Quantitative techniques by amit ramawat
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Transcript of Quantitative techniques by amit ramawat
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Module 6
1. Normal distribution: Importance & Properties
2. Sampling : Random & Non-random Sampling
3. Sampling distribution: Central limit theorem
4. Sampling Distribution of Mean & Proportion
5. Sampling Distribution of the difference of two
means & two proportions.
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1. Normal distribution
The normal distribution wasdiscovered by De Moivre as the
limiting case of binomial model in
1733. It was known to Laplace no later than
1774.
But through a historical error it hasbeen credited to Gauss, who first
made reference to it in 1809.
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Properties
The normal distribution is a continuousdistribution plays a very important &pivotal role in statistical theory &
practice. The experimental results, very often
seem to follow the normal distributionor the bell shaped curve.
The normal curve is symmetrical & isdefined by its mean() & its standarddeviation ( ).
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Characteristics of the normal
curveAll normal curves are symmetrical
about the mean.
The height of the normal curve is at its
maximum at the mean value.
The height of the curve declines as we
go in either direction from the mean,
but never touches the base, so thatthe tails of the curve on both the sides
of the mean extend indefinitely.
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Q.1
The average daily sales of 500 branchoffices was Rs. 150 thousand & the
standard deviation Rs. 15 thousand.
Assuming the distribution to benormal, indicate how many branches
have sales between:
Rs. 120 thousand & Rs. 145thousand.
Rs. 140 thousand & Rs. 165
thousand.
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Q.2 (2008 question paper)
In a normal distribution, 31% of theitems are under 45 & 8% are over 64.
Find the mean & standard deviation of
the distribution.
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Q.3
The income of a group of 10,000persons was found to be normally
distributed with mean Rs. 1750 p.m. &
standard deviation Rs. 50.i. Show that in this group 95% had
income exceeding Rs. 1668 & only
5% had income exceeding Rs. 1832.ii. What was the lowest income among
the richest 100?
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Q.4
A workshop produces 2000 units perday. The average weight of units is
130 kg with a standard deviation of 10
kg. Assuming normal distribution, howmany units are expected to weigh less
than 142 kg?
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Q.5
Assume that the test scores from a collegeadmissions test are normally distributed witha mean of 450 & a standard deviation of 100.
a. What percentage of the people taking the
test score between 400 & 500?b. Suppose someone received a score of 630.
What percentage of the people taking thetest score better? What percentage score
worse?c. If a particular university will not admit
anyone scoring below 480, what percentageof the persons taking the test would be
acceptable to the university?
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Sampling
In the sampling method, instead of everyunit of the population only a part of thepopulation is studied & the conclusionsare drawn on that basis for the entire
population.
Example 1: a housewife examines only2or 3 grains of boiling rice to know,
whether the pot is ready or not. Example 2: a doctor examines a few
drops of the blood & draws conclusionabout the blood constitution of the whole
body.
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Purpose of sampling
A sample is not studied for its own sake.The basic objective of its study is to drawinference about the population.
In other words, sampling is only a tool
which helps to know the characteristicsof the universe or population byexamining a small part of it.
The values obtained from the study of
sample, such as the average & variance,are known as statistic. On the otherhand, such values for the population arecalled parameters.
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Random sampling methods
I. Simple random samplingIt refers to the sampling technique in
which each & every item of the
population is given an equal chanceof being included in the sample.
The selection is thus free from personalbias because the investigator does
not exercise his preference in thechoice of items.
Its also known as Representative
sampling.
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Methods of obtaining a Simple
Random Samplea. Lottery method:
All items of the population are
numbered or named on separate
slips of paper of identical size, color& shape. These slips are then folded
& mixed up in a container or a drum.
A blindfold selection is then made ofthe number of slips required to
constitute desired sample.
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b. Table of Random numbers:
A number of random tables are
available such as:
i. Tippettstable of random numbers.
ii. Fisher & Yates numbers.
iii. Kendall & Babington Smith numbers.
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II. Stratified Sampling
The process of stratification requires
that the population may be divided into
classes called strata.Then a sample may be taken from each
group by simple random method & the
resulting sample is called a stratifiedsampling.
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III. Systematic Sampling
The method is to select every kth item
from the list where k refers to the
sampling interval. The starting pointbetween the first & the kth item is
selected at random.
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IV. Multi-Stage SamplingAs the name implies, this method refers
to a sampling procedure which is
carried out in several stages.At first, the first stage units are sampled
by some suitable method such asrandom sampling. Then, a sample of
second stage units are selected fromeach of the selected first stage unitsagain by some suitable method
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Non-Random Sampling
MethodsI. Judgment SamplingIn this method of sampling, the choice
of sample items depends exclusively
on the judgment of the investigator.For example: if a sample of 10 students
is to be selected from a class of sixtyfor analyzing the spending habits of
students, the investigator wouldselect 10 students who, in hisopinion, represent the class.
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II. Quota SamplingIt is a type of judgment sampling.
Quotas are set up according to given criteria,but within the quotas the selection of
sample items depends on personaljudgment.
For example: in a radio listening survey, theinterviewers may be told to interview 500people living in a certain area & that out of
every 100 persons interviewed 60 are to behousewives, 25 farmers & 15 childrenunder the age of 15.
Within these quotas, the interviewer is free toselect the people interviewed.
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III. Convenience sampling
The method of convenience sampling is
also called the chunk.
A chunk is a fraction of one populationtaken for investigation because of its
covenient availability.
Thus, a chunk is selected neither byprobability nor by judgment but by
convenience.
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Sampling Distributions
Much of the information used in thebusiness & industry is gathered by
means of sampling.
It has been pointed out earlier that notonly it is often impossible either
physically or because of limitations
imposed by time to take a census ofall the items in the population.
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Central Limit Theorem
According to this theorem, if we selecta large number of simple random
samples, say, from any population
distribution & determine the mean ofeach sample, the distribution of theses
sample means will tend to be
described by the normal probabilitydistribution.
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Q1. Sample Distribution of the
Mean The time between 2 arrivals in a
queuing model is normally distributed
with a mean 2 minutes & standard
deviation 0.25 minute. If a randomsample of size 36 is drawn, what is the
probability that the sample mean will
be greater than 2.1 minutes?
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Q2. Sampling distribution of the
difference of the 2Means Strength of the wire produced by
company A has a mean of 4500 kg & astandard deviation of 200 kg.Company B has a mean of 4000 kg &a standard deviation of 300 kg. If 50wires of company A & 100 wires ofCompany B are selected at random &
tested for strength, what is theprobability that the sample mean ofstrength of A will be at least 600 kgmore than that of B?
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Q.3 Sampling distribution of
Proportions In a quality department of
manufacturing paints, at the time of
dispatch of decorative paints, 30% of
the containers are found to bedefective. If a random sample of 500
is drawn with replacement from the
population, what is the probability thatthe sample proportion will be less than
or equal to 25% defective?
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Q.4 Sampling distributions of the
difference of two proportions 10% of machines produced by
company A are defective & 5% of
those produced by company B are
defective. A random sample of 250machines is taken from company A &
a random sample of 300 machines is
taken from company B.What is the probability that the
difference in sample proportion is less
than or equal to 0.02?