Download Chapter 8

Business Statistics, 4e
by Ken Black
Chapter 8
Discrete Distributions
Statistical Inference:
Estimation for
Single Populations
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-1
Learning Objectives
• Know the difference between point and interval
estimation.
• Estimate a population mean from a sample mean
when s is known.
• Estimate a population mean from a sample mean
when s is unknown.
• Estimate a population proportion from a sample
proportion.
• Estimate the population variance from a sample
variance.
• Estimate the minimum sample size necessary to
achieve given statistical goals.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-2
Statistical Estimation
• Point estimate -- the single value of a statistic
calculated from a sample
• Interval Estimate -- a range of values calculated
from a sample statistic(s) and standardized
statistics, such as the Z.
– Selection of the standardized statistic is
determined by the sampling distribution.
– Selection of critical values of the standardized
statistic is determined by the desired level of
confidence.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-3
Confidence Interval to Estimate 
when n is Large
• Point estimate
• Interval
Estimate
X

X
n
XZ
n
or
XZ
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
s
s
n
 XZ
s
n
8-4
Distribution of Sample Means
for (1-)% Confidence


2
2


 Z
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0
X
Z
Z
2
8-5
Distribution of Sample Means
for (1-)% Confidence


2
.5

.5
2
2

2

 Z
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0
X
Z
Z
2
8-6
Distribution of Sample Means
for (1-)% Confidence


2
1
2
2
1
2

 Z
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0
X
Z
Z
2
8-7
Probability Interpretation
of the Level of Confidence
Pr ob[ X  Z 
s
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
n
   X  Z
2
s
n
]  1 
8-8
Distribution of Sample Means
for 95% Confidence
.025
.025
95%
.4750
.4750

X
Z
-1.96
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0
1.96
8-9
95% Confidence Interval for 
X  4.26, s  11
. , and n  60.
X Z
s
  X Z
s
n
n
46
46
153  1.96
   153  1.96
85
85
153  9.78    153  9.78
143.22    162.78
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-10
95% Confidence Intervals for 
95%

X
X
X
X
X
X
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-11
95% Confidence Intervals for 
Is our interval,
95%

X
143.22 
162.78, in the
red?
X
X
X
X
X
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-12
Demonstration Problem 8.1
X  10.455, s  7.7, and n  44.
90% confidence  Z  1645
.
X Z
s
 X Z
s
n
n
7.7
7.7
10.455  1645
.
   10.455  1645
.
44
44
10.455  191
.    10.455  191
.
8.545    12.365
Pr ob[8.545    12.365]  0.90
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-13
Demonstration Problem 8.2
X  34.3, s  8, N = 800 and n  50.
98% confidence  Z  2.33
s N n
s N n
X Z
 X Z
n N 1
n N 1
8
800  50
8
800  50
34.3  2.33
   34.3  2.33
50 800  1
50 800  1
34.3  2.554    34.3  2.554
3175
.    36.85
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-14
Confidence Interval to Estimate 
when n is Large and s is Unknown
S
X  Z
n
or
S
S
X  Z
   X  Z
n
n
2
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
8-15
Car Rental Firm Example
X  85.5, S  19.3, and n  110.
99% confidence  Z  2.575
S
S
X Z
  X Z
n
n
19.3
19.3
85.5  2.575
   85.5  2.575
110
110
85.5  4.7    85.5  4.7
80.8    90.2
Pr ob[80.8    90.2]  0.99
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-16
Z Values for Some of the More
Common Levels of Confidence
Confidence
Level
Z Value
90%
1.645
95%
1.96
98%
2.33
99%
2.575
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-17
Estimating the Mean of a Normal
Population: Small n and Unknown s
• The population has a normal distribution.
• The value of the population standard
deviation is unknown.
• The sample size is small, n < 30.
• Z distribution is not appropriate for these
conditions
• t distribution is appropriate
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-18
The t Distribution
• Developed by British statistician, William
Gosset
• A family of distributions -- a unique
distribution for each value of its parameter,
degrees of freedom (d.f.)
• Symmetric, Unimodal, Mean = 0, Flatter
than a Z
X


• t formula t 
S
n
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-19
Comparison of Selected t Distributions
to the Standard Normal
Standard Normal
t (d.f. = 25)
t (d.f. = 5)
t (d.f. = 1)
-3
-2
-1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0
1
2
3
8-20
Table of Critical Values of t
df
1
2
3
4
5
t0.100 t0.050 t0.025 t0.010 t0.005
3.078
1.886
1.638
1.533
1.476
6.314
2.920
2.353
2.132
2.015
12.706
4.303
3.182
2.776
2.571
31.821
6.965
4.541
3.747
3.365
63.656
9.925
5.841
4.604
4.032
1.714
25
1.319
1.318
1.316
1.708
2.069
2.064
2.060
2.500
2.492
2.485
2.807
2.797
2.787
29
30
1.311
1.310
1.699
1.697
2.045
2.042
2.462
2.457
2.756
2.750
40
60
120
1.303
1.296
1.289
1.282
1.684
1.671
1.658
1.645
2.021
2.000
1.980
1.960
2.423
2.390
2.358
2.327
2.704
2.660
2.617
2.576
23
24

1.711
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.


t
With df = 24 and = 0.05,
t = 1.711.
8-21
Confidence Intervals for  of a Normal
Population: Small n and Unknown s
S
X t
n
or
S
S
X t
   X t
n
n
df  n  1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-22
Solution for Demonstration Problem 8.3
X  2.14, S  1.29, n  14, df  n  1  13

1.99

 0.005
2
2
t .005,13  3.012
S
S
X t
   X t
n
n
1.29
1.29
2.14  3.012
   2.14  3.012
14
14
2.14  1.04    2.14  1.04
110
.    318
.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-23
Solution for Demonstration Problem 8.3
S
S
X t
   X t
n
n
1.29
1.29
2.14  3.012
   2.14  3.012
14
14
2.14  1.04    2.14  1.04
110
.    318
.
Pr ob[110
.    318
. ]  0.99
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-24
Comp Time: Excel Normal View
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-25
Comp Time: Excel Formula View
B
A
C
D
E
F
1
Comp Time Data
2
6
21
17
20
7
0
3
8
16
29
3
8
12
4
5
11
9
21
25
15
16

=B7+B13*B9
6
n=
7
Mean =
8
S=
9
10
Std Error =
11
=
0.1
12
df =
=B6-1
13
14
t=
15
=COUNT(A2:F4)
=AVERAGE(A2:F4)
=STDEV(A2:F4)
=B8/SQRT(B6)
=TINV(B11,B12)
=B7-B13*B9
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-26
Confidence Interval to Estimate
the Population Proportion
p  Z 
2

pq
 P  p  Z 
n
2

pq
n
where:
p = sample proportion
q = 1 - p
P = population proportion
n = sample size
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-27
Solution for Demonstration Problem 8.5
X
34
n  212, X  34, p 

 016
.
n 212
q = 1 - p  1  016
.  0.84
90% Confidence  Z  1645
.

pq
 P  p  Z
n
p  Z

pq
n
(0.16)(0.84)
(0.16)( 0.84)
0.16  1.645
 P  0.16  1.645
212
212
0.16  0.04  P  0.16  0.04
0.12  P  0.20
Pr ob[012
.  P  0.20]  0.90
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-28
Population Variance
• Variance is an inverse measure of the group’s
homogeneity.
• Variance is an important indicator of total quality
in standardized products and services. Managers
improve processes to reduce variance.
• Variance is a measure of financial risk. Variance of
rates of return help managers assess financial and
capital investment alternatives.
• Variability is a reality in global markets.
Productivity, wages, and costs of living vary
between regions and nations.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-29
Estimating the Population Variance
• Population Parameter s
• Estimator of s
 X  X 
2
S
2

n 1
•  formula for Single Variance


n

1
S
 
2
2
s
2
degrees of freedom = n - 1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-30
Confidence Interval for s2
 n  1 S

2
2

2
s 
2
 n  1 S

2
2
1

2
df  n  1
  1  level of confidence
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-31
Selected 2 Distributions
df = 3
df = 5
df = 10
0
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-32
2 Table
df
0.975
0.950
1 9.82068E-04 3.93219E-03
2
0.0506357
0.102586
3
0.2157949
0.351846
4
0.484419
0.710724
5
0.831209
1.145477
6
1.237342
1.63538
7
1.689864
2.16735
8
2.179725
2.73263
9
2.700389
3.32512
10
3.24696
3.94030
0.100
2.70554
4.60518
6.25139
7.77943
9.23635
10.6446
12.0170
13.3616
14.6837
15.9872
0.050
3.84146
5.99148
7.81472
9.48773
11.07048
12.5916
14.0671
15.5073
16.9190
18.3070
0.025
5.02390
7.37778
9.34840
11.14326
12.83249
14.4494
16.0128
17.5345
19.0228
20.4832
df = 5
0.10
0
20
21
22
23
24
25
9.59077
10.28291
10.9823
11.6885
12.4011
13.1197
10.8508
11.5913
12.3380
13.0905
13.8484
14.6114
28.4120
29.6151
30.8133
32.0069
33.1962
34.3816
31.4104
32.6706
33.9245
35.1725
36.4150
37.6525
34.1696
35.4789
36.7807
38.0756
39.3641
40.6465
70
80
90
100
48.7575
57.1532
65.6466
74.2219
51.7393
60.3915
69.1260
77.9294
85.5270
96.5782
107.5650
118.4980
90.5313
101.8795
113.1452
124.3421
95.0231
106.6285
118.1359
129.5613
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
5
10
15
20
9.23635
With df = 5 and  =
0.10, 2 = 9.23635
8-33
Two Table Values of 2
df = 7
.05
.95
.05
0
2
4
6
8
10
12
14
2.16735
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
16
18
20
df
1
2
3
4
5
6
7
8
9
10
0.950
3.93219E-03
0.102586
0.351846
0.710724
1.145477
1.63538
2.16735
2.73263
3.32512
3.94030
0.050
3.84146
5.99148
7.81472
9.48773
11.07048
12.5916
14.0671
15.5073
16.9190
18.3070
20
21
22
23
24
25
10.8508
11.5913
12.3380
13.0905
13.8484
14.6114
31.4104
32.6706
33.9245
35.1725
36.4150
37.6525
14.0671
8-34
90% Confidence Interval for s2
S
.0022125, n  8, df  n  1  7,  .10
2

2



2

2
1

2

2
.1
2



2
1
.1
2
2
.05

 14.0671

2
.95
 n  1 S 2

2

 2.16735
s 
2
2
 8  1.0022125
14.0671
s 
2
 n  1 S 2

2
1

2
 8  1.0022125
2.16735
.001101  s .007146
2
Pr ob[0.001101  s  0.007146]  0.90 8-35
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Solution for Demonstration Problem 8.6
S


2
 1.2544, n  25, df  n  1  24,   .05

  .05  
2
2
2
2
2
1

2

2
1
.05
2
2
.025

n  1 S 2

2

 39.3641
2
.975
 12.4011
s 
2
2
25  1(1.2544) 
s
0.7648  s
2
39.3641
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2

n  1 S 2

2
1

2
25  1(1.2544)
12.4011
 2.4277
8-36
Determining Sample Size
when Estimating 
• Z formula
Z
X 
s
n
• Error of Estimation E  X  
(tolerable error)
2
2
2
• Estimated Sample Size
Z 2 s  Z 2 s 
n
• Estimated s
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
s
E
2


 E 
1
range
4
8-37
Sample Size When Estimating : Example
E  1, s  4
90% confidence  Z  1645
.
Zs
2
n
2
2
2
E
(
1645
.
)
(
4
)

1
2
2
2
 43.30 or 44
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-38
Solution for Demonstration Problem 8.7
E  2, range  25
95% confidence  Z  196
.
1
 1
estimated s :
range     25  6.25
 4
4
Zs
E
(196
. ) (6.25)

2
2
2
n
2
2
2
2
 37.52 or 38
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-39
Determining Sample Size
when Estimating P
•Z
formula
pP
Z
PQ
n
• Error of Estimation (tolerable
E  pP
error)
• Estimated Sample
2
PQ
Size
n Z 2
E
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-40
Solution for Demonstration Problem 8.8
E  0.03
98% Confidence  Z  2.33
estimated P  0.40
Q  1  P  0.60
2
PQ
Z
n
E
 0.40 0.60
(
2
.
33
)

.003
2
2
2
 1,447.7 or 1,448
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-41
Determining Sample Size when
Estimating P with No Prior Information
P
PQ
0.5
0.25
Z = 1.96
E = 0.05
400
350
300
0.4
0.24
0.3
0.21
250
n 200
150
0.2
0.16
0.1
0.09
100
50
0
0
Z
n
E
2
0.1
0.2
0.3
0.4
0.5
P
0.6
0.7
0.8
0.9
1
1
4
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-42
Example: Determining n when
Estimating P with No Prior Information
E  0.05
90% Confidence  Z  1645
.
with no prior estimate of P, use P  0.50
Q  1  P  0.50
2
PQ
Z
n
E
(1645
. )  0.50 0.50

.05
2
2
2
 270.6 or 271
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8-43