Statistical Inference and Sampling Introduction to Business Statistics

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, 5 e 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, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Random Sampling and Estimation Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College

Distribution of X • The mean of the probability distribution for X= x • Standard error of X = standard deviation of the probability distribution for X = Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Distribution of X Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Probabilities in the Sampling Distribution of X Figure 7. 8 Introduction to Business Statistics,

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 Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing x = n

Central Limit Theorem Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( known) Introduction to Business Statistics,

Confidence for the Mean of a Normal Population ( known) P(-1. 96 Z 1.

Confidence for the Mean of a Normal Population ( known) (1 - ) 100%

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, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Student’s t Distribution Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Confidence for the Mean of a Normal Population ( unknown) X– t = s/

Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

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 – Equation: Introduction to Business Statistics, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Selecting Necessary Sample Size Known æ ö E = Z / 2 ç ÷

Selecting Necessary Sample Size Unknown é Z / 2 × s ù n=ê ú

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 Introduction to Business Statistics, 5 e 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, 5 e 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, 5 e 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, 5 e 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, 5 e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing