Lecture 4 (SAMPLING AND SAMPLING DISTRIBUTIONS)

Question 1

SAMPLING

Answer 1

A means for gathering useful information about a population - information gathered and conclusions drawn

Question 2

What is the advantage of sampling over census?

Answer 2

Sampling saves money and time
Research process is sometimes destructive so can save product.
It is the only option when accessing a population is impossible.

Question 3

What are the reasons for taking a census over a sample?

Answer 3

Eliminates the possibility that a random sample is not representative of the population.
The person authorising the study is uncomfortable with sample information.
Safety of customer.

Question 4

POPULATION FRAME

Answer 4

A list, map, directory, or other source used to represent the population.

Question 5

OVER REGISTRATION (population frame)

Answer 5

The frame contains all members of the target population and some additional elements.

Question 6

UNDER REGISTRATION (population frame)

Answer 6

The frame does not contain all members of the target population.
The goal is to minimise differences between target population and frame.

Question 7

RANDOM SAMPLING

Answer 7

Every unit of the population has the same probability of being included in the sample.
A chance mechanism is used in the selection process.
Eliminates bias
Also known as probability sampling.

Question 8

NON-RANDOM SAMPLING

Answer 8

Every unit of the population does not have the same probability of being included in the sample.
Open to selection bias
Not appropriate data collection method for most statistical methods
Non-probability sampling.

Question 9

What are the 4 random sampling techniques?

Answer 9

Simple Random Sampling
Stratified Random Sampling
Systematic Random Sampling
Cluster (or Area) Sampling

Question 10

SIMPLE RANDOM SAMPLE

Answer 10

Basis for other random sampling techniques
Each unit is numbered 1 to n
A random number generator can be used to select n items from the sample.
Easier to perform for small populations
Cumbersome for large populations.

Question 11

STRATIFIED RANDOM SAMPLE

Answer 11

Population is divided into non-overlapping sub populations called strata
A random sample is selected from each stratum
Proportionate (% of the sample taken from each stratum is proportionate to the % that each stratum is within the whole population)
Disproportionate (when the % of the sample taken from each stratum is not proportionate to the % that each stratum is within the whole population.

has the potential to match the sample closely to the population
Stratified sampling is more costly
Stratum should be relatively homogeneous (i.e. race, gender, religion

Question 12

SAMPLING ERROR

Answer 12

A sample does not represent the population.

Question 13

SYSTEMATIC RANDOM SAMPLING

Answer 13

Convenient and relatively easy to administer.
Population elements are an ordered sequence (at least conceptually)
The first sample element is selected randomly from the first k population elements.
Thereafter, sample elements are selected at a constant interval, k, from the ordered sequence frame.
k = N/n
n = sample size
N = population size
k = size of selection interval

Question 14

Advantages/Disadvantages of systematic sampling

Answer 14

Ad: Convenience
Speed
Evenly distributed sampling across frame.
Dis: It is biased if the samples are ranked.

Question 15

CLUSTER (AREA) SAMPLING

Answer 15

Population is divided into non-overlapping clusters or areas
Each cluster is a miniature, or microcosm, of the population.
A subset of the clusters is selected randomly for the sample
If the number of elements in the subset of clusters is larger than the desired value of n, these clusters may be subdivided to form a new set of clusters and subjected to a random selection process.

Question 16

ADVANTAGES OF CLUSTER SAMPLING

Answer 16

More convenient for geographically dispersed populations.
Reduced travel costs to contact sample elements.
Simplified administration of the survey.
Unavailability of sampling frame prohibits using other random sampling methods.

Question 17

DISADVANTAGES OF CLUSTER SAMPLING

Answer 17

Statistically less efficient when the cluster elements are similar.
Costs and problems of statistical analysis are greater than for simple random sampling.

Question 18

NON-SAMPLING ERRORS

Answer 18

All errors other than sampling errors.
e.g.
Missing data, recording, data entry, and analysis errors.
Poorly conceived concepts, unclear definitions, and defective questionnaires.
Response errors occur when people do not know, will not say, or overstate in their answers.

Question 19

CENTRAL LIMIT THEOREM

Answer 19

Allows one to study populations with differently shaped distributions.
Creates the potential for applying the normal distribution to many problems when sample size is sufficiently large.
As sample size increases, the distribution narrows.
Due to the std dev of the mean.
Std Dev of mean decreases as sample size increases.

Question 20

What are the advantages of the central limit theorem

Answer 20

Advantage when sample data is drawn from populations not normally distributed or populations of unknown shape can also be analysed because the sample means are normally distributed due to large sample sizes.

Question 21

Properties of the Central Limit Theorem - sampling from a normal population

Answer 21

For sufficiently large sample sizes (n>30)
The distribution of sample means Xbar, is approximately normal;
The mean of this distribution is equal to u, the population mean.
Its standard deviation is o/sqrt(n)
regardless of the shape of the population distribution

Question 22

Z formula for sample means

Answer 22

z = (Xbar - u) / (o/sqrt(n))

Question 23

POINT ESTIMATE

Answer 23

A static taken from a sample that is used to estimate a population parameter.
xbar = Σx/n

Question 24

INTERVAL ESTIMATE

Answer 24

A range of values within which the analyst can declare, with some confidence, the population lies; also known as the confidence interval.
xbar +/- zɑ/2 * o/sqrt(n)

Question 25

What is aloha for a 95% confidence interval?

Answer 25

0.05

ɑ/2 = 0.025

Question 26

How do you find the z value from a confidence interval?

Answer 26

E.g. value of ɑ/2 or z0.025 look at table under:
0.500 - 0.0250 = 0.4750
look up 0.4750 and read 1.96 as the z value from the row and column.

Question 27

What is the purpose of the confidence interval?

Answer 27

Yields a range within which the researchers feel with some confidence the population mean is located.
If a researcher were to randomly select 100 samples of size n and use the results of each sample to construct 95% confidence intervals, approximately 95% out of 100 would contain the population mean.

Question 28

What is used when the sample size is > 5% of the population?

Answer 28

use the finite population correction factor.

Formula in booklet.

Question 29

Estimating the population mean using the z statistic when the sample size is small.

Answer 29

The central limit theorem applies only when the sample size is large (n>30)
However, the z formuals can still be used when the sample size is small (n<30) if it is known that the population from which the sample is drawn is normally distributed.

Question 30

How do you estimate the mean of a normal population when the standard deviation is unknown?

Answer 30

The population has a normal distribution.
The value of the population Std Dev is unknown; the sample Std Dev must be used in the estimation process.
z distribution is not appropriate for these conditions, t distribution is appropriate, and you use the sample Std Dev in the t formula.

Question 31

t DISTRIBUTION

Answer 31

A family of distributions - a unique distribution for each value of its parameter, degrees of freedom (d.f.)
Symmetrical, unimodal, mean = 0, flatter than z

t = (xbar - u) / (s/sqrt(n))

t distributions approach the normal curve as n becomes larger.

Question 32

How is the t distribution read?

Answer 32

T table uses the area in the tail of the distribution.
Emphasis is on ɑ, and each tail of the distribution contains ɑ/2 of the area under the curve when confidence intervals are constructed.
t values are located at the intersection of the df value and the selected ɑ/2 value.

Question 33

What are the confidence intervals for the t distribution?

Answer 33

xbar +/- tɑ/2,n-1 * (s/sqrt(n)

df = n-1

Question 34

Determining the sample size when estimating u.

Answer 34

It may be necessary to estimate the sample size when working on a project. 
In studies where u is being estimated, the size of the sample can be determined by using the z formula for sample means to solve for n.
z formula:
z = (xbar - u) / (o/sqrt(n))
Error of estimation (tolerable error):
E = xbar - u
Estimated sample size:
n = (z^2ɑ/2 * o^2) / E^2) 
= (zɑ/2 * o / E)^2
Estimated o = 1/4*range

Question 35

ERROR OF ESTIMATION

Answer 35

Difference between xbar and u.

xbar - u