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# 14 sampling distribution

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### 14 sampling distribution

1. 1. 14-04-2012 1 Research Methodology Dr. NimitChowdhary,Professor © Dr. Nimit Chowdhary Research Methodology Workshop p. 2 Saturday, April 14, 2012  There exists a populationwith its parameters  A number of samples can be drawn from this population  Each sample will have its own sample statistics like sample’s mean standard deviation,etc.
2. 2. 14-04-2012 2 © Dr. Nimit Chowdhary Research Methodology Workshop p. 3 Saturday, April 14, 2012  Distributionof a sample statistic is called a sampling distribution  Samplingdistribution is different from sampledistribution- distribution of variables withinthe sample © Dr. Nimit ChowdharySaturday, April 14, 2012 A baby sitter has five children under her supervision. The average age of these five children is 6 years. However, the age of each child individually is as follows: Child (X) Age Child (X) Age 1 2 4 8 2 4 5 10 3 6
3. 3. 14-04-2012 3 © Dr. Nimit Chowdhary Research Methodology Workshop p. 5 N = 5, 5 1 5 2 4 6 8 10 5 30 6 5 i i X years              © Dr. Nimit ChowdharySaturday, April 14, 2012 2 5 ( ) 40 8 5 2.83 N i i X N years            X 2 6 16 4 6 4 6 6 0 8 6 4 10 6 16  2 ( )X  2 ( ) 40X   
4. 4. 14-04-2012 4 © Dr. Nimit Chowdhary Research Methodology Workshop p. 7 Let us take all the samples of size 2 from this population. There will be 10 samples 1 2 1 1 3 2 1 4 3 1 5 4 2 3 5 2 4 6 1 5 7 3 4 8 3 5 9 4 5 10 , (2, 4) 3 , (2, 6) 4 , (2,8) 5 , (2,10) 6 , (4, 6) 5 , (4,8) 6 , (4,10) 7 , (6,8) 7 , (6,10) 8 , (8,10) 9 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X                               © Dr. Nimit Chowdhary Research Methodology Workshop p. 8 Saturday, April 14, 2012 10 1 10 3 4 5 6 5 6 7 7 8 9 10 60 6 10 i i X X X X years               
5. 5. 14-04-2012 5 © Dr. Nimit ChowdharySaturday, April 14, 2012 Sample mean Frequency Relative Frequency Probability 3 1 1/10 .1 4 1 1/10 .1 5 2 2/10 .2 6 2 2/10 .2 7 2 2/10 .2 8 1 1/10 .1 9 1 1/10 .1 1.0 © Dr. Nimit ChowdharySaturday, April 14, 2012 Sample mean Probability 3 .1 4 .1 5 .2 6 .2 7 .2 8 .1 9 .1 1.0 X ( )P x
6. 6. 14-04-2012 6 © Dr. Nimit Chowdhary Saturday, April 14, 2012 . ( ) (3 .1) (4 .1) (5 .2) (6 .2) (7 .2) (8 .1) (9 .1) 6 x x x x P x years                     © Dr. Nimit Chowdhary Research Methodology Workshop p. 12 Saturday, April 14, 2012  Samplingdistribution of means  Samplingdistribution of proportions
7. 7. 14-04-2012 7 © Dr. Nimit Chowdhary Saturday, April 14, 2012 1. Mean of sampling distribution and mean of populationis the same 2. Spread of the sample means in the distributionis smaller than the spread in the samplevalues 3. Samplingdistribution of samplingmeans tend to be bell shaped © Dr. Nimit Chowdhary Research Methodology Workshop p. 14 Regardless of the shape of the distribution of the population, the distribution of the sample means approaches the normal probability distribution as the sample size increases. Thus, we can use our knowledge of normal distributions to arrive at conclusions about distribution of sample means
8. 8. 14-04-2012 8 © Dr. Nimit Chowdhary Research Methodology Workshop p. 15 x  Standard error of mean is the standard deviation (a measure of dispersion) of the distribution of sample means ( ) around mean of sampling distribution. This mean can be considered as same as population mean x © Dr. Nimit Chowdhary Research Methodology Workshop p. 16 Saturday, April 14, 2012 2 ( )x x x N      N= is the number of samples (not individual sample size)
9. 9. 14-04-2012 9 3 6 9 4 6 4 5 6 1 6 6 0 7 6 1 8 6 4 9 6 9 X x  2 ( )x X  2 ( )x x   = 28 2 ( ) 28 4 7 2 x x x x x N           Note: Standard error (2) is smaller than population standard deviation (2.83)
10. 10. 14-04-2012 10 © Dr. Nimit ChowdharySaturday, April 14, 2012 1x x N n Nn n         1 N n N   x  For large samples when N>>n would approach 1 Then, N = Population size N = sample size  = populations standard deviation = standard error of the mean Note: decreases as sample size increases x  © Dr. Nimit Chowdhary Research Methodology Workshop p. 20 1 N n N   Is the finite correction factor • Must be used in case of large samples • Used when sample size is more that 5% of the population size
11. 11. 14-04-2012 11 The IQ scores of college students are normally distributed with a mean  of 120 and standard deviation of 10.  What is the probability that the IQ scores of any one student chosen at random is between 120 and 125 120  = 120  = 10 125 ( ) (125 120) 10 5 0.5 10 x Z Z         Calculating for Z, we get Area for Z=0.5, from Z tables is 0.1915. Therefore, there are 19.15 % chance that student picked up randomly will have an IQ score between 120 and 125.
12. 12. 14-04-2012 12  If a random sample of 25 students is taken, what is the probability that the mean of this sample will be between 120 and 125. We know, 1.This is the distribution of sample means. One of these samples (of size 25) has a mean of 120 2.The mean of the distribution will be same as population mean 120 3.The standard deviation of means around population mean (S.E.) will have to be calculated © Dr. Nimit Chowdhary Research Methodology Workshop p. 24 . . x S E n    10 10 2 525 x     2.5 ( ) 125 120 2 5 2.5 2 0.4938 x x z x Z Z Area           2x   Therefore, 120 125 120x  
13. 13. 14-04-2012 13 © Dr. Nimit Chowdhary Research Methodology Workshop p. 25 Saturday, April 14, 2012  Thus there are 49.38% chance that the samplemean would be between 120 and 125.  One can see that as sample size increases, s.d. reduces further and this chance will further increase.  The previous case can be considered as a limiting casewhen sample size = 1 (the samplesize reduces and so does the chance). …can be defined as a distribution of proportions of all possible random samples of a fixed size n… p = sample proportion  = populationproportion