Statistical Inference and Sampling

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

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.Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Estimation

• The population consists of every item of interest.

• The sample is randomly drawn from the population.

• Sample values should be selected randomly, one at a time, from the population.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Random Sampling and Estimation

Figure 7.1Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Distribution of X

• The mean of the probability distribution for X =

• Standard error of X = standard deviation of the probability distribution for X = / n.

x

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Distribution of X

Figure 7.6Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Distribution of X

mean x

standard deviation (standard error) x n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Probabilities in the Sampling Distribution of X

Figure 7.8

P(X 22) PX – 20

.77 22 – 20

.77

P(Z 2.60)

.5 – .4953 .0047

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Central Limit Theorem

When obtaining large samples 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 and a standard deviation (standard error) of

x n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Central Limit Theorem

Figure 7.10Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Central Limit Theorem

Figure 7.11Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Central Limit Theorem

Figure 7.12Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( known)

Figure 7.16Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( known)

ZX – / n

P –1.96 X – / n

1.96

.95

P X –1.96n

X 1.96n

.95

P(-1.96 Z 1.96)

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( known)

(1-) 100% Confidence Interval

x – Z /2n

, x Z /2

n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( unknown)

Student’s t Distribution

• Population variance unknown

• Degrees of freedom = n - 1

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Student’s t Distribution

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( unknown)

X – s/ n

t =

x – t / 2,n–1s

n

to x t / 2,n–1

s

n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Selecting Necessary Sample SizeKnown

• 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– Equation:

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Selecting Necessary Sample SizeKnown

E Z / 2

n

nZ / 2

E

2

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Selecting Necessary Sample SizeUnknown

H – L

4

nZ / 2s

E

2

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

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 elementsIntroduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Other Sampling Procedures

• Strata: are nonoverlapping subpopulations.

• Sampling design: specifies the manner in which the sampling units are to be selected.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

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.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Stratified Sampling

• Stratified sampling obtains more information due to the homogenous nature of each strata.

• Stratified sampling obtains a cross section fo the entire population.

• Obtain a mean within each strata as well as an estimate of .

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

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

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing