Statistics and Probability

1.  Illustrates random sampling
 Distinguishes between parameter and statistic.
 Identifies sampling distributions of statistics
(sample mean).

2. A consumer is interested in buying grapes. Before deciding, the consumer
requests a piece from the bunch of grapes shown by the seller. Based upon
this piece, the consumer decided to buy the bunch of grapes.

3.  The consumer’s decision to buy the grapes was
based only on a piece, or sample, of the bunch.
Obviously, it was not needed for the consumer to
buy and eat the whole bunch of grapes before
determining whether these grapes tasted good
enough to purchase. This idea of selecting a
portion, or sample, to determine the taste or
characteristics of all the grapes, or population, is
the concept of sampling.

4. SAMPLING IN EVERYDAY LIFE

5. A population refers to the entire group that is under study or
investigation.
A sample is a subset taken from a population, either by random or
non-random sampling techniques. A sample is a representation of the
population where one hopes to draw valid conclusions from about the
population.
Sampling is the process of selecting a portion, or sample, of the entire
population.
A simple random sampling or random sampling is a selection of
elements derived from a population , which is the subject of the
investigation or experiment, where each sample point has an equal
chance of being selected using the appropriate sampling technique.

6. SAMPLING…….
6
TARGET POPULATION
STUDY POPULATION
SAMPLE

7. TYPES OF
RANDOM
SAMPLING

8. Simple Random/Lottery Sampling
 A sampling technique where every
member of the population has an
equal chance of being selected. The
procedure is carried out by
randomly picking numbers, with
each number corresponds to each
member of the population.
 Example. Drawing of winning prizes
from the tambiolo.

9. SIMPLE RANDOM SAMPLING
9
• Applicable when population is small, homogeneous
& readily available
• All subsets of the frame are given an equal
probability. Each element of the frame thus has
an equal probability of selection.
• It provides for greatest number of possible
samples. This is done by assigning a number to
each unit in the sampling frame.
• A table of random number or lottery system is
used to determine which units are to be selected.

10. SIMPLE RANDOM SAMPLING……..
10
 Estimates are easy to calculate.
 Disadvantages
 If sampling frame large, this method
impracticable.
 Minority subgroups of interest in population
may not be present in sample in sufficient
numbers for study.

11. Systematic Sampling
 A sampling technique in which
members of the population are ordered
in some way such as alphabetically or
numerically and samples are selected
in intervals called sample intervals. In
this technique, a starting point is
randomly selected from the first k
positions, and then, every kth number,
is selected from the sample. Since k is
the ratio of the population size to
sample size, to find the k use the
formula:
 𝒌 = 𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒔𝒊𝒛𝒆/𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆

12. SYSTEMATIC SAMPLING……
12
 ADVANTAGES:
 Sample easy to select
 Suitable sampling frame can be identified easily
 Sample evenly spread over entire reference population
 DISADVANTAGES:
 Sample may be biased if hidden periodicity in population
coincides with that of selection.
 Difficult to assess precision of estimate from one survey.

13. Example. A Science teacher decides to select a sample of 10 students from her large lecture class
containing 300 students to be part of an experiment using the systematic sampling procedure. If each
student has an assigned number from 1 to 300 and she randomly selects 3 as her starting point,
identify the students selected for the experiment.
Solution.
Step 1: Identify the value of .
𝒌=𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒔𝒊𝒛𝒆/𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆
𝑘 =
300
10
𝒌=𝟑𝟎
Step 2: Since 𝒌=𝟑𝟎 and the starting point given is 3, the sample of selected students are:
1st student: 3rd 6th student: 123 + 30 = 153rd
2nd student: 3+30 = 33rd 7th student: 153 + 30 = 183rd
3rd student: 33+30 = 63rd 8th student: 183 + 30 = 213rd
4th student: 63+30 = 93rd 9th student: 213 + 30 = 243rd
5th student: 93+30 = 123rd 10th student: 243 + 30 = 273rd

14. Stratified Random Sampling
 A sampling procedure wherein the
members of the population are grouped
based on their homogeneity. This
technique is used when there are a
number of distinct subgroups in the
population, within each of which is
required that there is full
representation. The sample is
constructed by classifying the
population into subpopulations or
strata, based on some characteristics
of the population such as age, gender,
or socio-economic status. The
selection of elements is then made
separately from within each stratum,
usually by random or systematic
sampling methods.

15. STRATIFIED SAMPLING
15
Where population embraces a number of distinct categories, the
frame can be organized into separate "strata." Each stratum is then
sampled as an independent sub-population, out of which individual
elements can be randomly selected.
 Every unit in a stratum has same chance of being selected.
 Using same sampling fraction for all strata ensures proportionate
representation in the sample.
 Adequate representation of minority subgroups of interest can be
ensured by stratification & varying sampling fraction between strata
as required.

16. STRATIFIED SAMPLING……
16
 Finally, since each stratum is treated as an independent
population, different sampling approaches can be applied to
different strata.
 Drawbacks to using stratified sampling.
 First, sampling frame of entire population has to be prepared
separately for each stratum
 Second, when examining multiple criteria, stratifying variables
may be related to some, but not to others, further complicating
the design, and potentially reducing the utility of the strata.
 Finally, in some cases (such as designs with a large number of
strata, or those with a specified minimum sample size per group),
stratified sampling can potentially require a larger sample than
would other methods

17. STRATIFIED SAMPLING…….
17
Draw a sample from each stratum

18. Example. Using stratified random sampling,
select a sample of 387 students from the
population which are grouped according to the
cities they come from. The table below shows
the number of students per city.
Solution.
To determine the number of students to be taken as
sample from each City, divide the number of students
per city (stratum) by the total population (which is 36
500) and multiply the result by the total sample size
(which is 387).
Based on the table, 138 students will be drawn from City A, 111 students from City B, 85
students from City C, and 53 students from D. The selection of students from each City
may use random or systematic sampling methods.

19. Cluster Sampling
 It is sometimes called area sampling, the population is divided into groups or clusters, usually
based upon geographic location, and these clusters contain data values which are
heterogenous. A simple random sample of clusters is selected to represent the population.
Ideally, the clusters should be similar and be a representative small-scale version of the
overall population.

20. CLUSTER SAMPLING…….
20
 Advantages :
 Cuts down on the cost of preparing a sampling frame.
 This can reduce travel and other administrative costs.
 Disadvantages: sampling error is higher for a simple random sample of
same size.
 Often used to evaluate vaccination coverage in EPI

21. CLUSTER SAMPLING…….
21
• Identification of clusters
– List all cities, towns, villages & wards of cities with their population falling
in target area under study.
– Calculate cumulative population & divide by 30, this gives sampling interval.
– Select a random no. less than or equal to sampling interval having same no.
of digits. This forms 1st cluster.
– Random no.+ sampling interval = population of 2nd cluster.
– Second cluster + sampling interval = 4th cluster.
– Last or 30th cluster = 29th cluster + sampling interval

22. Difference Between Strata and Clusters
22
 Although strata and clusters are both non-overlapping subsets of the
population, they differ in several ways.
 All strata are represented in the sample; but only a subset of clusters are
in the sample.
 With stratified sampling, the best survey results occur when elements
within strata are internally homogeneous. However, with cluster sampling,
the best results occur when elements within clusters are internally
heterogeneous

23.  Example. A statistician wants to determine the
number of children per family in Tarlac City. To
minimize the cost of selecting a sample, the
statistician decides to divide the city of Tarlac into
Barangays (clusters) and use the cluster sampling
procedure to select the sample. A random sample
of the Barangays are selected and every family on
the Barangay are interviewed to determine the
number of children in the family.

24. Multi-Stage Sampling
 It is done using a combination of
different sampling techniques.
 Example. When selecting
respondents for a national election
survey, lottery method may be used
to select regions and cities. Then,
utilize stratified sampling to
determine the number of
respondents from the chosen areas
and clusters.

25. A nonrandom sampling is used when the sample is not a proportion of
the population and when there is no system in selecting a sample. This
is often used by the researchers to elicit and gather quick responses for
questions which do not require confidentiality. The researcher states
prejudice in the choice of the sample giving the members of the
population unequal chances to be selected.

26. TYPES OF
NONRANDOM
SAMPLING

27. Quota Sampling
 The researcher limits the number of his samples based on the required
number of the subject under investigation. The population is first segmented
into mutually exclusive subgroups, then judgement used to select subjects or
units from each segment is based on the specified proportion.
 For example, an interviewer may be told to sample 200 females and 300
males between age 45 and 60.

28. Convenience Sampling
 The researcher conducts a study at his convenient time, preferred place, or venue. It is the
most convenient and fastest sampling technique that make use of telephone, mobile
phones, or the internet. It simply uses results that are readily available.

29. Purposive Sampling
 It is used in very small sample sizes. Choosing samples is based on a certain criteria and
rules laid down by the researcher.
 For example, this can be used if the sample of the study are deans of universities or area
managers of certain institutions.
 Since the different random sampling techniques were presented, the next thing to be
determined is the sample size.

31. Identify which sampling technique was used in the
following situations?

32. A group of test subjects is
divided into twelve
groups; four of the groups
are chosen at random.
Cluster Sampling

33. A market researcher
polls every tenth
person who walks into
a store.
Systematic Random Sampling

34. A computer generates 100
random numbers, and 100
people whose names
correspond with numbers
on the list are chosen.
Simple Random Sampling/Lottery Sampling

35. The first 50 people who
walk into sporting event
are polled on their
television preferences.
Quota Sampling

36. A researcher conducted his
survey by randomly choosing
one town in the province of
Tarlac by using a fishbowl and
then randomly picking 40
samples from that town.
Multi-Stage

37. A researcher conducted his
survey by choosing 40
persons he knew. What
sampling method that the
researcher used?
Convenience Sampling

38. A study was done to determine the age, number of times
per week and duration of residents using a local park in
Victoria, Tarlac. The first house in the neighborhood
around the park was selected randomly, and then the
resident of every eight house in the neighborhood
around the park was interviewed. What sampling
method was used?
Systematic Random Sampling

39.  In most applications of Statistics, researchers
use a sample rather than the entire population
since it is usually impractical or impossible to
obtain all the population observations or
measurements; thus, sample information are
used to estimate the characteristics of a
population. That is, the use of a statistic to
make inferences about the corresponding
population parameter is being done.

40.  A statistic is a number which describes a
characteristic of a sample. It can be
directly computed and observed. It
serves as estimator of the population
parameter.
 A parameter is a number which
describes a characteristic of a
population. While statistic can be
directly computed and observed, the
value of a parameter can be
approximated and is not necessarily
equal to the statistic of a sample.

42. In calculating a statistic, such as a sample mean, from a random
sample of the population, the computed statistic is not necessarily
equal to the population parameter. Furthermore, taking another
random sample from the same population may result to a different
computed statistic. But both are estimates of the parameter. This
clearly shows that the statistics which can be computed from a
randomly selected sample of the given population are distinct. If
so, what could be the distribution of values that can be computed
for the statistics? What is the frequency with which different
values for the statistic will be computed to estimate the
parameter?

43. A sampling distribution is the probability distribution
when all possible samples of size are repeatedly
drawn from a population.

44. Illustrative Example 1
Construct a sampling distribution of the mean and a histogram for the set of data below.
86 89 92 95 98

48. Problem: Find the mean of the set of data below and construct a sampling distribution, without replacement
and repetition, by selecting 4 samples at a time ( = 4). Construct a histogram of the sample means. 25, 28,
24, 27, 30, 20
Random Sample Sample Mean
20,24,25,27 24
20,24,25,28 24.25
20,24,25,30 24.75
20,24,27,28 24.5
20,24,27,30 25.25
20,24,28,30 25.5
20,25,27,28 25
20,27,28,30 26.25
24,25,27,28 26
24,25,27,30 26.5
24,27,28,30 27.25
25,27,28,30 27.5
24,25,28,30 26.75
20,25,28,30 25.75
20,25,27,30 25.5
Sample Mean P(x)
24 0.067
24.25 0.067
24.5 0.067
24.75 0.067
25 0.067
25.25 0.067
25.5 0.13
25.75 0.067
26 0.067
26.25 0.067
26.5 0.067
26.75 0.067
27.25 0.067
27.5 0.067

49. PERFORMANCE TASK # 4 (Problem Solving )
Direction: Answer the given problem below in a separate sheet of paper and
show your complete solution.
Problem: Find the mean of the set of data below and construct a sampling
distribution, without replacement and repetition, by selecting 5 samples at a
time ( = 5). Construct a histogram of the sample means. 5 8 11 14 17 20 23