SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL
description
Transcript of SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL
![Page 1: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/1.jpg)
SAMPLING, SAMPLING DISTRIBUTION &
CONFIDENCE INTERVALPresented By
Abid Nawaz MeraniNida Sohail
Farzah SiddiquiSoaiyba Jabeen Ahmed
![Page 2: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/2.jpg)
INTRODUCTIONGiven By:
FARZAH SIDDIQUI
![Page 3: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/3.jpg)
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.
FARZAH SIDDIQUI 01
![Page 4: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/4.jpg)
ADVANCE BUSINESS STATISTICS
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.
FARZAH SIDDIQUI 02
![Page 5: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/5.jpg)
ADVANCE BUSINESS STATISTICS
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”.
FARZAH SIDDIQUI 03
![Page 6: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/6.jpg)
ADVANCE BUSINESS STATISTICS
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.
FARZAH SIDDIQUI 04
![Page 7: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/7.jpg)
ADVANCE BUSINESS STATISTICS
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.
FARZAH SIDDIQUI 05
![Page 8: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/8.jpg)
ADVANCE BUSINESS STATISTICS
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.
FARZAH SIDDIQUI 06
![Page 9: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/9.jpg)
QUESTION ON SAMPLING
Explained By:NIDA SOHAIL
![Page 10: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/10.jpg)
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.
![Page 11: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/11.jpg)
ADVANCE BUSINESS STATISTICS
FINDING POSSIBLE SAMPLES :-
N n
5 2
5 2
Nida Sohail 08
![Page 12: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/12.jpg)
ADVANCE BUSINESS STATISTICS
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)
Nida Sohail 09
![Page 13: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/13.jpg)
ADVANCE BUSINESS STATISTICS
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
Nida Sohail 10
![Page 14: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/14.jpg)
FINDING MEAN OF SAMPLE MEANS :-
Where,X = Mean of Sample Means
X = 2
Nida Sohail ADVANCE BUSINESS STATISTICS 11
![Page 15: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/15.jpg)
ADVANCE BUSINESS STATISTICS
Where, µ = Population Mean x1 = 3 x2 = 0 x3 = 1 x4 = 2 x5 = 4 N = No. of Observations of the Population
µ = 2
Nida Sohail 12
FINDING POPULATION MEAN :-
![Page 16: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/16.jpg)
CONFIDENCE INTERVAL
Explained By:SOAIYBA JABEEN AHMED
![Page 17: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/17.jpg)
ADVANCE BUSINESS STATISTICS
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
Soaiyba Jabeen Ahmed 13
![Page 18: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/18.jpg)
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.
Soaiyba Jabeen Ahmed 14
![Page 19: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/19.jpg)
ADVANCE BUSINESS STATISTICS
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 15
![Page 20: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/20.jpg)
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:
![Page 21: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/21.jpg)
ADVANCE BUSINESS STATISTICS
DATA :-
Soaiyba Jabeen Ahmed 17
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
![Page 22: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/22.jpg)
ADVANCE BUSINESS STATISTICS
REQUIRED :-
Draw a sample of size n = 10 from the data.
Construct a 95% confidence interval for population parameter µ.
Soaiyba Jabeen Ahmed 18
![Page 23: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/23.jpg)
ADVANCE BUSINESS STATISTICS
SOLUTION :-FORMULA FOR CALCULATING
CONFIDENCE INTERVAL
< µ <
Lower Limit Upper Limit
Soaiyba Jabeen Ahmed 19
![Page 24: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/24.jpg)
ADVANCE BUSINESS STATISTICS
Where,
x = Sample Mean
σ = Population Standard Deviation
n = Sample Size
(1 – α) = Co-efficient of Confidence Interval
Soaiyba Jabeen Ahmed 20
![Page 25: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/25.jpg)
CONFIDENCE INTERVAL
FURTHER EXPLAINED By:
ABID NAWAZ MERANI
![Page 26: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/26.jpg)
ADVANCE BUSINESS STATISTICS
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.
Abid Nawaz Merani 21
![Page 27: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/27.jpg)
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
![Page 28: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/28.jpg)
ADVANCE BUSINESS STATISTICS
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
Abid Nawaz Merani 23
![Page 29: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/29.jpg)
ADVANCE BUSINESS STATISTICS
FINDING X :-
Abid Nawaz Merani 24
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
![Page 30: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/30.jpg)
ADVANCE BUSINESS STATISTICS
RANDOM SAMPLE :-
Abid Nawaz Merani 25
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
![Page 31: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/31.jpg)
ADVANCE BUSINESS STATISTICS
FINDING SAMPLE MEAN X :-X = ∑X n X = 1700
10
X = 170
Abid Nawaz Merani 26
![Page 32: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/32.jpg)
ADVANCE BUSINESS STATISTICS
FINDING CONFIDENCE INTERVAL :-Now we have,X = 170σ = 28.56Zα/2 = -1.96n = 10
By using the formula, < µ <
Abid Nawaz Merani 27
![Page 33: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/33.jpg)
ADVANCE BUSINESS STATISTICS
By putting the values in the formula,
158.49 < 168.65 < 181.50
Abid Nawaz Merani 28
![Page 34: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/34.jpg)
ADVANCE BUSINESS STATISTICS
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.
Abid Nawaz Merani 28
![Page 35: SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL](https://reader035.fdocuments.net/reader035/viewer/2022062411/56816305550346895dd382af/html5/thumbnails/35.jpg)
STATISTICS IS A LOGICAL SUBJECT AND WE ARE REALLY PROUD TO BE
THE PART OF SIXTY SEVEN LOGICAL MINDS.
THANK YOU SO MUCH FOR YOUR GENEROSITY, SUPPORT, ATTENTION AND FOR MAINTAINING DISCIPLINE
IN OUR PRESENTATION.