Agricultural and Biological Statistics

Sampling and Sampling DistributionsChapter 5

Two types of sampling procedures 1. Probability based procedures 2. Convenience Sampling

Probability Based Sampling Set up sampling procedure so that every

element in the population has a known chance of being included in the sample.

Allows use of probability statements with the analysis of the sample data.

Several Methods Simple Random Sampling Systematic Sampling Stratified Sampling Cluster Sampling Sequential Sampling

Choice of Method depends on several factors.. Whether the population is homogeneous.

Homogeneous means you can use random or systematic sampling

Stratified or cluster sampling is necessary when attributes you are looking at are concentrated at different levels among different groups of the population.

Factors continued The degree of accuracy required

Select sample design that make variance of sample statistics as small as possible for the size of sample we have chosen.

Sampling Error is the difference between the value of the statistic or sample variable.

Factors continued The Cost of the Sampling Plan

Increase reliability, increase size, but increase costs as well.

Efficiency- A sample design is more efficient than another if it results in lower costs, but the same degree of reliability.

Simple Random Sampling Every element in the population available for

sampling has an equal probability of being selected.

Randomness is CRUCIAL Example: Survey students at the door of the

Student Union. Not Random. Number of items placed in container at door, one

unit at a time.

Systematic Sampling Used if we have access to a list of the

population. With this procedure we obtain the sample by

taking every kth unit in the population, where k stands for some whole number that is approx. the sampling ratio N/n. N= 10,000 listed names n= 500 K=N/n=20 thus we select a sample by taking every 20th

unit in the population.

Systematic Sampling Cont’d Must choose a random starting point however. Randomly select a number between 1 and 20 If it is 8 then the sample consists of 8, 28, 48,

until we get all 500 values. Don’t use it for daily grocery store sales.

Stratified Sampling Know something in advance about the population. More efficient in this case than simple random

sampling. Strata are chosen so that they contain similar

characteristics and are more homogeneous than the population as a whole.

Then sample each stratum using a simple random approach.

Proportionate Stratified Sample Assures number in the sample from each

stratum is proportional to the number of items in the population for the stratum. Example: Soil classes for stratum when looking at

yields.

Disproportionate Sample Smaller proportion from lower standard

deviation. Useful in handling heterogeneous

populations.

Cluster Sampling Is diametrically opposed to stratified sampling Select groups of individual items called clusters from the

population at random, and then choose all or a sub sample of the items within each cluster to make up the overall sample.

Want differences between clusters to be as small as possible and differences between items within the cluster to be as large as possible.

We want a cluster to be a miniature of the population, so that any cluster is a representative sample.

Advantages: Low cost for a given degree of reliability

Sequential Sampling Widely used in quality control Involves testing a relatively small sample in

QC, and on the basis of the sample outcome, deciding whether to accept or reject the lot.

Keeps cost low. If small sample size does not lead to a clear

decision then we just increase then sampling size.

Nonprobability Samples Employ these for cost or efficiency reasons.

Convenience Sampling- men on the street. Judgment Sampling- use judgment, farmers with

best management practices Quota Sampling- construct sample to look like

population 20 - 60 - 20

wheat cotton livestockfarmers farmers farmers

Sampling Distribution Draw repeated samples from the same population. Sampling distribution of the mean and variance

Calves Birth Weights

X= {90 80 100 80 90 100}

Mean µ is 90 pounds

Variance 2 is 66.67

Draw randomly all different combinations of 2 (sample size) there are:

6C2=6!/(6-2)!(2!)=15

Total (Σx) Sample Mean ( x )x1x2 90,80 170 85

x1x3 90,100 190 85

x1x4 90,80 170

x1x5

x1x6

x2x3

x2x4

Using the sample mean as a random variable to construct a sampling distribution of the mean.

Since the sample means have frequencies of occurrence we can place them in a probability distribution.

Sample m x Freq. (f) Probability P(x)

x· P(x)

80 1 1/15 80/15

85 4 4/15 340/15

90 5 5/15 450/15

95 4 4/15 380/15

100 1 1/15 100/15

Totals 15 15/15 =1 1350/15=90

Use Expected Value Formula

This gives the same value as the population mean. This relationship always holds and is formally stated by the central limit theorem.

Central Limit Theorem If random samples of size n observations are drawn

from a population with a finite mean µ and standard deviation , then the sample mean x is approximately normally distributed with mean µ and standard deviation /√n when n is large.

Example

90

10.3x

3267686.

70.7170.72 z

Sample population

x

xxz

486833.9

1.3

Z=1.22 P=.3888