Chapter07 - Sampling and Sampling Distributions

  Statistics for Managers Using Microsoft® Excel 5th Edition

Learning Objectives

  In this chapter, you will learn:

   To distinguish between different survey sampling methods

   The concept of the sampling distribution

   To compute probabilities related to the sample mean and the sample proportion

   The importance of the Central Limit Theorem

Why Sample?

   than selecting every item in the population (census). Selecting a sample is less costly than

  Selecting a sample is less time-consuming

   selecting every item in the population.

An analysis of a sample is less cumbersome

   and more practical than an analysis of the entire population.

  Types of Samples Quota Samples Non-Probability Samples Judgment Chunk Probability Samples Simple Random Systematic Stratified Cluster Convenience

  Types of Samples In a nonprobability sample , items included

   are chosen without regard to their probability of occurrence.

   based only on the fact that they are easy, inexpensive, or convenient to sample. In a judgment sample, you get the opinions of

  In convenience sampling, items are selected

   pre-selected experts in the subject matter. Types of Samples In a probability sample, items in the sample

  

are chosen on the basis of known probabilities.

  Probability Samples Simple Systematic Stratified Cluster Random

Simple Random Sampling

  Every individual or item from the frame has an

   equal chance of being selected Selection may be with replacement (selected

   individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame).

Samples obtained from table of random numbers

   or computer random number generators.

  Systematic Sampling

   Decide on sample size: n

   Divide frame of N individuals into groups of k individuals: k=N/n

   Randomly select one individual from the 1st group

   Select every kth individual thereafter For example, suppose you were sampling n = 9 individuals from a population of N = 72. So, the population would be divided into k = 72/9 = 8 groups. Randomly select a member from group 1, say individual 3. Then, select every 8 ^th individual thereafter (i.e. 3, 11, 19, 27, 35, 43, 51, 59, 67)

Stratified Sampling

   (called strata) according to some common characteristic. A simple random sample is selected from each

  Divide population into two or more subgroups

   subgroup, with sample sizes proportional to strata sizes. Samples from subgroups are combined into one.

   This is a common technique when sampling

   population of voters, stratifying across racial or socio-economic lines.

Cluster Sampling

  Population is divided into several “clusters,” each

   representative of the population. A simple random sample of clusters is selected.

   All items in the selected clusters can be used, or items

   can be chosen from a cluster using another probability sampling technique. A common application of cluster sampling involves

   election exit polls, where certain election districts are selected and sampled.

  

Comparing Sampling Methods

   Simple random sample and Systematic sample Simple to use

  

May not be a good representation of the population’s

   underlying characteristics Stratified sample

    Ensures representation of individuals across the entire population

   More cost effective

  Cluster sample

   Less efficient (need larger sample to acquire the same

   level of precision)

Evaluating Survey Worthiness

  What is the purpose of the survey?

   Were data collected using a non-probability

   sample or a probability sample? Coverage error – appropriate frame?

   Nonresponse error – follow up

   Measurement error – good questions elicit good

   responses Sampling error – always exists 

Types of Survey Errors

  Coverage error or selection bias

   Exists if some groups are excluded from the frame

   and have no chance of being selected Non response error or bias

   People who do not respond may be different from

   those who do respond Sampling error

   Chance (luck of the draw) variation from sample to

   sample.

  Measurement error

   Due to weaknesses in question design, respondent

   error, and interviewer’s impact on the respondent

Sampling Distributions

   possible values of a statistic for a given size sample selected from a population.

  A sampling distribution is a distribution of all of the

   For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are

interested in the distribution of all potential mean GPA

we might calculate for any given sample of 50 students.

  Sampling Distributions Sample Mean Example Suppose your population (simplified) was

   four people at your institution. Population size N=4

   Random variable, X, is age of individuals

   Values of X: 18, 20, 22, 24 (years)  Sampling Distributions Sample Mean Example Summary Measures for the Population Distribution:

  21

  4

  24

  22

  20

  X μ ⁱ     

   

  2.236 N μ) (X σ ^{2 i} 

    

  .3 .2 .1 18 20 22 24 A B C D P(x) x

  Uniform Distribution

Sampling Distributions Sample Mean Example

  23

  21

  22

  22

  20

  21

  22

  24

  19

  21

  22

  23

  24

  16 Sample Means 16 possible samples

  (sampling with replacement)

  20

  1 ^st Obs.

  2 ^nd Observation

  2 ^nd Observation

  18

  20

  22

  24 18 18,18 18,20 18,22 18,24 20 20,18 29,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24

  Now consider all possible samples of size n=2

  1 ^st Obs.

  18

  21

  20

  22

  24

  18

  18

  19

  20

  20 Sampling Distributions Sample Mean Example Sampling Distribution of All Sample Means

  1 ^st Obs

  20

  Distribution

  (no longer uniform) Sample Means

   X

  18 19 20 21 22 23 24 .1 .2 .3

  16 Sample Means

  24

  23

  22

  21

  24

  23

  22

  21

  22

  2 ^nd Observation

  22

  21

  20

  19

  20

  21

  20

  19

  18

  18

  24

  22

  20

  18

  _

Sampling Distributions Sample Mean Example

  21

  1.58

  Summary Measures of this Sampling Distribution:

   

    

  2 ^{2 2 2 X i X} 

  

  

16

21) - (24 21) - (19 21) - (18 N ) μ

  

  16

    

  X     

  X μ i

  19

  21

  24

X ( σ

  Sampling Distributions Sample Mean Example Population N = 4

  1.58 σ 21 μ

  X  

  X 2.236 σ 21 μ

    Sample Means Distribution n = 2

   18 20 22 24 A B C D .1 .2 .3

  P(X) X 18 19 20 21 22 23 24

  .1 .2 .3

  P(X)

  X _

  _ Sampling Distributions Standard Error Different samples of the same size from the same population

   will yield different sample means. A measure of the variability in the mean from sample to

   sample is given by the Standard Error of the Mean: σ σ

  

  X n

   sample size increases.

  Note that the standard error of the mean decreases as the

  Sampling Distributions

Standard Error: Normal Pop.

  If a population is normal with mean μ and standard

   deviation σ, the sampling distribution of the mean is also normally distributed with

  σ σ

  and 

  μ μ 

  X X n

  (This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

  Sampling Distributions Z Value: Normal Pop.

   Z-value for the sampling distribution of the sample mean:

  ( X μ )  ( X μ) _X  Z

    σ σ _X n where: = sample mean

  X μ = population mean

  

= population standard deviation

σ

  n = sample size Sampling Distributions Properties: Normal Pop.

  Normal Population Distribution

  μ μ  x

  μ x x

  (i.e. is unbiased )

  Normal Sampling Distribution (has the same mean)

  μ x x Sampling Distributions Properties: Normal Pop.

   For sampling with replacement:

   As n increases,

   decreases ^x

  σ Larger sample size Smaller sample size x

  μ Sampling Distributions Non-Normal Population

  The Central Limit Theorem states that as the sample

   size (that is, the number of values in each sample) gets large enough, the sampling distribution of the mean is approximately normally distributed. This is true regardless of the shape of the distribution of the individual values in the population. Measures of the sampling distribution:

   σ σ μ μ

   

  x

  x n Sampling Distributions Non-Normal Population

  Population Distribution

  μ x

  Sampling Distribution (becomes normal as n increases)

  Larger Smaller sample sample size size x

  μ _x

Sampling Distributions Non-Normal Population

  For most distributions, n > 30 will give a

   sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 will

   give a sampling distribution that is nearly normal For normal population distributions, the

   sampling distribution of the mean is always normally distributed

Sampling Distributions Example

  Suppose a population has mean μ = 8 and standard

   deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between

   7.75 and 8.25? Even if the population is not normally distributed, the

   central limit theorem can be used (n > 30).

   So, the distribution of the sample mean is approximately normal with

  σ

  3 μ

  8

_{x   }

 σ _x

  0.5 n

  36

Sampling Distributions Example

  5 .

  36

  3 8 - 8.25 5 .

  36

  3 8 -

  7.75     

  Z Z First, compute Z values for both 7.75 and 8.25.

  0.3830 0.5) Z P(-0.5 8.25) μ P(7.75 _X      

  Now, use the cumulative normal table to compute the correct probability. Sampling Distributions Example Population Distribution

  = 2(.5000-.3085) = 2(.1915) = 0.3830

  μ  8 _Sample

  X Sampling Standardized Normal

  Distribution Distribution 7.75 8.25 -0.5 0.5

  μ Z  x z

  μ

  8 _X  Sampling Distributions The Proportion The proportion of the population having some

   characteristic is denoted π.

   Sample proportion ( p ) provides an estimate of π: X number of items in the sample having the characteri stic of interest p

    n sample size

   p has a binomial distribution

  0 ≤ p ≤ 1

   (assuming sampling with replacement from a finite population or without replacement from an infinite population)

  Sampling Distributions The Proportion

   Standard error for the proportion: n ) (1

  σ _p     n

  

) (1

σ Z _{p  }

     

     p p

   Z value for the proportion:

  Sampling Distributions The Proportion: Example

   If the true proportion of voters who support Proposition A is π = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45?

   In other words, if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ?

  Sampling Distributions The Proportion: Example (1 ) .4(1 .4)  

    σ

  .03464 _p    Find : σ p n 200

  Convert to .40 .40 .45 .40  

    P(.40 .45) P Z

   p       standardized

  .03464 .03464   normal:

  P(0 Z 1.44)    Sampling Distributions The Proportion: Example Use cumulative normal table: P(0 ≤ Z ≤ 1.44) = P(Z ≤ 1.44) – 0.5 = .4251

  Standardized Sampling Distribution

  Normal Distribution _Standardize .4251

  .40 .45

  1.44

  p Z Chapter Summary

   Described different types of samples

   Examined survey worthiness and types of survey errors

   Introduced sampling distributions

   Described the sampling distribution of the mean

   For normal populations

   Using the Central Limit Theorem

   Described the sampling distribution of a proportion

   Calculated probabilities using sampling distributions In this chapter, we have