Agricultural and Biological Statistics

Sampling and Sampling Distributions Chapter 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 k th 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 20 th 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 wheat cottonlivestock farmers farmers farmers

Sampling Distribution  Draw repeated samples from the same population.  Sampling distribution of the mean and variance Calves Birth Weights X= { } Mean µ is 90 pounds Variance 2 is 66.67

Draw randomly all different combinations of 2 (sample size) there are: 6 C 2 = 6! / (6-2)!(2!) =15 Total (Σx) Sample Mean ( x ) x1x2x1x2 90, x1x3x1x3 90, x1x4x1x4 90,80170 x1x5x1x5 x1x6x1x6 x2x3x2x3 x2x4x2x4

 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 xFreq. (f)Probability P(x) x· P(x) 8011/1580/ /15340/ /15450/ /15380/ /15100/15 Totals1515/15 =11350/15=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 Sample population Z=1.22P=.3888