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World Tour Essay

1024 words - 5 pages

Sampling Distributions
Suppose that we draw all possible samples of size n from a given population. Suppose further that we compute a statistic (e.g., a mean, proportion, standard deviation) for each sample. The probability distribution of this statistic is called a sampling distribution.
Variability of a Sampling Distribution
The variability of a sampling distribution is measured by its variance or its standard deviation. The variability of a sampling distribution depends on three factors:
* N: The number of observations in the population.
* n: The number of observations in the sample.
* The way that the random sample is chosen.
If the population size is much larger than the ...view middle of the document...

When a population element can be selected more than one time, we are sampling with replacement
2) Without replacement
Suppose we have a bowl of 100 unique numbers from 0 to 99. We want to select a random sample of numbers from the bowl. After we pick a number from the bowl, we can put the number aside or we can put it back into the bowl. If we put the number back in the bowl, it may be selected more than once; if we put it aside, it can selected only one time.
When a population element can be selected only one time, we are sampling without replacement.

THE TOPIC WHICH WE HAVE SELECTED IS WITHOUT REPLACEMENT SAMPLING DISTRIBUTION
Example:
Draw all possible samples of size 2 without replacement from a population consisting of 3, 6, 9, 12, 15. Form the sample distribution of sample means and verify the results.
Solution:
We have population values 3, 6, 9, 12, 15, population size N=5 and sample size n=2. Thus, the number of possible samples which can be drawn without replacement is
= =10

Sample No. | Sample Values | Sample Mean
(X) | Sample No. | Sample Values | Sample Mean
(X) |
1 | 3, 6 | 4.5 | 6 | 6, 12 | 9.0 |
2 | 3, 9 | 6.0 | 7 | 6, 15 | 10.5 |
3 | 3, 12 | 7.5 | 8 | 9, 12 | 10.5 |
4 | 3, 15 | 9.0 | 9 | 9, 15 | 12.0 |
5 | 6, 9 | 7.5 | 10 | 12, 15 | 13.5 |
The sampling distribution of the sample mean X and its mean and standard deviation are:
X | f | f(X) | Xf(X) | X2f(X) |
4.5 | 1 | 1/10 | 4.5/10 | 20.25/10 |
6.0 | 1 | 1/10 | 6.0/10 | 36.00/10 |
7.5 | 2 | 2/10 | 15.0/10 | 112.50/10 |
9.0 | 2 | 2/10 | 18.0/10 | 162.00/10 |
10.5 | 2 | 2/10 | 21.0/10 | 220.50/10 |
12.0 | 1 | 1/10 | 12.0/10 | 144.00/10 |
...

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