1. Chapter 7 – Statistical Inference and Sampling
Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 7 – Statistical Inference and Sampling
Slides prepared by Jeff Heyl, Lincoln University
©2003 South-Western/Thomson Learning™

2. Simple Random Sampling
All items in the population have the same probability of being selected
Finite Population: To be sure that a simple random sample is obtained from a finite population the items should be numbered from 1 to N
Nearly all statistical procedures require that a random sample is obtained

3. Estimation The population consists of every item of interest
Population mean is µ and is generally not known
The sample is randomly drawn from the population
Sample values should be selected randomly, one at a time, from the population

4. Random Sampling and Estimation
Figure 7.1
Population (mean = µ)
Sample (mean = X)
X estimates µ

5. Distribution for Everglo Bulb Lifetime|
µ = 400 X
 = 50
300
350
450
500
Figure 7.2

6. Sample Means
Figure 7.3

7. Excel Histogram Frequency Histogram Class Limits Figure 7.4 8 7 6 5 43 2 1
377 and 384 and 391 and 398 and 405 and 412 and 419 and 426 and
under 384 under 391 under 398 under 405 under 412 under 419 under 426 under 433
Class Limits
Figure 7.4

8. Distribution of X
The mean of the probability distribution for X = µX = µ
Standard error of X = standard deviation of the probability distribution for X = X =  n

9. Normal Curves x X Population (mean = µ, standard deviation = )
Random sample (mean = X,
standard deviation = s
 = 50 x
X = value from this population
x = 50 10 X
X follows a normal distribution, centered at µ with a standard deviation  / n
µx = 400
Assumes the individual observations follow a normal distribution
Figure 7.5

10. Central Limit Theorem
When obtaining large samples (n > 30) from any population, the sample mean X will follow an approximate normal distribution
What this means is that if you randomly sample a large population the X distribution will be approximately normal with a mean µ and a standard deviation (standard error) of
x =  n

11. Distribution of X Figure 7.6  = 50 µ = 400 Population X µx = 400
= 15.81 50 10
(n = 50)
= 7.07
(n = 20)
= 11.18 20
(n = 100)
= 5
100
Figure 7.6

12. Distribution of X Mean = µx = µ Standard deviation = x = 
(standard error)  n

13. Assembly Time | 14 17 µ = 20 23 26 22  = 3 Area = P(X > 22)
X = assembly line
Figure 7.7

14. Assembly Time | 14 19.23 µx 20.77 22 x = .77 Area = P(X > 22) X
Figure 7.8

15. Assembly Time | 19
µx = 20 21
Area = P(19 < X < 21) X
Figure 7.9

16. Central Limit Theorem Figure 7.10 Uniform population µ = = 100
µ = = 100
 = = 28.87
a + b 2
b - a 12
a = 50
µ = 100
b = 150 X
(n = 2)
(n = 5)
(n = 30) µx
By the CLT, µx = µ = 100
x = = = 5.27  n
28.87 30
Figure 7.10

17. Exponential population
Central Limit Theorem
µ = 100 =  X
Exponential population
(n = 2)
(n = 5)
(n = 30) µx
By the CLT, µx = µ = 100
x = = = 18.26  n
100 30
Figure 7.11

18. Central Limit Theorem Figure 7.12 | µ X U-shaped population (n = 2)

19. Sampling Without Replacement
Mean = µx = µ
Standard deviation = x = •
(standard error)  n
N - n
N - 1

20. Distribution of Sample Mean
µx = 48,000
X = average income of 45 female managers
x = =
= ( )(.935)
= $  n
N - n
N - 1
8500 45
Observed value of X = $43,900
Figure 7.13

21. Confidence Intervals X
µ = ?
 known
Figure 7.14

22. Confidence Intervals 3 x = = .6 minute 25 Area = P(X > 20) = .5
X = average of 25 assembly lines
x = = .6 minute 3 25
Area = P(X > 20) = .5
µx = 20
Figure 7.15

23. Confidence Intervals Area = .475 -1.96 1.96 Total area = .95 Z
-1.96
1.96
Total area = .95 Z
Figure 7.16

24. Confidence for the Mean of a Normal Population ( known)
P(-1.96  Z  1.96) = .95
Z =
X - µ
 / n
P ≤ ≤ = .95
P X ≤ µ ≤ X = .95  n

25. Confidence for the Mean of a Normal Population ( known)
(1 - ) • 100% Confidence Interval
x - Z/ , x + Z/2  n
E = margin of error = Z/2  n

26. Confidence for the Mean of a Normal Population ( known)
Area = .1
1.96 Z
Area = .05
Area = .025
1.645
1.28
Figure 7.17

27. Excel Screens
Figure 7.18

28. Excel Screens
Figure 7.19

29. Excel Screens
Figure 7.20

30. Confidence for the Mean of a Normal Population ( unknown)
Student’s t Distribution
Population variance unknown
Degrees of freedom = n - 1
x - t/2, n to x + t/2, n - 1 s n

31. Student’s t Distributiont
Standard normal, Z
t curve with 20 df
t curve with 10 df
Figure 7.21

32. Confidence Interval
Figure 7.22

33. Confidence Interval
Figure 7.23

34. Selecting Necessary Sample Size Known 
Sample size based on the level of accuracy required for the application
Maximum error: E
Used to determine the necessary sample size to provide the specified level of accuracy
Specified in advance

35. Selecting Necessary Sample Size Known 
E = Z/2  n
n =
Z/2 •  E 2

36. Selecting Necessary Sample Size Unknown 
To obtain a rough approximation, ask someone who is familiar with the data to be collected:
What do you think will be the highest value in the sample (H)?
What will be the lowest value (L)?

37. Selecting Necessary Sample Size Unknown 
Z/2 • s E 2
 
H - L 4

38. Other Sampling Procedures
Population: the collection of all items about which we are interested
Sampling Unit: a collection of elements selected from the population
Cluster: a sampling unit that is a group of elements from the population, such as all adults in a particular city block
Sampling frame: a list of population elements

39. Other Sampling Procedures
Strata: are nonoverlapping subpopulations
Sampling design: specifies the manner in which the sampling units are to be selected

40. Simple Random Sampling
Population mean: µ
Estimator:
Estimated standard error of X:
Approximate confidence interval:
X = ∑x n 
N - n
N - 1
sx = •
X ± Z/2sx

41. Systematic Sampling The sampling frame consists of N records
The sample of n is obtained by sampling every kth record, where k is an integer approximately equal N/n
The sampling frame should be ordered randomly

42. Stratified Sampling
Stratified sampling obtains more information due to the homogenous nature of each strata
Stratified sampling obtains a cross section of the entire population
Obtain a mean within each strata as well as an estimate of 

43. Stratified Sampling Use the following notation:
ni = sample size in stratum i
Ni = number of elements in stratum i
N = total population size = ∑Ni
n = total sample size = ∑ni
Xi = sample mean in stratum i
si = sample standard deviation in stratum i

44. Stratified Sampling Population mean: µ Estimator:
Estimated standard error of X:
Approximate confidence interval:
Xst =
∑NiXi N
sx = ∑ st Ni 2
Ni - ni si ni
Xst ± Z/2sx

45. Cluster Sampling
Single-stage cluster sampling: randomly select a set of clusters for sampling
Include all elements in the cluster in your sample
Two-stage cluster sampling: randomly select a set of clusters for sampling
Randomly select elements from each sampled cluster

46. Cluster Sampling Population mean: µ Estimator: ∑Ti
Estimated standard error of Xc:
Approximate confidence interval:
Xc =
∑Ti
∑ni
Xc ± Z/2sx c
sx =
M - m
mMN2
∑(Ti - Xcni)2
m - 1

47. obtain approximate confidence
Confidence Interval
Constructing a Confidence Interval for a Population Mean
 known
 unknown
Use Table A-4 (Z)
Use Table A-5 (t)
Can use Table A-4 (Z) to
obtain approximate confidence
interval if n > 30
Figure 7.25