SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docx

SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe
the properties of statistical distributions. In statistical terms, the
sample meanfrom a group of observations is an estimate of the
population mean. Given a sample of size n, consider n
independent random variables X1, X2... Xn, each corresponding
to one randomly selected observation. Each of these variables
has the distribution of the population, with mean and standard
deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a
database. It can also be said that it is nothing more than a
balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by
how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷
3 = 6
So the Mean is 6

SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a
sample is. A sample is a select number of items taken from a
population. For example, if you are measuring American
people’s weights, it wouldn’t be feasible (from either a time or
a monetary standpoint) for you to measure the weights of every
person in the population. The solution is to take a sample of the
population, say 1000 people, and use that sample size to
estimate the actual weights of the whole population.
The sample variance helps you to figure out the spread out in
the data you have collected or are going to analyze. In
statistical terminology, it can be defined as the average of the
squared differences from the mean.
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the
number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use

the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we
want.
· Next we need to divide by the number of data points, which is
simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of
all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9,
3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 =
140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the
result
This is the part of the formula that says:
So what is xi? They are the individual x values 9, 2, 5, 4, 12, 7,
etc...
In other words x1 = 9, x2 = 2, x3 = 5, etc.
So it says "for each value, subtract the mean and square the

result", like this
Example (continued):
(9 - 7)2 = (2)2 = 4
(2 - 7)2 = (-5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(7 - 7)2 = (0)2 = 0
(8 - 7)2 = (1)2 = 1
We need to do this for all the numbers
Step 3. Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by how
many.
First add up all the values from the previous step.
= 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9 = 178
But that isn't the mean yet, we need to divide by how many,
which is simply done by multiplying by "1/N":
Mean of squared differences = (1/20) × 178 = 8.9
This value is called the variance.
STANDARD DEVIAITON:
DEFINITION:
This descriptor shows how much variation or dispersion from
the average exists.
The symbol for Standard Deviation is σ (the Greek letter
sigma).
It is calculated using:
In case of a sample the ‘N’ in this formula is replaced by n-1.

It is used to determine the expected value. A low standard
deviation indicates that the data points tend to be very close to
the mean (also called expected value); a high standard deviation
indicates that the data points are spread out over a large range
of values.
To determine the standard deviation, you need to take the
square root of the variance.
EXAMPLE PROBLEM:
Let’s look at the previous problem and compute the standard
deviation. The standard deviation as mentioned earlier is
nothing more than the measure of dispersion (spread). It can be
calculated by taking the square root of the variance. In case of
the previous problem where the variance was 8.9, its
corresponding standard deviation would be the square root of
8.9 which is 2.983
σ = √(8.9) = 2.983...
HYPOTHESES TESTING:
DEFINITION:
Hypothesis testing is a topic at the heart of statistics. This
technique belongs to a realm known as inferential statistics.
Researchers from all sorts of different areas, such as
psychology, marketing, and medicine, formulate hypotheses or
claims about a population being studied.

Hypothesis testing is used to determine the validity of these
claims. Carefully designed statistical experiments obtain sample
data from the population. The data is in turn used to test the
accuracy of a hypothesis concerning a population. Hypothesis
tests are based upon the field of mathematics known as
probability. Probability gives us a way to quantify how likely it
is for an event to occur. The underlying assumption for all
inferential statistics deals with rare events, which is why
probability is used so extensively. The rare event rule states
that if an assumption is made and the probability of a certain
observed event is very small, then the assumption is most likely
incorrect.
The basic idea here is that we test a claim by distinguishing
between two different things:
1. An event that easily occurs by chance
2. An event that is highly unlikely to occur by chance.
If a highly unlikely event occurs, then we explain this by stating
that a rare event really did take place, or that the assumption we
started with was not true.
HOW TO USE THE TEST FOR DECISION MAKING
PURPOSES:
1. Formulate the null hypothesis(commonly, that the
observations are the result of pure chance) and the alternative
hypothesis(commonly, that the observations show a real effect
combined with a component of chance variation).
2. Identify a test statistic that can be used to assess the truth of
the null hypothesis.
3. Compute the P-value, which is the probability that a test
statistic at least as significant as the one observed would be
obtained assuming that the null hypothesis were true. The
smaller the -value, the stronger the evidence against the null
hypothesis.
4. Compare the -value to an acceptable significance value
(sometimes called an alpha value). If , that the observed effect
is statistically significant, the null hypothesis is ruled out, and

the alternative hypothesis is valid.
EXAMPLE OF HYPOTHESIS TESTING (TWO-TAIL TEST)
If you are told that the mean weight of 3rd graders is 85 pounds
with a standard deviation of 20 pounds, and you find that the
mean weight of a group of 22 students is 95 pounds, do you
question that that group of students is a group of third graders?
· The z-score is ((x-bar) - µ)/(*sigma*/(n^.5)); the numerator is
the difference between the observed and hypothesized mean, the
denominator rescales the unit of measurement to standard
deviation units. (95-85)/(20/(22^.5)) = 2.3452.
· The z-score 2.35 corresponds to the probability .9906, which
leaves .0094 in the tail beyond. Since one could have been as
far below 85, the probability of such a large or larger z-score is
.0188. This is the p-value. Note that for these two tailed tests
we are using the absolute value of the z-score.
· Because .0188 < .05, we reject the hypothesis (which we shall
call the null hypothesis) at the 5% significance level; if the null
hypothesis were true, we would get such a large z-score less
than 5% of the time. Because .0188 > .01, we fail to reject the
null hypothesis at the 1% level; if the null hypothesis were true,
we would get such a large z-score more than 1% of the time.
DECISION TREE:
DEFINITION:
A schematic tree-shaped diagram used to determine a course of
action or show a statistical probability. Each branch of the
decision tree represents a possible decision or occurrence. The
tree structure shows how one choice leads to the next, and the
use of branches indicates that each option is mutually exclusive.
A decision tree can be used to clarify and find an answer to a
complex problem. The structure allows users to take a problem
with multiple possible solutions and display it in a simple, easy-
to-understand format that shows the relationship between

different events or decisions. The furthest branches on the tree
represent possible end results.
HOW TO APPLY IT:
1. As a starting point for the decision tree, draw a small square
around the center of the left side of the paper. If the description
is too large to fit the square, use legends by including a number
in the tree and referencing the number to the description either
at the bottom of the page or in another page.
2. Draw out lines (forks) to the right of the square box. Draw
one line each for each possible solution to the issue, and
describe the solution along the line. Keep the lines as far apart
as possible to expand the tree later.
3. Illustrate the results or the outcomes of the solution at the
end of each line. If the outcome is uncertain, draw a circle
(chance node). If the outcome leads to another issue, draw a
square (decision node). If the issue is resolved with the
solution, draw a triangle (end node). Describe the outcome
above the square or circle, or use legends, as appropriate.
4. Repeat steps 2 through 4 for each new square at the end of
the solution lines, and so on until there are no more squares,
and all lines have either a circle or blank ending.
5. The circles that represent uncertainty remain as they are. A
good practice is to assign a probability value, or the chance of
such an outcome happening.
Since it is difficult to predict at onset the number of lines and
sub-lines each solution generates, the decision tree might
require one or more redraws, owing to paucity of space to
illustrate or represent options and/or sub-options at certain
spaces.
It is a good idea to challenge and review all squares and circles
for possible overlooked solutions before finalizing the draft.
EXAMPLE:
Your company is considering whether it should tender for two
contracts (MS1 and MS2) on offer from a government
department for the supply of certain components. The company
has three options:

· tender for MS1 only; or
· tender for MS2 only; or
· tender for both MS1 and MS2.
If tenders are to be submitted, the company will incur additional
costs. These costs will have to be entirely recouped from the
contract price. The risk, of course, is that if a tender is
unsuccessful, the company will have made a loss.
The cost of tendering for contract MS1 only is $50,000. The
component supply cost if the tender is successful would be
$18,000.
The cost of tendering for contract MS2 only is $14,000. The
component supply cost if the tender is successful would be
$12,000.
The cost of tendering for both contracts MS1 and MS2 is
$55,000. The component supply cost if the tender is successful
would be $24,000.
For each contract, possible tender prices have been determined.
In addition, subjective assessments have been made of the
probability of getting the contract with a particular tender price
as shown below. Note here that the company can only submit
one tender and cannot, for example, submit two tenders (at
different prices) for the same contract.
Option Possible Probability
tender of getting
prices ($) contract
MS1 only 130,000 0.20
115,000 0.85
MS2 only 70,000 0.15
65,000 0.80
60,000 0.95
MS1 and MS2 190,000 0.05
140,000 0.65
In the event that the company tenders for both MS1 and MS2 it

will either win both contracts (at the price shown above) or no
contract at all.
· What do you suggest the company should do and why?
· What are the downside and the upside of your suggested
course of action?
· A consultant has approached your company with an offer that
in return for $20,000 in cash, she will ensure that if you tender
$60,000 for contract MS2, only your tender is guaranteed to be
successful. Should you accept her offer or not and why?
Solution
The decision tree for the problem is shown below.
Below we carry out step 1 of the decision tree solution
procedure which (for this example) involves working out the
total profit for each of the paths from the initial node to the
terminal node (all figures in $'000).
Step 1
· path to terminal node 12, we tender for MS1 only (cost 50), at
a price of 130, and win the contract, so incurring component
supply costs of 18, total profit 130-50-18 = 62
a price of 130, and lose the contract, total profit -50

· path to terminal node 22, we tender for MS1 and MS2 (cost
55), at a price of 190, and win the contract, so incurring
component supply costs of 24, total profit 190-55- 24=111
55), at a price of 190, and lose the contract, total profit -55

55), at a price of 140, and win the contract, so incurring
component supply costs of 24, total profit 140-55- 24=61
55), at a price of 140, and lose the contract, total profit -55
Hence we can arrive at the table below indicating for each
branch the total profit involved in that branch from the initial
node to the terminal node.
Terminal node Total profit $'000
12 62
13 -50
14 47
15 -50
16 44
17 -14
18 39
19 -14
20 34
21 -14
22 111
23 -55
24 61
25 -55
We can now carry out the second step of the decision tree
solution procedure where we work from the right-hand side of
the diagram back to the left-hand side.

Step 2
· For chance node 5 the EMV is 0.2(62) + 0.8(-50) = -27.6
· For chance node 6 the EMV is 0.85(47) + 0.15(-50) = 32.45
Hence the best decision at decision node 2 is to tender at a price
of 115 (EMV=32.45).
of 60 (EMV=31.6).
of 140 (EMV=20.4).
Hence at decision node 1 we have three alternatives:
· tender for MS1 only EMV=32.45
· tender for MS2 only EMV=31.6
· tender for both MS1 and MS2 EMV = 20.4
Hence the best decision is to tender for MS1 only (at a price of
115) as it has the highest expected monetary value of 32.45
($'000).
INFLUENCE OF SAMPLE SIZE:
DEFINITION:
Sample size is one of the four interrelated features of a study

design that can influence the detection of significant
differences, relationships, or interactions. Generally, these
survey designs try to minimize both alpha error (finding a
difference that does not actually exist in the population) and
beta error (failing to find a difference that actually exists in the
population).
The sample size used in a study is determined based on the
expense of data collection and the need to have sufficient
statistical power.
HOW TO USE IT:
We already know that the margin of error is 1.96 times the
standard error and that the standard error is sq.rt ^p(1�^p)/n. In
general, the formula is ME = z sq.rt ^p(1-^p)/n
where
*ME is the desired margin of error
*z is the z-score, e.g., 1.645 for a 90% confidence interval, 1.96
for a 90% confidence interval, 2.58 for a 99% confidence
interval
_ ^p is our prior judgment of the correct value of p.

_ n is the sample size (to be found)
EXAMPLE:
If ^p =0.3 and Z=1.96 and ME =0.025 then the necessary sample
size is:
ME= Z sq.rt (^p*1-^p)/n
0.025 = 1:96 sq.rt (0.3*0.7)/n
n=1291 or 1300 students
POPULATION MEAN:
DEFINITION:
The population mean is the mean of a numerical set that
includes all the numbers within the entire group.
In most cases, the population mean is unknown and the sample

mean is used for validation purposes. However, if we want to
calculate the population mean, we will have to construct the
confidence interval. This can be achieved by the following
steps:
HOW TO USE IT:
· The sample statistic is the sample mean x¯
· The standard error of the mean is s/sq.rt n where s is the
standard deviation of individual data values.
· The multiplier, denoted by t*, is found using the t-table in the
appendix of the book. It's a simple table. There are columns for
.90, .95, .98, and .99 confidence. Use the row for df = n − 1.
· Thus the formula for a confidence interval for the mean is
x¯±t∗ (s/sq.rt n)
EXAMPLE:
In a class survey, students are asked if they are sleep deprived
or not and also are asked how much they sleep per night.
Summary statistics for the n = 22 students who said they are
sleep deprived are:
· Thus n = 22, x¯ = 5.77, s = 1.572, and standard error of the
mean = 1.572/sq.rt 22=0.335
· A confidence interval for the mean amount of sleep per night

is 5.77 ± t* (0.335) for the population that feels sleep deprived.
· Go to the t-table in the appendix of the book and use the df =
22 – 1 = 21 row. For 95% confidence the value of t* = 2.08.
· A 95% confidence interval for μ is 5.77 ± (2.08) (0.335),
which is 5.77 ± 0.70, or 5.07 to 6.7
· Interpretation: With 95% confidence we estimate the
population mean to be between 5.07 and 6.47 hours per night.
RANDOM SAMPLING
Random sampling is a sampling technique where we select a
group of subjects (a sample) for study from a larger group (a
population). Each individual is chosen entirely by chance and
each member of the population has a known, but possibly non-
equal, chance of being included in the sample.
By using random sampling, the likelihood of bias is reduced.
WHEN RANDOM SAMPLING IS USED:
Random sampling is used when the researcher knows little
about the population.
THE STEPS ASSOCIATED WITH RANDOM SAMPLING:
1. Define the population
2. Choose your sample size
3. List the population

4. Assign numbers to the units
5. Find random numbers
6. Select your sample
EXAMPLE:
In a study, 10,000 students will be invited to take part in the
research study. The selection was limited to 200 randomly
selected students. In this case, this would mean selecting 200
random numbers from the random number table. Imagine the
first three numbers from the random number table were:
0011
(the 11th student from the numbered list of 10,000 students)
9292
(the 9,292nd student from the list)
2001
(the 2,001st student from the list)
We would select the 11th, 9,292nd, and 2,001st students from
our list to be part of the sample. We keep doing this until we
have all 200 students that we want in our sample.
SAMPLING DISTRIBUTION:
DEFINITION:

The sampling distribution is a theoretical distribution of a
sample statistic. There is a different sampling distribution for
each sample statistic. Each sampling distribution is
characterized by parameters, two of which are and. The latter is
called the standard error.
It is used for making probability statements in inferential
statistics.
HOW IS SAMPLING DISTRIBUTION USED?
Step 1: Obtain a simple random sample of size n.
Step 2: Compute the sample mean.
Step 3: Assuming we are sampling from a finite population,
repeat Steps 1 and 2 until all simple random samples of size n
have been obtained.
EXAMPLE OF SAMPLING DISTRIBUTION:
THE SAMPLE DISTRIBUTION
The sample distribution is the distribution resulting from the
collection of actual data. A major characteristic of a sample is
that it contains a finite (countable) number of scores, the
number of scores represented by the letter N. For example,
suppose that the following data were collected:
32
35
42
33

36
38
37
33
38
36
35
34
37
40
38
36
35
31
37
36
33
36
39
40
33
30
35
37
39

32
39
37
35
36
39
33
31
40
37
34
34
37
These numbers constitute a sample distribution. Using the
procedures discussed in the chapter on frequency distributions,
the following relative frequency polygon can be constructed to
picture this data:
SAMPLING ERROR:
DEFINITION:
The error that arises as a result of taking a sample from a
population rather than using the whole population.
It is used to detect the difference between the sample and the
true, but unknown value of population parameter.

HOW TO USE IT/CALCULATE IT:
· Determine the level of confidence followed by the critical
values
· Calculate the sample standard deviation
· Calculate the margin of error using
E = Critical value * sample standard deviation/sq.rt of sample
size
EXAMPLE:
1. What is the margin of error for a simple random sample of
900 people at a 95% level of confidence? The sample standard
deviation is 2.
By use of the table we have a critical value of 1.96, and so the
margin of error is 1.96/(2 √ 900 = 0.03267, or about 3.3%.
2. What is the margin of error for a simple random sample of
1600 people at a 95% level of confidence and a sample standard
deviation of 2?
At the same level of confidence as the first example, increasing
the sample size to 1600 gives us a margin of error of 0.0245, or
about 2.5%.
This shows that by increasing the sample size, the margin of
error decreases.
PROBABILITY:
DEFINITION:

Probability is the chance that something will happen — how
likely it is that some event will occur.
Probability is used in various areas, such as assessing risks in
medical treatment, forecasting weather, what to sell at a
discount and when to sell it, determining car insurance rates,
determining future commercial and manufacturing construction,
in developing other real estate, in municipal planning for such
things as placing new roads, and in financial planning at home
and in the business world.
HOW TO CALCUATE IT:
1. Count the number of all distinctive and equally likely
outcomes of the experiment. Let that be n.
2. Count the number of distinctive outcomes that represent the
occurrence of the event in question. Let that be ne.
3. Calculate the result of the division ne/n. That is the
probability of the event.
EXAMPLE:
Find the probability of getting an even number after rolling a
die.

· Event: Getting an even number
· Steps above:
· Distinctive outcomes: 1, 2, 3, 4, 5, 6 are all the outcomes,
their count n=6
· Outcomes representing the event: 2, 4, 6 are all the even
numbers you can get, their count ne=3
· Probability: P = ne/n = 3/6 = 0.5 or 1/2
POWER CURVE:
DEFINITION:
Power curves illustrate the effect on power of varying the
alternate hypothesis.
The curve illustrates how a sample of observations with a
defined variance is quite powerful in correctly rejecting the null
hypothesis (for example, if m0=8) when the true mean is less
than 6 or greater than 10. The curve also illustrates that the test
is not powerful — it may not reject the null hypothesis even
when the true mean differs from m0 — when the difference is
small. This is also extensively used in testing the relationship

between power and sample size.
HOW IT IS USED:
See example below.
EXAMPLE:
If the researcher learns from literature that the population
follows a normal distribution with mean of 100 and variance of
100 under the null hypothesis and he/she expects the mean to be
greater than 105 or less than 95 under the null hypothesis and
he/she wants the test to be significant at 95% level, the
resulting power function would be:
Power=1-Φ[1.96-(105-100)/(10/n)]+Φ[-1.96-(95-100)/(10/n)],
which is,
Power=1-Φ[1.96-n/2]+Φ[-1.96+n/2].
That function shows a relationship between power and sample
size. For each level of sample size, there is a corresponding
sample size. For example, if n=20, the corresponding power
level would be about 0.97, or, if the power level is 0.95, the
corresponding sample size would be 16.
PROBABILITY DISTRIBUTION:

DEFINITION:
A statistical function that describes all the possible values and
likelihoods that a random variable can take within a given
range. At times it is presented in the form of a table or an
equation that links the outcome of a statistical experiment with
its probability of occurrences.
HOW IT IS USED:
It establishes a range that will be between the minimum and
maximum statistically possible values, but where the possible
values are likely to be plotted on the probability distribution
depends on a number of factors, including the distribution
mean, standard deviation, skewness, and kurtosis.
HOW TO USE IT:
· Identify the event
· Create a table showing the possibility of its occurrence
EXAMPLE:
An example will make clear the relationship between random
variables and probability distributions. Suppose you flip a coin

two times. This simple statistical experiment can have four
possible outcomes: HH, HT, TH, and TT. Now, let the variable
X represent the number of Heads that result from this
experiment. The variable X can take on the values 0, 1, or 2. In
this example, X is a random variable because its value is
determined by the outcome of a statistical experiment.
A probability distribution is a table or an equation that links
each outcome of a statistical experiment with its probability of
occurrence. Consider the coin flip experiment described above.
The table below, which associates each outcome with its
probability, is an example of a probability distribution.
Number of heads
Probability
0
0.25
1
0.50
2
0.25
The above table represents the probability distribution of the
random variable X.
EXPECTED VALUE OF SAMPLE INFORMATION:
DEFINITION:
In decision theory, the expected value of sample information is

the expected increase in utility that you could obtain from
gaining access to a sample of additional observations before
making a decision.
WHAT IS IT USED FOR?
Calculate the Expected Monetary Value (EMV) of each
alternative action.
HOW TO USE IT/CALCULATE IT?
1.Determine the optimal decision and its expected return for the
possible outcomes of the sample using the posterior
probabilities for the states of nature
2. Calculate the values of the optimal returns
3.Subtract the EV of the optimal decision obtained without
using the sample information from the amount determined in
step (2)
EXAMPLE:
The expected value of sample information is computed as
follows:

Suppose you were going to make an investment into only one of
three investment vehicles: stock, mutual fund, or certificate of
deposit (CD). Further suppose that the market has a 50% chance
of increasing, a 30% chance of staying even, and a 20% chance
of decreasing. If the market increases, the stock investment will
earn $1500 and the mutual fund will earn $900. If the market
stays even, the stock investment will earn $300 and the mutual
fund will earn $600. If the market decreases, the stock
investment will lose $800 and the mutual fund will lose $200.
The certificate of deposit will earn $500 independent of the
market's fluctuation.
Question:
What is the expected value of perfect information?

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