1. Daniela Stan Raicu School of CTI, DePaul University
CSC 323 Quarter: Spring 02/03
Daniela Stan Raicu
School of CTI, DePaul University
11/28/2018
Daniela Stan - CSC323

2. Outline Chapter 5: Sampling Distributions Population and sample
Sampling distribution of a sample mean
Central limit theorem
Examples
11/28/2018
Daniela Stan - CSC323

3. Introduction Population Sample
This chapter begins a bridge from the study of probabilities to the study of statistical inference, by introducing the sampling distribution.
Quality of sample data:
The quality of all statistical
analysis depends on the quality
of the sample data
If the data sample is not representative, analyzing the data and drawing conclusions will be unproductive-at best.
Sample
Population
Random Sampling: every unit in the
population has an equal chance to be
chosen
11/28/2018
Daniela Stan - CSC323

4. Some definitions  2 Parameter: A number describing a population.
Statistic: A number describing a sample.
1. A random sample should represent the population well, so sample statistics from a random sample should provide reasonable estimates of population parameters.
Sample statistics
Population parameter
Sample mean x 
Sample proportion p_hat p
Sample variance s2 2
11/28/2018
Daniela Stan - CSC323

5. Some definitions (cont.)
2. All sample statistics have some error in estimating population parameters.
3. If repeated samples are taken from a population and the same statistic (e.g. mean) is calculated from each sample, the statistics will vary, that is, they will have a distribution.
4. A larger sample provides more information than a smaller sample so a statistic from a large sample should have less error than a statistic from a small sample.
11/28/2018
Daniela Stan - CSC323

6. Describing the Sample Mean
Let us assume that we want to estimate the mean  of the population since usually this is the first piece of information that an analyst wants to analyze:
Since the value of the sample mean depends on the particular sample we draw, the sample mean is a variable with a huge number of possible values.
The sample mean is a random variable because the samples are drawn randomly.
The best way to summarize this vast amount of information is to describe it with a probability distribution.
11/28/2018
Daniela Stan - CSC323

7. The Distribution of the Sample Mean
Problem:
Population:
{A,B,C,D,E,F}
Population mean:
 = .1483
Population Variance:
 =
11/28/2018
Daniela Stan - CSC323

8. The Distribution of the Sample Mean
Assumptions:
What is the central value of the variable x?
What is its variability?
Is there a familiar pattern in the variability?
 = .1483
 =
11/28/2018
Daniela Stan - CSC323

9. What is the central value of the sample mean?
For large samples, the distribution of x should be symmetrical: x should be larger than  about 50% of the time and x should be smaller than  about 50% of the time.
It can be shown theoretically (Central Limit theorem) that the mean of the sample means equals the population mean:
E(x) = 
In our example, E(x)= = 
x is an unbiased estimator
11/28/2018
Daniela Stan - CSC323

10. What is the variance of the sample mean?
An estimator variance reveals a great deal about the quality of the estimator.
The variance of the sample mean
s2 = 2/n
Where 2 = variance of the population
n = sample size
Increase of the sample size n
Decrease of the variance s2
Better accuracy of the estimator
11/28/2018
Daniela Stan - CSC323

11. Accuracy of the Estimator
As in many problems, there
is a trade off between
accuracy and dollars.
What we will get from
our money if we invest
dollars in obtaining a larger
size?
n = 100?
n = 200?
11/28/2018
Daniela Stan - CSC323

12. Is there a familiar pattern in the data?
As the sample size becomes larger, the distribution of the sample mean becomes closer to a normal distribution, regardless the distribution of the population from which the sample is drawn.
The central limit theorem summarizes the distribution of the
sample mean.
11/28/2018
Daniela Stan - CSC323

13. The Central Limit Theorem
11/28/2018
Daniela Stan - CSC323

14. Importance of the central limit theorem
The most important feature is that it can be applied to
any population as long as the sample size n is large enough.
How large is large?
n >= 30
11/28/2018
Daniela Stan - CSC323

15. Importance of the central limit theorem
Examples:
11/28/2018
Daniela Stan - CSC323

16. Is x normal distributed?
Is the population normal?
Yes No
Is ?
Is ?
Yes No
Yes No
is normal
has
t-student
distribution
is considered
to be normal
may or
may not be
considered normal
(We need more info)
11/28/2018
Daniela Stan - CSC323