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14 sampling distribution

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    14 sampling distribution 14 sampling distribution Document Transcript

    • 14-04-2012 Research Methodology Dr. Nimit Chowdhary, Professor There exists a population with 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.© Dr. Nimit Chowdhary Research Methodology Workshop p. 2 Saturday, April 14, 2012 1
    • 14-04-2012 Distribution of a sample statistic is called a sampling distribution Sampling distribution is different from sample distribution- distribution of variables within the sample© Dr. Nimit Chowdhary Research Methodology Workshop p. 3 Saturday, April 14, 2012A 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 asfollows: Child (X) Age Child (X) Age 1 2 4 8 2 4 5 10 3 6Saturday, April 14, 2012 © Dr. Nimit Chowdhary 2
    • 14-04-2012 5N = 5, X i 1 i  5 2  4  6  8  10  5 30   6 years 5© Dr. Nimit Chowdhary Research Methodology Workshop p. 5 N X  ( X   )2  ( X i   )2 i 5 2 6 16  N 4 6 4 40 6 6 0   8 8 6 4 5 10 6 16   2.83 years ( X   ) 2  40Saturday, April 14, 2012 © Dr. Nimit Chowdhary 3
    • 14-04-2012 X 1 , X 2  (2, 4)  X 1  3 Let us take all the samples of size 2 X 1 , X 3  (2, 6)  X 2  4 from this X 1 , X 4  (2, 8)  X 3  5 population. X 1 , X 5  (2,10)  X 4  6 There will be 10 X 2 , X 3  (4, 6)  X 5  5 samples X 2 , X 4  (4, 8)  X 6  6 X 1 , X 5  (4,10)  X 7  7 X 3 , X 4  (6, 8)  X 8  7 X 3 , X 5  (6,10)  X 9  8 X 4 , X 5  (8,10)  X 10  9© Dr. Nimit Chowdhary Research Methodology Workshop p. 7 10 X i 1 i X 10 3 4 5 6  5 6  7  7 89 X 10 60 X  6 years 10© Dr. Nimit Chowdhary Research Methodology Workshop p. 8 Saturday, April 14, 2012 4
    • 14-04-2012 Sample Relative mean Frequency 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 1.0Saturday, April 14, 2012 © Dr. Nimit Chowdhary Sample mean Probability X P ( x) 3 .1 4 .1 5 .2 6 .2 7 .2 8 .1 9 .1 1.0Saturday, April 14, 2012 © Dr. Nimit Chowdhary 5
    • 14-04-2012 x   x.P ( x) x  (3  .1)  (4  .1)  (5  .2)  (6  .2)  (7  .2)  (8  .1)  (9  .1) x  6 years© Dr. Nimit Chowdhary Saturday, April 14, 2012 Sampling distribution of means Sampling distribution of proportions Research Methodology Workshop p.© Dr. Nimit Chowdhary 12 Saturday, April 14, 2012 6
    • 14-04-20121. Mean of sampling distribution and mean of population is the same2. Spread of the sample means in the distribution is smaller than the spread in the sample values3. Sampling distribution of sampling means tend to be bell shaped© Dr. Nimit Chowdhary Saturday, April 14, 2012 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 normaldistributions to arrive at conclusions aboutdistribution of sample means© Dr. Nimit Chowdhary Research Methodology Workshop p. 14 7
    • 14-04-2012 xStandard error of mean is the standarddeviation (a measure of dispersion) of thedistribution of sample means ( x ) aroundmean of sampling distribution.This mean can be considered as same aspopulation mean© Dr. Nimit Chowdhary Research Methodology Workshop p. 15 ( x   x ) 2 x  N N= is the number of samples (not individual sample size) Research Methodology Workshop p.© Dr. Nimit Chowdhary 16 Saturday, April 14, 2012 8
    • 14-04-2012 X x ( X   x )2 3 6 9 4 6 4 5 6 1 6 6 0 7 6 1 8 6 4 9 6 9 ( x   x ) 2= 28 ( x   x ) 2x  N 28 Note: Standard error (2)x   4 7 is smaller than population standardx  2 deviation (2.83) 9
    • 14-04-2012  N n N = Population sizex  N = sample size n N 1  = populations standardFor large samples when N>>n deviation N n  x= standard error of the would approach 1 mean N 1Then, Note:  x decreases as  sample size increasesx  nSaturday, April 14, 2012 © Dr. Nimit Chowdhary Is the finite correction factor N n • Must be used in case of large N 1 samples • Used when sample size is more that 5% of the population size© Dr. Nimit Chowdhary Research Methodology Workshop p. 20 10
    • 14-04-2012The IQ scores of college students are normallydistributed with a mean  of 120 and standarddeviation  of 10. What is the probability that the IQ scores of any one student chosen at random is between 120 and 125 125 120  = 120  = 10Calculating for Z, we get Area for Z=0.5, from Z tables is 0.1915. ( x   ) (125  120)Z  Therefore, there are 19.15 %  10 chance that student picked 5 up randomly will have an IQZ  0.5 10 score between 120 and 125. 11
    • 14-04-2012 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 1202. The mean of the distribution will be same as population mean 1203. The standard deviation of means around population mean (S.E.) will have to be calculated 125 120  x  120 Therefore,  x  2 S .E.   x  n (x  x ) 125  120 10 10 Z  x   2 x 2 25 5 5 Z  2.5 2 Areaz  2.5  0.4938© Dr. Nimit Chowdhary Research Methodology Workshop p. 24 12
    • 14-04-2012 Thus there are 49.38% chance that the sample mean 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 case when sample size = 1 (the sample size reduces and so does the chance).© Dr. Nimit Chowdhary Research Methodology Workshop p. 25 Saturday, April 14, 2012…can be defined as a distribution of proportions of all possible random samples of a fixed size n… p = sample proportion  = population proportion 13