quantitative techniques by amit ramawat

8/13/2019 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?

quantitative techniques by amit ramawat

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