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Given By: F ARZAH S IDDIQUI

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ADVANCE BUSINESS STATISTICS01

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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 SIDDIQUIADVANCE BUSINESS STATISTICS02

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FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS03

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FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS04

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FARZAH SIDDIQUIADVANCE BUSINESS STATISTICS05

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Consider 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 STATISTICS06

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Explained By: N IDA S OHAIL

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Nida SohailADVANCE BUSINESS STATISTICS07

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N n 5 2 5 2 Nida SohailADVANCE BUSINESS STATISTICS08

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SAMPLE NUMBERALL 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 SohailADVANCE BUSINESS STATISTICS09

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ALL POSSIBLE SAMPLESSAMPLE MEAN X = ( x 1 + x 2 ) / 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 SohailADVANCE BUSINESS STATISTICS10

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Where, X = Mean of Sample Means X = 2 Nida SohailADVANCE BUSINESS STATISTICS11

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Where, µ = Population Mean x 1 = 3 x 2 = 0 x 3 = 1 x 4 = 2 x 5 = 4 N = No. of Observations of the Population µ = 2 Nida SohailADVANCE BUSINESS STATISTICS12

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Explained By: S OAIYBA J ABEEN A HMED

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Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS13

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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 AhmedADVANCED BUSINESS STATISTICS14

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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 that the parameter lies in the interval. The confidence percentage is called confidence level. It is abbreviated by CI. Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS15

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Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS16

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Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS17 115125 130 135 140 145 150 155160 165 170 175 180 185 200 105110115125 130135145 150 160165 170 175 180 185 190 195200 205 210 215220230 240

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Draw a sample of size n = 10 from the data. Construct a 95% confidence interval for population parameter µ. Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS18

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FORMULA FOR CALCULATING CONFIDENCE INTERVAL < µ < Lower Limit Upper Limit Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS19

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Where, x = Sample Mean σ = Population Standard Deviation n = Sample Size (1 – α) = Co-efficient of Confidence Interval Soaiyba Jabeen AhmedADVANCE BUSINESS STATISTICS20

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FURTHER EXPLAINED By: A BID N AWAZ M ERANI

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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 MeraniADVANCE BUSINESS STATISTICS21

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Abid Nawaz MeraniADVANCE BUSINESS STATISTICS22 Condition # 2 :- Population normal Sample size n < 30 Population σ known Condition # 3 :- Population normal Sample size n > 30 Population σ unknown

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FINDING α :- 1 - α = 95% α = 1 – 0.95 α = 0.05 FINDING Z α/2 :- Z α/2 = Z 0.05/2 Z α/2 = Z 0.025 (From the Z-Table) Z α/2 = - 1.96 Abid Nawaz MeraniADVANCE BUSINESS STATISTICS23

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Abid Nawaz MeraniADVANCE BUSINESS STATISTICS24 01115125 130 135 140 02140 145 150 155160 03160 165 04165170 175 05175180 185 200 06105110115125 130135145 150 07150160165 170 08175 180 185 190 09190 195200 205 10210 215220230 240

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RANDOM SAMPLE :- Abid Nawaz MeraniADVANCE BUSINESS STATISTICS25 Serial No.Random Nos.Random Sample 010.38 4X 1 = 180 020.84 6X 2 = 175 030.46 7X 3 = 125 040.76 2X 4 =145 050.54 1X 5 = 170 060.45 5X 6 = 180 070.48 3X 7 = 180 080.11 5X 8 = 125 090.65 3X 9 = 185 100.03 9X 10 = 160 ----- ∑X = 1700

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X = ∑X n X = 1700 10 X = 170 Abid Nawaz MeraniADVANCE BUSINESS STATISTICS26

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Now we have, X = 170 σ = 28.56 Z α/2 = -1.96 n = 10 By using the formula, < µ < Abid Nawaz MeraniADVANCE BUSINESS STATISTICS27

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By putting the values in the formula, 158.49 < 168.65 < 181.50 Abid Nawaz MeraniADVANCE BUSINESS STATISTICS28

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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 MeraniADVANCE BUSINESS STATISTICS28

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