OPRE Chp 7

Random Sample (infinite population):

A _______________ from an infinite population is a sample selected such that the following conditions are satisfied: (1) Each element selected comes from the same population (2) each element is selected independently.

Steps to minimise nonsampling Errors

Carefully define the target population and design the data collection procedure. Carefully design the data collection process and train the data collectors Pretest the data collection procedure Use stratified random sampling when population-level information about an important qualitative characteristic is available. Use systematic sampling when population-level information about an important quantitative characteristic is available.

Central Limit Theorem

In selecting random samples of size n from a population, the sampling distribution of the sample mean 𝑥 ̅ can be approximated by a normal distribution as the sample size becomes large.

Cluster Sampling - the population is first divided into separate groups of elements called:

clusters

Stratified Random Sampling - formulas are available for:

combining the stratum sample results into one population parameter estimate

you can sample form a ____________ or _____________ population.

finite or infinite

√(N-n)/(N-1)

finite population correction factor

√((𝑁−𝑛)/(𝑁−1))

finite population correction factor.

Convenience Sampling

It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected. The sample is identified primarily by convenience. Example: A professor conducting research might use student volunteers to constitute a sample.

Recommendation

It is recommended that probability sampling methods (simple random, stratified, cluster, or systematic) be used. For these methods, formulas are available for evaluating the "goodness" of the sample results in terms of the closeness of the results to the population parameters being estimated. An evaluation of the goodness cannot be made with non-probability (convenience or judgment) sampling methods.

The number of different simple random samples of size n that can be selected from a finite population of size N is:

N! / n!(N - n)! Number of combinations NCn

Cluster Sampling - Disadvantage:

This method generally requires a larger total sample size than simple or stratified random sampling.

Systematic Sampling - Advantage

This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. Example: Selecting every 100th listing in a telephone book after the first randomly selected listing

Element

is the entity on which data are collected

sampled population

is the population from which the sample is drawn

Populations are often generated by an *ongoing process* where there is:

no upper limit on the number of units that can be generated.

When these conditions are satisfied, the probability distribution of x in the sample proportion, 𝑝 ̅ = x/n, can be approximated by a:

normal distribution (because n is a constant, the sampling distribution of 𝑝 ̅ can also be approximated by a normal distribution).

When the population has a normal distribution, the sampling distribution of 𝑥 ̅ is ___________________ for any sample size.

normally distributed

The sampling distribution of 𝑝 ̅ can be approximated by a normal distribution whenever the sample size is large enough to satisfy the two conditions:

np > 5 and n(1 - p) > 5

Stratified Random Sampling - Each element in the population belongs to:

one and only one stratum.

To estimate the value of a population parameter, we compute a corresponding characteristic of the sample, referred to as a:

sample statistic

to estimate the value of population parameter, we can compute the corresponding characteristic of the sample, referred to as:

sample statistic

Different random numbers will identify a different ________ which would result in different ______________. This difference is to be expected because:

sample, point estimates a sample, not a census of the entire population, is being used to develop the point estimates.

the population from which the sample is actually taken.

sampled population

the probability distribution of all possible values of the sample mean 𝑥 ̅ .

sampling distribution of 𝑥 ̅.

With an infinite population we cannot:

select a simple random sample because we cannot construct a frame consisting of all the elements.

Cluster Sampling - a ___________________ of the clusters are then taken.

simple random sample

Stratified Random Sampling - a _____________ is taken from each stratum

simple random sample

𝜎 𝑥 ̅ = 𝜎 = n = N =

standard deviation of 𝑥 ̅ standard deviation of the population sample size population size

in general, the ________________ refers to the st.dev of the point estimator.

standard error

𝜎 𝑥 ̅ (st.dev) is referred to as the:

standard error of the mean

𝜎_𝑝 ̅ is referred to as the

standard error of the proportion.

Stratified Random Sampling - The population is first divided into groups of elements called:

strata

Often the cost of collecting information from a sample is:

substantially less than from a population, especially when personal interviews must be conducted to collect the information.

the population we want to make inferences about.

target population

The procedures used to select a simple random sample from a finite population are based upon:

the use of random numbers

whenever a sample is used to amok inferences about a population, we should make sure that the ______________ and the _____________ are in ___________ ________________.

targeted population, sampled population, close agreement. (sample needs to be representative of the population.

Why do we select a sample?

to collect data to answer a research question about a population

The purpose of the second condition of the random sample selection procedure (each element is selected independently) is:

to prevent selection bias

When the expected value of the point estimator equals the population parameter, we say that the point estimator is:

unbiased

Infinite population case

when the population is infinitely large or the elements of the population are being generated by an ongoing process for which there is no limit on the number of elements that can be generated. Thus, it is not possible to develop a list or frame of all the elements in the population.

a finite population is treated as being infinite if *n/N* is

≤ .05

Standard deviation of 𝑥 ̅ - finite population =

𝜎 x bar = √N-n / n-1 X (𝜎/√n)

Standard Deviation of 𝑝 ̅ - infinite population

𝜎 𝑝 ̅ =√((𝑝(1−𝑝))/𝑛)

Standard deviation of 𝑥 ̅ - infinite population =

𝜎 𝑥 ̅ = 𝜎/√n

Standard Deviation of 𝑝 ̅ - Finite population

𝜎_𝑝 ̅ = √((𝑁−𝑛)/(𝑁−1)) √((𝑝(1−𝑝))/𝑛)

with proper sampling methods, the sample results can provide:

"good" estimates of the population characteristics.

normalcdf(____,____,_____,____)

(lower bound, upper bound, mean, st. error)

Finite populations are often defined by lists such as:

- organization membership roster - credit card account numbers - inventory product numbers - registered voters

Examples of ongoing processes, with infinite populations are:

- parts being manufactured on a production line - transactions occurring at a bank - telephone calls arriving at a technical help desk - customers entering a store

The random numbers generated using Excel's RAND function follow a uniform probability distribution between:

0 and 1

In most applications, the sampling distribution of 𝑥 ̅ can be approximated by a normal distribution whenever the sample size is

30 or more

in cases where the population is *highly skewed* or *outliers* are present, samples of size ______ may be needed.

50

Simple random sample (finite population)

A _________________ of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. The simplest type of probability sample is one in which each sample of size n has the same probability of being selected. It is called a simple random sample. A simple random sample of size n from a finite population of size N is defined as follows.

Parameters

A numerical characteristic of a population, such as a population mean µ, a population standard deviation, a population proportion p, and so on.

cluster sampling - example:

A primary application is area sampling, where clusters are city blocks or other well-defined areas.

Sample statistic

A sample characteristic, such as a sample mean (x ̅), a sample standard deviation s, a sample proportion (p ̅) , and so on. The value of the sample statistic is used to estimate the value of the corresponding population parameter.

Judgment Sampling Advantage and Disadvantage

Advantage: It is a relatively easy way of selecting a sample. Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample.

Convenience Sampling Advantage and Disadvantage

Advantage: Sample selection and data collection are relatively easy. Disadvantage: It is impossible to determine how representative of the population the sample is.

Reasons for Nonsampling Errors

Coverage error Non-response error - Interviewer error - Processing error Measurement error

Expected Value of 𝑝 ̅

E(𝑝 ̅) = 𝑝 p = the population proportion

Expected value of 𝑥 ̅

E(𝑥 ̅ ) = µ µ = population mean

Systematic Sampling

If a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population. We randomly select one of the first n/N elements from the population list. We then select every n/Nth element that follows in the population list.

Stratified Random Sampling - Best results are obtained when the elements within each stratum are:

as much alike as possible (i.e. a homogeneous group)

the probability distribution of all possible values of the sample proportion 𝑝 ̅.

Sampling Distribution of 𝑝 ̅

Our procedure for selecting a simple random sample of size n from a population of size N involves two steps.

Step 1: Assign a random number to each element of the population. Step 2: Select the *n* elements corresponding to the *n* smallest random numbers. Because each set of n elements in the population has the same probability of being assigned the n smallest random numbers, each set of n elements has the same probability of being selected for the sample. If we select the sample using this two-step procedure, every sample of size n has the same probability of being selected; thus, the sample selected satisfies the definition of a simple random sample.

Stratified Random Sampling - Example:

The basis for forming the strata might be department, location, age, industry type, and so on.

Cluster Sampling - Advantage:

The close proximity of elements can be cost effective (i.e. many sample observations can be obtained in a short time)

Errors in Sampling

The difference between the value of sample statistic and the corresponding value of the population parameters is called the sampling error. Deviations of the sample from the population that occur for reasons other than random sampling are referred to as nonsampling errors. Nonsampling error can occur in a sample or a census.

Judgment Sampling

The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population. It is a nonprobability sampling technique. Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

Situations involving sampling from an infinite population are usually associated with:

a process that operates over time

Cluster Sampling - ideally, each cluster is:

a representative small-scale version of the population

Stratified Random Sampling - Advantage: If strata are homogeneous, this method is:

as "precise" as simple random sampling but with a smaller total sample size.

When the population from which we are selecting a random sample does not have a normal distribution, the ________________________ is helpful in identifying the shape of the sampling distribution of 𝑥 ̅.

central limit theorem

population

is a collection of all the elements of interest

Point estimation:

is a form of statistical inference

frame

is a list of the elements that the sample will be selected from

sample

is a subset of the population

what does the sample result provide? and why?

only estimates of the values of the population characteristics. because the sample contains only a portion of the population.

The value of a point estimator used in a particular instance as an estimate of a population parameter.

point estimate

In _________________ we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter.

point estimation

The sample statistic, such as 𝑥 ̅, s, or p ̅, that provides the point estimate of the population parameter.

point estimator

𝑥 ̅ is the point estimator of the

population mean µ

𝑝 ̅ is the point estimator of the

population proportion p

s is the point estimator of the

population standard deviation 𝜎

The sampling distribution of 𝑥 ̅ can be used to:

provide probability information about how close the sample mean 𝑥 ̅ is to the population mean µ.

Sampling from a Finite Population - sampling with replacement

replacing each sampled element before selecting subsequent elements

Cluster Sampling - all elements within each sample (chosen) cluster form:

the sample