Random Sampling and Sampling Distributions

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Random Sampling and Sampling Distributions
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Statistical Inference

• The purpose of statistical inference is to obtain
  information about a population from information
  contained in a sample.
• A population is the set of all the elements of
  interest.
• A sample is a subset of the population.
• The sample results provide only estimates of the
  values of the population characteristics.
• With proper sampling methods, the sample results
  will provide good estimates of the population
  characteristics.

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Population and Sample
Sample
Population
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Parameter
A number describing a population
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Statistic
A number describing a sample
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Random Sampling from a Population

• To make an inference about a population parameter
  (characteristic), we draw a random sample from
  the population
• Suppose we select a sample of size n from a
  population of size N
• A random sampling procedure is one in which every
  possible sample of n observations from the
  population is equally likely to occur

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Note-I

• A random sample should represent the population
  well, so sample statistics from a random sample
  should provide reasonable estimates of population
  parameters
• All sample statistics have some error in
  estimating population parameters

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Note-II

• 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
• 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

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Point Estimation

• In point estimation we use the data from the
  sample to compute a value of a sample statistic
  that serves as an estimate of a population
  parameter.
• We refer to as the point estimator of the
  population mean ?.
• s is the point estimator of the population
  standard deviation ?.
• is the point estimator of the population
  proportion p.

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Sampling Error

• The absolute difference between an unbiased point
  estimate and the corresponding population
  parameter is called the sampling error.
• Sampling error is the result of using a subset of
  the population (the sample), and not the entire
  population to develop estimates.
• The sampling errors are
• for sample mean
• s - s for sample standard deviation
• for sample proportion

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Sampling Distribution

• A statistic is any function of observations in a
  random sample
• A statistic is a random variable with a
  probability distribution
• The probability distribution of a statistic is
  called its sampling distribution. Its standard
  deviation is called the standard error of the
  statistics.

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Sampling Distribution of the Sample Mean

• Suppose we attempt to make an inference about the
  population mean by drawing a sample from the
  population and calculating the sample mean
• The sample mean of a random sample of size n from
  a population is given by

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Sampling Distribution of the Sample Mean

• Making Inferences about a Population Mean

Population with mean m ?
A simple random sample of n elements is
selected from the population.
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Sampling Distribution of the Sample Mean

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .
• Expected Value of
• E ( ) ?
• where ? the population mean

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Sampling Distribution of the Sample Mean

• Central Limit Theorem
• Suppose X1, X2, , Xn are n independent random
  variables from a population with mean ? and
  variance ?2. Then the sum or average of those
  variables will be approximately normal with mean
  ? and variance ?2/n as the sample size becomes
  large
• Implication
• If we view each member of a random sample as an
  independent random variable, then the mean of
  those random variables, meaning the sample mean,
  will be normally distributed as the sample size
  gets large

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Sampling Distribution of the Sample Mean

• Implication The variance of the sampling
  distribution of the sample mean decreases as the
  sample size n increases
• The larger is the sample drawn from a population,
  the more certain is the inference made about the
  population mean based on sample information, such
  as the sample mean

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Sampling Distribution of the Sample Mean

• The CLT applies when sample size is greater or
  equal than 30
• Note In most applications with financial data,
  sample size will be significantly greater than 30
• Using the results of the CLT, the sampling
  distribution of the sample mean will have a mean
  equal to ? and a variance equal to ?2/n
• The corresponding standard deviation of the
  sample mean, called the standard error of the
  sample mean, will be

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Does have a normal distribution?
Is the population normal?
Yes
No
is normal
Is ?
Yes
No
may or may not be considered normal
is considered to be normal
(We need more info)
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Sampling Distribution of a Sample Proportion

• If X follows a binomial distribution, then to
  find the probability of a certain number of
  successes in n trials, we need to know the
  probability of a success p
• To make inferences about the population
  proportion p (the probability of a success as
  described above), we use the sample proportion
• The sample proportion is the ratio of the number
  of successes (X) in a sample of size n

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Sampling Distribution of a Sample Proportion

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample proportion .
• Expected Value of
• where
• p the population proportion
• Thus, is an unbiased estimate of the
  population proportion, p.

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Sampling Distribution of a Sample Proportion

• Making Inferences about a Population Proportion

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Sampling Distribution of the Sample Variance

• Suppose we draw a random sample n from a
  population and want to make an inference about
  the population variance
• This inference can be based on the sample
  variance defined as follows
• The mean of the sampling distribution of the
  sample variance is equal to the population
  variance

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Population Variance

• s2 Is the population variance, s is the
  population standard deviation
• Note that the divisor of sample variance is the
  sample size minus one (n-1), while for the
  population variance it is the population size N.

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Statistical Inference

• There are two procedures for making inferences
• Estimation
• There are 2 types of estimators point
  estimator and interval estimator
• 2. Confidence Intervals
• 3. Hypotheses testing

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Point Estimation

• The functional form of the pdf is known
• The distribution depends on an unknown parameter,
  say , that may have any value in the
  parameter space
• Point estimation is to choose a member from the
  family by guessing a value of

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• Point Estimator
• A point estimator draws inference about a
  population by estimating the value of an unknown
  parameter using a single value or a point.

Parameter
Population distribution
?
Sample distribution
Point estimator
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Point Estimate

• The statistic is the point estimator

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Point Estimation
The point estimator is an unbiased estimator
for the parameter if If the estimator is
not unbiased, then the difference Is called the
bias of the estimator