Define the following: variable of interest sampling units population population size sampling frame
variable of interest: the variable of interest on a specific measurement occasion
sampling units: objects on which you measure the variables of interest
population: the collection of all the elements of interest
population size: the number of elements in the population
sampling frame: a list of all the elements in a population
what is a sample
a group of elements (a subset of the population) selected in some manner from the population, size of the sample is n
random selection v. systematic selection
random: choosing the elements that will be in the sample using some procedure that depends on random chance
- allows us to use probability theory to make inferences about the population
systematic selection: choosing elements according to some pattern or system
- widely used in forestry
what are parameters? give examples
characteristics of a population
- typically denoted in greek letters
- exampled include: population standard deviation (sigma), population variance (sigma squared), population mean (mu), population total (tan)
What is a statistic? give examples
a characteristic of a sample that may be used to estimate a population parameter, its a point estimate of the corresponding parameter
- mean of y (y-bar)
- standard deviation of y (sy)
- variance of y (sy squared)
- total of y (yT)
What are interval estimates
an interval around the point estimate with a width determined so that the probability that the interval will contain the parameter of interest matches some desired level
- ex. 95% confidence interval for Uy is an interval estimate
Define bias, precision and accuracy in terms of statistics
bias: the difference between the expected value of a statistic used to estimate a parameter and that parameter
precision: the spread of a statistic, calculated from repeated samples, about its long-run means
accuracy: how close a statistics calculated from a particular sample might be to its associated parameter
- can be measured using mean squared error (MSE)
What is relative efficiency
relative efficiency (RE): the efficiency of one sampling method compared to another sampling method - ratio of variances of a given statistic obtained on the same population using an identical sample size
Why do we sample?
- considerably quicker and cheaper than a census
- information required may involve destructive measures
- sampling may be more accurate than census
what is simple random sampling
SRS: based on a selection procedure where every combination of n elements in the population has an equal chance of bein the sample
What are the steps involved in SRS
- obtain a sampling frame
- select the sampling plots
- measure the elements selected, record the measurements, and produce summaries
- always sum all the different values for y, and sum all the different squared values for y
- calculate point estimates
- sample mean
- population total
- variance of the observations (which estimates the population variance)
- standard deviation of observations (which estimates the population sd)
- square root of the variance! - standard deviation of the mean, variance of the sample mean
- calculate interval estimates
- for estimates of the mean and total
- Yt +/- t(n-1,1-a/2) x Syt
What is AE
Allowable sampling error: half the width of the widest confidence interval that you would be willing to accept
- used to determine sample size
- t-value times the standard deviation of the mean
When should SRS be used?
- population is relatively homogeneous
- a sampling frame is available
- no other information is available
What are the difficulties of SRS?
- time consuming
- does not ensure complete coverage of the population
- does not take advantage of other information that may be available
What is STRS
Stratified random sampling: when the population is not homogenous, they can be split up into different strata, each stratum is then treated as a separate sub population and sampled separately, the results are then combined using appropriate weights to obtain overall estimates for the population
What is the STRS procedure
- obtain a sampling frame and stratify the population
- select the sample plots
- measure the elements selected, record the measurements, and produce summaries
- sum of the y’s and sum of the squared ys for each stratum
- mean y, variance y, standard deviation of y, standard deviation of the mean for each stratum - calculate point and interval estimates for the parameters of interest within each stratum
- equations are the same as for SRS - calculate point and interval estimates for the population as a whole
- this is where equations start to differ
explain effective degrees of freedom
before we construct confidence limits we need to determine the appropriate degrees of freedom for our estimate of the overall mean, EDF is a good approximation of the degrees of freedom and it falls between over approximating and under approximating
What are four ways we can allocate the sampling units among the different strata?
- equal allocation: each stratum gets an equal number of sampling units
- does not take into account the size of the stratum or variability of the strata - proportional allocation: assigns sample size proportionally to the relative sizes of the strata
- frequently used if total sample size is given - Neyman allocation: takes the variation within the different strata into account, as well as the size of the strata
- requires standard deviation for each stratum - optimum allocation: takes the cost of sampling into account
- will achieve a given error level (AE) for the lowest cost
When should STRS be used?
- the population is heterogeneous, but can be separated into distinctive, more homogeneous groups
- the strata size are known in advance of sampling
- a sampling frame is available
- we are interested in estimating parameters for the various strata
What is systematic sampling?
selecting sampling units according to some system
- ex. select every tenth element in a population
- provides unbiased estimates of the mean and total
What are the advantages of systematic sampling over random sampling
- the location of elements is cheaper (faster)
- a detailed sampling frame is not required
- ensures complete coverage of the population
- the element included in any sample can be relocated easily
What are two types of systematic sampling, explain them
strip cruises: long rectangle plots, strip cruise consists of selecting strips systematically from all the strips that could be established on the land area
line plots: plots (point centres) are located at equal intervals along lines that are also located at equal intervals
