Emil Milewski - Statistics II Essentials
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REAs Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Statistics II discusses sampling theory, statistical inference, independent and dependent variables, correlation theory, experimental design, count data, chi-square test, and time series.
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For a statistician, the relationship between samples and population is important. This branch of statistics is called sampling theory. We gather all pertinent information concerning the sample in order to make statements about the whole population.
Sample quantities such as sample mean, deviation, etc., are called sample statistics or statistics. Based on these quantities we estimate the corresponding quantities for population, which are called population parameters or parameters. For two different samples the difference between sample statistics can be due to chance variation or some significant factor. The latter case should be investigated and possible mistakes corrected. The statistical inference is a study of inferences made concerning a population and based on the samples drawn from it.
Probability theory evaluates the accuracy of such inferences. The most important initial step is the choice of samples which are representative of a population. The methods of sampling are called the design of the experiment. One of the most widely used methods is random sampling.
A sample of n measurements chosen from a population N (N > n) is said to be a random sample if every different sample of the same size n from the population has an equal probability of being selected.
One way of obtaining a random sample is to assign to each member of the population a number. The population becomes a set of numbers. Then, using the random number table, we can choose a sample of desired size.
EXAMPLE:
Suppose 1,000 voters are registered and eligible to vote in an upcoming election. To conduct a poll you need a sample of fifty persons. To each voter you assign a number between one and 1,000. Then, using the random number table or a computer program, you choose at random fifty numbers, which are fifty voters. This is your required sample.
From a bag containing ten numbers from 1 to 10 we have to draw three numbers. As the first step, we draw a number. Now we have the choice of replacing or not replacing the number into the bag. If we replace the number, then this number can come up again. If the number is not replaced, then it can come up only once.
Sampling where each element of a population may be chosen more than once (i.e., where chosen element is replaced) is called sampling with replacement. Sampling without replacement takes place when each element of a population can be chosen only once.
Remember that populations can be finite or infinite.
EXAMPLE:
A bag contains ten numbers. We choose two numbers without replacement. This is sampling from a finite population.
EXAMPLE:
A coin is tossed ten times and the number of tails is counted. This is sampling from an infinite population.
A population is given from which we draw samples of size n , with or without replacement. For each sample we compute a statistic such as the mean, standard deviation, variance, etc. These numbers will depend on the sample and they will vary from sample to sample. In this way we obtain a distribution of the statistic which is called sampling distribution.
For example, if for each sample we measure its mean, then the distribution obtained is the sampling distribution of means. In the same way we obtain the sampling distributions of variances, standard deviations, medians, etc.
A population is given with a finite mean and a standard deviation . Random samples of n measurements are drawn. If the population is infinite or if sampling is with replacement, then the relative frequency histogram for the sample means will be approximately normal with mean and standard deviation 
.
Suppose the distribution of x for the population is as shown in , with the mean .
The standard deviation is .
shows the relative frequency histogram, called the sampling distribution, for the sample mean 
. The samples with replacement of size n are measured and sample mean 
is computed. The mean for the sampling distribution of 
is , the same as for the whole population. The standard deviation of the sampling distribution is equal to the standard deviation of the x measurements divided by n , that is 
.
If the samples of size n are drawn without replacement from a finite population of size N , then
where and denote the population mean and standard deviation; while n and n denote the mean and standard deviation respectively of the sampling distribution.
EXAMPLE:
Suppose a population consists of the five numbers 2, 4, 5, 6, 8. All possible samples of size two are drawn with replacement. Thus, there are 5 
5 = 25 samples. The mean of the population is
and the standard deviation of the population is
We shall list all 25 samples and their corresponding sample means
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