# SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVAL

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SAMPLING, SAMPLING DISTRIBUTION & CONFIDENCE INTERVALPresented By

Abid Nawaz MeraniNida SohailFarzah SiddiquiSoaiyba Jabeen Ahmed

1INTRODUCTIONGiven By:FARZAH SIDDIQUISAMPLING :-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 SIDDIQUIADVANCE BUSINESS STATISTICS01ADVANTAGES OF SAMPLING :-Sampling method is cheaper to collect information as compared to census(i.e. complete enumeration).

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

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

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

FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS02POPULATION :-

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 SIDDIQUIADVANCE BUSINESS STATISTICS03PARAMETER :-

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 and the population variance is denoted by . FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS04STATISTIC :-

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 s.FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS05SAMPLING 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 SIDDIQUIADVANCE BUSINESS STATISTICS06QUESTION ON SAMPLINGExplained By:NIDA SOHAILNida SohailADVANCE BUSINESS STATISTICS07QUESTION :-

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

Nida SohailADVANCE BUSINESS STATISTICS08

FINDING SAMPLE SIZE :-SAMPLE NUMBERALL POSSIBLE SAMPLES01(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 SohailADVANCE BUSINESS STATISTICS09FINDING SAMPLE MEANS :-ALL POSSIBLE SAMPLESSAMPLE 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 = 3.0Nida SohailADVANCE BUSINESS STATISTICS10FINDING MEAN OF SAMPLE MEANs :-Where,X = Mean of Sample Means

X = 2

Nida SohailADVANCE BUSINESS STATISTICS11

Where, = Population Meanx1 = 3x2 = 0x3 = 1x4 = 2x5 = 4N = No. of Observations of the Population

= 2Nida SohailADVANCE BUSINESS STATISTICS12FINDING POPULATION MEAN :-

Confidence intervalExplained By:SOAIYBA JABEEN AHMEDESTIMATION :-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 5Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS13x = 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 .Soaiyba Jabeen AhmedADVANCED BUSINESS STATISTICS14CONFIDENCE 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 AhmedADVANCE BUSINESS STATISTICS15Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS16QUESTION :-

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 168.65 and Standard Deviation 28.56, is:

DATA :-Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS17115125125130130130130135135140140140140145145150150150155160160160160160165165165165165165165170170170170170170170175175175180180180180180185185185200105110115125125130135145145150150160165165165170170170170170175175175180180180180185185190190190190195200200200200200205210210210210215220230230240240REQUIRED :-Draw a sample of size n = 10 from the data.

Construct a 95% confidence interval for population parameter .

Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS18SOLUTION :-FORMULA FOR CALCULATING CONFIDENCE INTERVAL

< 30Population unknown

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

FINDING Z/2 :-Z/2 = Z0.05/2Z/2 = Z0.025 (From the Z-Table)Z/2 = - 1.96Abid Nawaz MeraniADVANCE BUSINESS STATISTICS23FINDING X :-Abid Nawaz MeraniADVANCE BUSINESS STATISTICS2401115125125130130130130135135140021401401401451451501501501551600316016016016016516516516516516504165170170170170170170170175175051751801801801801801851851852000610511011512512513013514514515007150160165165165170170170170170081751751751801801801801851851900919019019019520020020020020020510210210210210215220230230240240RANDOM SAMPLE :-

Abid Nawaz MeraniADVANCE BUSINESS STATISTICS25Serial No.Random Nos.Random Sample010.38 4X1 = 180020.84 6X2 = 175030.46 7X3 = 125040.76 2X4 =145050.54 1X5 = 170060.45 5X6 = 180070.48 3X7 = 180080.11 5 X8 = 125090.65 3 X9 = 185100.03 9 X10 = 160----------X = 1700FINDING SAMPLE MEAN X :-X = X n X = 1700 10 X = 170 Abid Nawaz MeraniADVANCE BUSINESS STATISTICS26FINDING CONFIDENCE INTERVAL :-Now we have,X = 170 = 28.56Z/2 = -1.96n = 10By using the formula, <