Sampling Distribution

1
Sampling Distribution

• Tripthi M. Mathew, MD, MPH

2
Objectives

• Learning Objective
• To understand the topic on Sampling
  Distribution and its importance in different
  disciplines.
• Performance Objectives
• At the end of this lecture the student will be
  able to
• Apply the basic knowledge of sampling
  distribution to solve problems.
• Interpret the results of the problems.

3
Types of Distribution

• Frequency Distribution
• Normal (Gaussian) Distribution
• Probability Distribution
• Poisson Distribution
• Binomial Distribution
• Sampling Distribution
• t distribution
• F distribution

4
What is Sampling Distribution?

• Sampling is defined as the process of selecting a
  number of observations (subjects) from all the
  observations (subjects) from a particular group
  or population.
• Sampling distribution is defined as the frequency
  distribution of the statistic for many samples.
• It is the distribution of means and is also
  called the sampling distribution of the mean.

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Features of Sampling Distribution

• The 4 features of sampling distribution include
• 1) The statistic of interest (Proportion, SD, or
  Mean)
• 2) Random selection of sample
• 3) Size of the random sample (very important)
• 4) The characteristics of the population being
  sampled.

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Characteristics of Sampling Distribution

• Central Limit Theorem
• When random samples of size is taken from a
  population, the distribution of sample means will
  approach the normal distribution.
• When the Sampling distribution of the mean has
  sample sizes of 30 or more then it is said to be
  normally distributed.

7
Statistical Characteristics of Sampling
Distribution

• The major statistics are
• Mean
• Standard deviation
• Standard error
• The standard error (SE or SEM) of the sampling
  distribution is given by the formula
• s
• vn
• Where, n - sample size
• s- standard deviation of the sample
• x sample mean

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Statistical Characteristics of Sampling
Distribution Contd

• a) SE of a proportion v p (1-p)/n
• Where, p is the sample proportion
• b) SE of a percentage v p (100-p)/n
• Where, p is the sample percentage

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Statistical Characteristics of Sampling
Distribution Contd

• Confidence Interval
• a) CI p z a/2 v p (1-p)/n
• b) CI p z a/2 v p (100-p)/n

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Statistical Characteristics of Sampling
Distribution Contd

• Z Score (Standard Score)
• Z x- µ
• s /vn
• Where, X is the sample mean
• µ is the mean of the sampling
  distribution
• s is the SE of the sampling
  distribution
• vn

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Exercises

• An Epidemiologist studied a randomly selected
  group of 25 individuals (men and women) between
  30- 49 years of age and finds that their mean
  heart rate is 70 beats per minute.

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG.
  Basic and Clinical Biostatistics, 2nd edition,
  1994.

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Exercise 1

• How frequently will the sample of 25 individuals
  have a mean heart rate of 74 beats per minute or
  higher?
• or in other words
• What proportion of samples will have mean values
  of 74 beats per minute or greater, if repeated
  samples of 25 individuals are randomly selected
  from the population?

• Exercises are modified from examples in
  Dawson-Saunders, B
  Trapp, RG. Basic and Clinical Biostatistics, 2nd
  edition, 1994.

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Exercise 2

• Further investigation revealed that the 25
  individuals appeared to have used a drug for
  treatment and now the epidemiologist (Epi) wants
  to detect the adverse effects of the drug on the
  heart rate. The Epidemiologist assumes that a
  mean heart rate in the upper 5 of the
  distribution will be cause for concern.
• Determine the value that divides the upper
  5 from the lower 95 of the sampling
  distribution.

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG.
  Basic and Clinical Biostatistics, 2nd edition,
  1994.

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The Use of Normal Curve to solve problems
95
5
73.29
µ
1
2

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG.
  Basic and Clinical Biostatistics, 2nd edition,
  1994.

15
Exercise 3

• The disease detective (Epi) wants to know how
  many patients should be included in the study to
  determine the drugs effect. The Epi assumes that
  the mean heart rate must not rise above 72 beats
  per minute, 90 of the time.
• or in other words
• To include individuals in the study, what
  should the random sample size be so that 90 of
  the mean samples of this size will be 72 beats
  per minute or less?

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG.
  Basic and Clinical Biostatistics, 2nd edition,
  1994.

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Solution/Answers

• 1) 2.3
• 2) 73.29

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG.
  Basic and Clinical Biostatistics, 2nd edition,
  1994.

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Solution/Answers

• 3) 40.96

• Exercises are modified from examples in
  Dawson-Saunders, B Trapp, RG. Basic and
  Clinical Biostatistics, 2nd edition, 1994.

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Other Types of Sampling Distribution

• F distribution
• This is a sampling distribution of the mean
  with an estimated standard deviation.
• t Distribution
• This is the sampling distribution of two
  variances (squared standard deviations).

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Application of Sampling distribution

• The sampling distribution like the normal
  distribution is a descriptive model, so it is
  used to describe real world situations.
• It is very useful to make statements about the
  probability of specific observations occurring.
• Investigators/researchers/modelers use it for
  estimation and hypothesis testing.

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References/Further Reading

• 1) Dawson-Saunders, B Trapp, RG. Basic and
  Clinical Biostatistics, 2nd edition, 1994.
• 2) Last, J. A Dictionary of Epidemiology. 3rd
  edition,1995.
• 3) Wisniewski, M. Quantitative Methods For
• Decision Makers, 3rd edition, 2002.
• 4) Pidd, M. Tools For Thinking. Modelling in
  Management Science. 2nd edition, 2003.