Chapter 7: Variation in repeated samples – Sampling distributions
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Transcript of Chapter 7: Variation in repeated samples – Sampling distributions
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Recall that a major objective of statistics is to make inferences about a population from an analysis of information contained in sample data.
Typically, we are interested in learning about some numerical feature of the population, such as
• the proportion possessing a stated characteristic;
• the mean and the standard deviation.
A numerical feature of a population is called a parameter.
The true value of a parameter is unknown. An appropriate sample-based quantity is our source about the value of a parameter.
A statistic is a numerical valued function of the sample observations.
Sample mean is an example of a statistic.
Chapter 7: Variation in repeated samples – Sampling distributions
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The sampling distribution of a statisticThree important points about a statistic:
• the numerical value of a statistic cannot be expected to give us the exact value of the parameter;
• the observed value of a statistic depends on the particular sample that happens to be selected;
• there will be some variability in the values of a statistic over different occasions of sampling.
Because any statistic varies from sample to sample, it is a random variable and has its own probability distribution.
The probability distribution of a statistic is called its sampling distribution.
Often we simply say the distribution of a statistic.
![Page 3: Chapter 7: Variation in repeated samples – Sampling distributions](https://reader036.fdocuments.us/reader036/viewer/2022080923/56812c61550346895d90f449/html5/thumbnails/3.jpg)
Statistical inference about the population mean is of prime practical importance. Inferences about this parameter are based on the sample mean and its sampling distribution.
Distribution of the sample mean
![Page 4: Chapter 7: Variation in repeated samples – Sampling distributions](https://reader036.fdocuments.us/reader036/viewer/2022080923/56812c61550346895d90f449/html5/thumbnails/4.jpg)
![Page 5: Chapter 7: Variation in repeated samples – Sampling distributions](https://reader036.fdocuments.us/reader036/viewer/2022080923/56812c61550346895d90f449/html5/thumbnails/5.jpg)
Figure 7.4 (p. 275)
Distributions of for n = 3 and n = 10 in sampling from an asymmetric population.X
An example illustrating the central limit theorem
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Consider a population with mean 82 and standard deviation 12.
If a random sample of size 64 is selected, what is the probability that the sample mean will lie between 80.8 and 83.2?
Solution: We have μ = 82 and σ = 12. Since n = 64 is large, the central limit theorem tells us that the distribution of the sample mean is approximately normal with
Converting to the standard normal variable:
Thus,
5.164
12)(,82)(
nXsdXE
5.1
82
X
n
XZ
Example on probability calculations for the sample mean
5762.2119.7881.]8.8.[
]5.1/)822.83(5.1/)828.80[(
]2.838.80[
ZP
ZP
XP