Download - SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL

SAMPLING, SAMPLING DISTRIBUTION &

CONFIDENCE INTERVALPresented By

Abid Nawaz MeraniNida Sohail

Farzah SiddiquiSoaiyba Jabeen Ahmed

INTRODUCTIONGiven By:

FARZAH SIDDIQUI

ADVANCE BUSINESS STATISTICS

SAMPLING :-The process of selecting a small part from a large collection, such that the selected part will show all the characteristics of the large collection is called Sampling.

SHORT EXPLANATION :-

Sampling is quite often used in our day-to-day practical life. For Example, in a shop we assess the quality of rice, sugar and any other commodity by taking a handful of it from the large and then decide to purchase it or not.

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ADVANTAGES OF SAMPLING :-o Sampling method is cheaper to collect information as

compared to census(i.e. complete enumeration).

o The data may be collected, classified and analyzed much more quickly with a sample than with a census enquiry.

o A sample is often used as a check to verify the accuracy of complete count.

o It provides greater accuracy because the volume of work is reduced in the sample survey.

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POPULATION :-A population may be defined as the large collection of similar units. The number of units in the population (i.e. size of a population) is always denoted by “N”.

SAMPLE :-A small part of a population is called Sample. The number of units in the sample (i.e. size of a sample) is always denoted by “n”.

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PARAMETER :-Any quantity calculated from a population is called a Parameter. For example, population mean, population variance, etc. are therefore parameters. The population mean is denoted by µ and the population variance is denoted by σ².

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STATISTIC :-Any characteristics estimate from a sample is called Statistic.For example, sample mean, sample variance, etc. are therefore statistic. The sample mean is denoted by x and the sample variance is denoted by s².

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SAMPLING DISTRIBUTIONConsider all possible samples of size n which can be drawn from a given population. For each sample we can compute a statistic. Which will vary from sample to sample. In this way we obtain a distribution of the statistic which is called its sampling distribution.

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QUESTION ON SAMPLING

Explained By:NIDA SOHAIL

ADVANCE BUSINESS STATISTICSNida Sohail 07

QUESTION :-Following is the data of Wickets taken by an Australian Bowler “Bret Lee” in his recent 5 matches against India.

Wickets : 3, 0, 1, 2, 4

Required: Draw a sample size n = 2. Find all the sample means. Find the mean of sample means. Find population mean.


FINDING POSSIBLE SAMPLES :-

N n

5 2

5 2

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FINDING SAMPLE SIZE :-SAMPLE NUMBER ALL POSSIBLE SAMPLES

01 (3 , 0)

02 (3 , 1)

03 (3 , 2)

04 (3 , 4)

05 (0 , 1)

06 (0 , 2)

07 (0 , 4)

08 (1 , 2)

09 (1 , 4)

10 (2 , 4)

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FINDING SAMPLE MEANS :-ALL POSSIBLE SAMPLES SAMPLE MEAN

X = (x1 + x2) / 2(3 , 0) X 1 = 1.5

(3 , 1) X 2 = 2.0

(3 , 2) X 3 = 2.5

(3 , 4) X 4 = 3.5

(0 , 1) X 5 = 0.5

(0 , 2) X 6 = 1.0

(0 , 4) X 7 = 2.0

(1 , 2) X 8 = 1.5

(1 , 4) X 9 = 2.5

(2 , 4) X 10 = 3.0

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FINDING MEAN OF SAMPLE MEANS :-

Where,X = Mean of Sample Means

X = 2

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Where, µ = Population Mean x1 = 3 x2 = 0 x3 = 1 x4 = 2 x5 = 4 N = No. of Observations of the Population

µ = 2

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FINDING POPULATION MEAN :-

CONFIDENCE INTERVAL

Explained By:SOAIYBA JABEEN AHMED


ESTIMATION :-Estimation means to determine the unknown value of a population parameter by the help of sample data.

FOR EXAMPLE :-Let 2, 4, 6, 8, 10 are the sample observations then x = 2 + 4 + 6 + 8 + 10 5

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ADVANCED BUSINESS STATISTICS

x = 6

Where 6 is an estimate where as the statistic X used as formula is called an estimator.The statistic X is said to be an unbiased estimator when the mean of all possible X is equal to the population mean µ.



CONFIDENCE INTERVAL :-A confidence -interval estimate of a parameter consist of an interval of numbers obtained from a point estimate of the parameter together with a percentage that specifies how confident we are thatthe parameter lies in the interval. The confidence percentage is called confidence level. It is abbreviated by CI.


Soaiyba Jabeen Ahmed ADVANCE BUSINESS STATISTICS 16

QUESTION :-

The data of Cholesterol Level of 100 Individuals belonging to the age group between 20 to 40 years that prefer Junk Food, is collected from a Heart Disease Hospital situated in Federal B. Area, has the Population Mean “µ” 168.65 and Standard Deviation “σ” 28.56, is:


DATA :-


115 125 125 130 130 130 130 135 135 140

140 140 140 145 145 150 150 150 155 160

160 160 160 160 165 165 165 165 165 165

165 170 170 170 170 170 170 170 175 175

175 180 180 180 180 180 185 185 185 200

105 110 115 125 125 130 135 145 145 150

150 160 165 165 165 170 170 170 170 170

175 175 175 180 180 180 180 185 185 190

190 190 190 195 200 200 200 200 200 205

210 210 210 210 215 220 230 230 240 240


REQUIRED :-

Draw a sample of size n = 10 from the data.

Construct a 95% confidence interval for population parameter µ.



SOLUTION :-FORMULA FOR CALCULATING

CONFIDENCE INTERVAL

< µ <

Lower Limit Upper Limit



Where,

x = Sample Mean

σ = Population Standard Deviation

n = Sample Size

(1 – α) = Co-efficient of Confidence Interval


CONFIDENCE INTERVAL

FURTHER EXPLAINED By:

ABID NAWAZ MERANI


CONDITION FOR APPLYING FORMULA :-Before solving the question we have to check the following conditions, if any one of these conditions are fulfilled then we apply the formula. The conditions are:

Condition # 1 :- Population normal. Sample size n ≥ 30. Population σ unknown.

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ADVANCE BUSINESS STATISTICSAbid Nawaz Merani 22

Condition # 2 :- Population normal Sample size n < 30 Population σ known

Condition # 3 :- Population normal Sample size n > 30 Population σ unknown


FINDING α :-1 - α = 95%α = 1 – 0.95α = 0.05

FINDING Zα/2 :-Zα/2 = Z0.05/2Zα/2 = Z0.025 (From the Z-Table)Zα/2 = - 1.96



FINDING X :-


01 115 125 125 130 130 130 130 135 135 140

02 140 140 140 145 145 150 150 150 155 160

03 160 160 160 160 165 165 165 165 165 165

04 165 170 170 170 170 170 170 170 175 175

05 175 180 180 180 180 180 185 185 185 200

06 105 110 115 125 125 130 135 145 145 150

07 150 160 165 165 165 170 170 170 170 170

08 175 175 175 180 180 180 180 185 185 190

09 190 190 190 195 200 200 200 200 200 205

10 210 210 210 210 215 220 230 230 240 240


RANDOM SAMPLE :-


Serial No. Random Nos. Random Sample

01 0.38 4 X1 = 180

02 0.84 6 X2 = 175

03 0.46 7 X3 = 125

04 0.76 2 X4 =145

05 0.54 1 X5 = 170

06 0.45 5 X6 = 180

07 0.48 3 X7 = 180

08 0.11 5 X8 = 125

09 0.65 3 X9 = 185

10 0.03 9 X10 = 160

----- ----- ∑X = 1700


FINDING SAMPLE MEAN X :-X = ∑X n X = 1700

10

X = 170



FINDING CONFIDENCE INTERVAL :-Now we have,X = 170σ = 28.56Zα/2 = -1.96n = 10

By using the formula, < µ <



By putting the values in the formula,

158.49 < 168.65 < 181.50



CONCLUSION :-Just as the Population Mean of 100 individuals lies between the Confidence Interval computed by taking the Sample of 10 observations from the given Population, the Population Mean of thousands of individuals also lies between the Confidence Interval if we take 100 observations as a Sample of that data.


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THE PART OF SIXTY SEVEN LOGICAL MINDS.

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