Download - 14 sampling distribution
14-04-2012
1
Research Methodology Dr. Nimit Chowdhary, Professor
© Dr. Nimit Chowdhary Research Methodology Workshop p. 2 Saturday, April 14, 2012
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.
14-04-2012
2
© Dr. Nimit Chowdhary Research Methodology Workshop p. 3 Saturday, April 14, 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 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) Age1 2 4 82 4 5 103 6
14-04-2012
3
© Dr. Nimit Chowdhary Research Methodology Workshop p. 5
N = 5,5
1
52 4 6 8 10
530 65
ii
X
years
© Dr. Nimit ChowdharySaturday, April 14, 2012
2
5( )
40 85
2.83
N
ii
X
N
years
X2 6 164 6 46 6 08 6 410 6 16
2( )X
2( ) 40X
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
103 4 5 6 5 6 7 7 8 9
1060 610
ii
XX
X
X years
14-04-2012
5
© Dr. Nimit ChowdharySaturday, April 14, 2012
Sample mean Frequency
Relative Frequency Probability
3 1 1/10 .14 1 1/10 .15 2 2/10 .26 2 2/10 .27 2 2/10 .28 1 1/10 .19 1 1/10 .1
1.0
© Dr. Nimit ChowdharySaturday, April 14, 2012
Sample mean Probability
3 .14 .15 .26 .27 .28 .19 .1
1.0
X ( )P x
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 ChowdharyResearch Methodology Workshop p.
12 Saturday, April 14, 2012
Sampling distribution of means Sampling distribution of proportions
14-04-2012
7
© Dr. Nimit Chowdhary Saturday, April 14, 2012
1. Mean of sampling distribution and mean of population is the same
2. Spread of the sample means in the distribution is smaller than the spread in the sample values
3. Sampling distribution of sampling means 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
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 ChowdharyResearch Methodology Workshop p.
16 Saturday, April 14, 2012
2( )xx
xN
N= is the number of samples (not individual sample size)
14-04-2012
9
3 6 94 6 45 6 16 6 07 6 18 6 49 6 9
X x2( )xX
2( )xx = 28
2( )
28 47
2
xx
x
x
xN
Note: Standard error (2) is smaller than population standard deviation (2.83)
14-04-2012
10
© Dr. Nimit ChowdharySaturday, April 14, 2012
1x
x
N nNn
n
1N nN
xFor 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
1N nN
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
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 0.510
xZ
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.
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
. . xS En
10 10 2525x
2.5
( ) 125 1202
5 2.52
0.4938
x
x
z
xZ
Z
Area
2x Therefore,120
125
120x
14-04-2012
13
© Dr. Nimit Chowdhary Research Methodology Workshop p. 25 Saturday, April 14, 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).
…can be defined as a distribution of proportions of all possible random samples of a fixed size n…
p = sample proportion = population proportion