Interval
Solution*
pˆ  qˆ
pˆ  qˆ
pˆ  z / 2 
 p  pˆ  z / 2 
n
n
.175  (.825)
.175  (.825)
.175  1.645 
 p  .175  1.645 
200
200
.1308  p  .2192
Finding Sample Sizes
Finding Sample Sizes
for Estimating 
SE = Sampling Error
(1)
(2)
X   SE
Z

x
x
SE  Z 2 x  Z 2
( Z 2 ) 
2
(3)
I don’t want to
sample too much
or too little!
n
( SE )
2
2

n
Sample Size Example
What sample size is needed to be 90% confident
the mean is within  5? A pilot study suggested
that the standard deviation is 45.
( Z 2 ) 
2
n
( SE )
2
2
1.645  45


2
 5
2
2
 219.2  220
Finding Sample Sizes
for Estimating p
SE = Sampling Error
(1)
(2)
pˆ  p SE
Z

 pˆ
 pˆ
SE  Z 2 pˆ  Z 2
pq
n
2
(3)
n
( Z 2 ) pq
( SE ) 2
If no estimate of p is
available, use p = q = .5
Sample Size Example
What sample size is needed to estimate p with
90% confidence and a width of .03?
width .03
SE 

 .015
2
2
( Z 2 ) 
2
n
( SE )
2
2
1.645 .5  .5


2
.015
2
2
 3006.69  3007
Thinking Challenge
You work in Human Resources
at Merrill Lynch. You plan to
survey employees to find their
average medical expenses. You
want to be 95% confident that
the sample mean is within ± $50.
A pilot study showed that  was
about $400. What sample size
do you use?
Sample Size Solution*
n
( Z 2 ) 2  2
( SE ) 2
1.96   400 


2
 50 
2
 245.86  246
2
Finite Population Correction
Factor
Finite Population Correction Factor
• Use when n, the sample size, is relatively large
compared to N, the size of the population
• If n/N > .05 use the finite population
correction factor
• Finite population correction factor:
N n
N
Finite Population Correction Factor
• Approximate 95% confidence interval for μ:
s
x 2
n
N n
s
   x 2
N
n
N n
N
• Approximate 95% confidence interval for p:
pˆ (1  pˆ ) N  n
pˆ (1  pˆ ) N  n
pˆ  2
 p  pˆ  2
n
N
n
N
Finite Population Correction Factor
Example
You want to estimate a population mean, μ, where
x =115, s =18, N =700, and n = 60. Find an approximate
95% confidence interval for μ.
Since
n
N
 60
700
 .086
is greater than .05 use the finite correction factor
Finite Population Correction Factor
Example
You want to estimate a population mean, μ, where
x =115, s =18, N =700, and n = 60. Find an approximate
95% confidence interval for μ.
s
x 2
n
18
115  2 
60
N n
s
   x 2
N
n
N n
N
700  60
18
   115  2 
700
60
110.6    119.4
700  60
700
Conclusion
1. Stated What Is Estimated
2. Distinguished Point & Interval Estimates
3. Explained Interval Estimates
4. Computed Confidence Interval Estimates for
Population Mean & Proportion
5. Computed Sample Size
6. Discussed Finite Population Correction Factor