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22S:
Statistical Methods and Computing
Sampling Distributions
Lecture 11 Mar. 5, 2008
Kate Cowles 374 SH, 335- kcowles@stat.uiowa.edu
Sampling distributions Suppose our sample size was 10.
What can we say about ¯x from a sample of size 10 as an estimate of μ?
We can get an idea of how good an estimator ¯x is likely to be by asking ”What would happen if we took many samples of 10 subjects from this population?”
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To answer this question, we would like to
- Take a large number of samples of size 10 from the same population
- Calculate the sample mean ¯x for each sample
- Make a histogram of the values of ¯x
- Examine the distribution displayed in the his- togram for shape, center, spread, and outliers
4 Simulating the sampling distribution of x ¯
In practice, it is too difficult and expensive to draw many samples from a large population such as all adult Chinese males. But we can imitate random sampling by using a computer to do simulation.
In this way we can study the distribution of sam- ple means.
- Imagine that we knew that upper arm skin- fold thicknesses in Chinese males followed a normal distribution with mean μ = 10 mm and standard deviation σ = 3 mm.
- Then we could use the computer to gener- ate many random samples of size 10 from this distribution and calculate X¯ from each of these samples
The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.
Our simulated distribution is only an approxi- mation to the true sampling distribution, but it gives us an idea of what it would look like.
The mean and standard deviation of x¯
Let ¯x be the sample mean of a simple random sample of size n drawn from a large population with mean μ and standard deviation σ.
- Then the mean of the sampling distribution of ¯x is μ.
- The standard deviation is of the sampling distribution of √σn. - Averages are less variable than individual observations. - The results of large samples are less vari- able than the results of small samples
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The central limit theorem
We have described the center and spread of the sampling distribution of ¯x. What about its shape?
The shape depends on the shape of the popula- tion distribution.
Special case:
- If a population has a normal distribution with mean μ and standard deviation σ, then the sample mean ¯x of a random sample of n observations has a normal distribution with mean μ and standard deviation √σn
8 The Central Limit Theorem says that
- Provided that n is large enough, the shape of the sampling distribution of ¯x is approxi- mately normal.
Now we have to unstandardize
x¯ − 120 3
− 1 .96(3) ≤ x¯ − 120 ≤ 1 .96(3) 120 − 1 .96(3) ≤ x¯ ≤ 120 + 1.96(3)
- 12 ≤ x¯ ≤ 125. 88
In other words, 95% of all the possible sam- ples of size 25 drawn from this population would have sample means between 114.12 and 125. oz.
